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# Energy is added to a piston-cylinder arrangement. The piston is withdrawn in such a way that the temperature remains constant. The initial pressure and volume are 200 KPa and 2m^3, respectively. If the final pressure is 100 KPa, determine the work done by the ideal gas on the piston. ( Assume T1 & T2 are constant)

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1. If temperature is constant,

P = Po*Vo/V

and Work out = Integral of P dV
(from V1 to V2)

= (Po*Vo) * Integral of dV/V

= Po*Vo ln(V2/V1)

= Po*Vo ln(P1/P2)

= 400 ln2 kJ

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2. Student

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# Calling \end{document} in a macro

I have a file, which is meant to be used as a fragment file inside a larger document:

\include{boilerplate}
content
\include{boilerplate}

The goal is to make a fragment file compilable. So inside boilerplate.tex, I have:

\ifx\maindocument\undefined
\documentclass{article}
\def\maindocument{}
\begin{document}
fragment is now standalone
\else
fragment is finished
\end{document}
\fi

When I compile the fragment, I get:

(\end occurred when \ifx on line 1 was incomplete)

The "cause" of this is that inside my \ifx, I end the document and so the \ifx never finishes. However, everything seems to compile properly. How can I keep this general format, but make the error message go away? I tried putting an \expandafter{\end{document}} but that didn't help.

After incorporating the suggestions about \enddocument (or another macro), I was able to remove the error. However, a few remain:

(\end occurred when \iftrue on line 3 was incomplete)
(\end occurred when \ifnum on line 3 was incomplete)

Here is my boilerplate.tex now:

\def\prestuff{%
\documentclass{article}
\def\maindocument{}
\begin{document}
fragment is now standalone
}

\def\poststuff{%
fragment is finished
\end{document}
}

\ifx\maindocument\undefined
\expandafter\prestuff
\else
\expandafter\poststuff
\fi

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I wouldn't use \include and prefer \input. The errors are indeed caused by \include. –  egreg Jul 28 '11 at 8:55
I've updated my answer in response to your updated question. –  Loop Space Jul 28 '11 at 9:00

Another strategy for avoiding too many open conditionals (incidentally, all of these techniques have uses beyond the given one wherever you'd rather not next conditionals unnecessarily) is to define an "execute after" command which is set inside the conditional. So in this case,

\ifx\maindocument\undefined
\documentclass{article}
\def\maindocument{}
\begin{document}
fragment is now standalone
\def\donext{}
\else
fragment is finished
\end{document}
\def\donext{\end{document}}
\fi
\donext

Obviously, you can add more stuff in to the \donext macro.

Update: This is in response to the new errors that you are getting. These come from the \include command. Basically, by ending the document in an included file, you aren't giving the \include command a chance to finish what it normally does. As this involves a few conditionals and other stuff (such as writing to aux files), TeX complains about this.

If you don't need all the fancy extra of \include then you can replace it by \input. One of the main things that \include gives you that \input doesn't is individual aux files for each included file. As you are including the file twice, I'd be concerned that the second one might overwrite the first anyway so would hesitate before using \include. If your "real life" situation is similar to the example in the question, then I don't see any benefit from using \include over \input so this replacement should be safe.

If you do still need to use \include then you could probably hack your way out of it by looking at the \include and the @include commands and putting everything after the \@input@ line in to your \poststuff macro. The following works for me, but I don't guarantee that there isn't something else I've overlooked!

\def\prestuff{%
\documentclass{article}
\def\maindocument{}
\begin{document}
fragment is now standalone
}

\makeatletter
\def\poststuff{%
fragment is finished
\clearpage
\@writeckpt{boilerplate}%
\if@filesw
\immediate\closeout\@partaux
\fi
\else
\@nameuse{cp@boilerplate}%
\fi
\let\@auxout\@mainaux
\fi
\end{document}
}
\makeatother

\ifx\maindocument\undefined
\expandafter\prestuff
\else
\expandafter\poststuff
\fi

(But as I said, using \input avoids this hassle.)

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Wehn TeX reads your input, it executes \end{document} before the \fi. Inside \end{document} there is the TeX \end primitive, and it should never be used without closing all conditionals. So you can either use the low-level \endocument macro (as suggested by mhp):

  ...
\expandafter\enddocument
\fi

or for a more general solution, use your own macros to deal with the conditional material

  ...
\def\mymacro{\end{document}}
\expandafter\mymacro
\fi

The later approach allows you to include multiple items to be inserted after the \fi.

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Don't use \include except for including independent chapters in the main text body. Use the raw inclusion command \input instead, then the remaining warnings will disappear.

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Instead of expandafter\end{document}, try \expandafter\enddocument.

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The standalone package could achieve what you want.

%boilerplate.tex
\documentclass{standalone}
\begin{document}
Content of boilerplate
\end{document}

%main.tex

\documentclass{article} %or whatever
\usepackage{standalone}

\begin{document}
\include{boilerplate} % Perhaps input would be better?
Other text
\include{boilerplate}
\end{document}

Make sure any packages you use in boilerplate are included in the main document. standalone automatically strips out the preambles of the included file.

-

format: plain


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# Proving $x_1 = (x_0+x_2)/2$ on a quadratic Bezier curve

Can someone please prove that $x_1 = (x_0 + x_2)/2$ on a quadratic Bezier curve.

Where: $x_0$ is the first control points, $x_2$ is the second control point, $x_1$ is the middle control point located where the tangents at $x_0$ and $x_2$ meet

For the function $f(x) = ax^2 + bx + c$ at points $P_0 = (x_0, y_0)$, $P_1 = (x_1, y_1)$, $P_2 = (x_2, y_2)$.

Thanks

• Maybe you mean a quadratic Bezier ? – Yves Daoust Oct 23 '16 at 14:44
• I changed $x_3$ to $x_1$. Is that what you meant ?? – bubba Oct 23 '16 at 14:49

The tangent line at $P_0$ is $y-y_0 = (2ax_0 + b)(x-x_0)$.
The tangent line at $P_2$ is $y-y_2 = (2ax_2 + b)(x-x_2)$.
Confirm that the $x$-coordinate of the intersection point is $x_1$.


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## Derived functors and Cech cohomology

Series
Other Talks
Time
Wednesday, November 11, 2009 - 1:00pm for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Farbod Shokrieh – Ga Tech
Organizer
John Etnyre
We will show that the construction of derived functors satisfy the required universal property.I will then show that, for any ringed space, the abelian category of all sheaves of Modules has enough injectives. We achieve this by first characterizing injective abelian groups (Baer's theorem).The relation with Cech cohomology will also be studied. In particular, I will show that the first Cech and Grothendieck sheaf cohomology groups are isomorphic for any topological space (without using spectral sequences).


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# Python Circle From 3 Points

Intersection of a circle and a line. The first line above makes all the types of object of Zelle's module accessible, as if they were already defined like built-in types str or list. Three Circles can have none/one/or an area as its solution. Is there any function in matlab that could find the distance between two points. Our answer is slightly different to the answers obtained by using a parabolic model and linear approximations. It can be used when the points are decimal. 3: With G02 we will give the coordinates of the circle end point, as for a full circle the start-point and end-point remains the same, so we will use same X,Y coordinate G02 X10 Y25 4: Now to complete G02 G-code we will have to give the circle-center-point coordinates, I - Distance from circle start-point to circle-center-point in X-axis. The three points will not be on a straight line. 3 older comments. Note: Because during the solution we get two values for x coordinate and two values for y coordinate, it is important to match the correct values for x and y. Circles-4 | Equation of Tangent at a point | JEE Main and Advanced | Powered by IITians - Duration: 26:43. I'll add this to the tinyCAD. I also try cvHoughCircles(), it give me a lot of circles but not the 3 circles I expected. Tangent Circles From 3 Points. Can a circle always be drawn which goes through these 3 points? we assumed yes. The default viewing volume in OpenGL is a cube, [-1, +1] x [-1, +1] x [-1, +1], with the origin in the middle of the current viewport (a rectangular area of our window, for our case the viewport has the same dimensions as our window), the positive x axis points to the right, the positive y axis points up and the positive z axis points toward. Please check your connection and try running the trinket again. Points defining the circumference are colored red; points in the interior are colored gray. KDTree is the way. I suppose that making the circles in 3D, instead of 2D, and letting the 4th circle pass through the 3 points would be so crazy, that the effect you got there would not be so smooth. Hi, I've been trying to deduce the general equation of a circle in 3D space, but without much luck. Unless you have some prior information as to what the form of the mathematical function "should be", there is no way of distinguishing which of those infinite number of functions is the "right" one. It can be used when the points are decimal. You can open/save every DXF file without losing any content (except comments), Unknown tags in the DXF file will be ignored but preserved for saving. Property #1: We know the dimensions of the object in some measurable unit (such as inches. Marker Reference¶. Two Python workers receive an input point within the circle, each estimates Pi as (4 * 1) / 1, or 4. NET This behaviour is probably fine for the majority of 2D drafting activities, but if you want to create circles from a point cloud - by selecting points from its perimeter using the Node object snap - then it's less than ideal. coord = 10, 50, 240, 210 arc = canvas. From the above we can find the coordinates of any point on the circle if we know the radius and the subtended angle. The points of this circle are represented in a string with the coordinates like so: "(x1,y1),(x2,y2),(x3,y3)" Where x1,y1,x2,y2,x3,y3 are digits. The radius is perpendicular to the tangent of the circle at a point $$D$$ so: $m_{AB} = - \frac{1}{m_{CD}}$ Write down the gradient-point form of a straight line equation and substitute $$m_{AB}$$ and the coordinates of $$D$$. # -*- coding: utf-8 -*- # Cercle sur 3 points. Now, create three equations in three cells, say C2:C4, with the formulas corr. The figure gives you a visual representation of this circle. The ceter point is 1 foot from a line drawn between the other 2 points. The average of the three Python's workers estimates is (4. A {\displaystyle A} Draw a another small arc intersecting the previous one. 3 Points and Pointclouds. The formula comes from cercle_3pts. Given three points p1, p2 and p3, find orientation. I suppose that making the circles in 3D, instead of 2D, and letting the 4th circle pass through the 3 points would be so crazy, that the effect you got there would not be so smooth. circle — draw a circle int or float)) -- center point of the circle as a sequence of 2 ints/floats, e. Some version of this problem often appears in the SAT Math 2 subject test! Problem. One Python worker receives an input point inside and an input point outside the circle, estimating Pi as (4 * 1) / 2, or 2. boundingRect (). Maths Wallah - Vineet Loomba [IITR] 14,253 views 26:43. My implementation will work on either selecting two adjacent edges, or 3 vertices (not necessarily adjacent) edit This is now added to the github version as 'Circle Centers', prior to updating the addons/contrib version. Three Circle Hypotrochoid with Python Turtle. Given three points p1, p2 and p3, find orientation. The algorithm starts with theta at zero. I also try cvHoughCircles(), it give me a lot of circles but not the 3 circles I expected. #N#def get_avg_bgr(in_img, in_cntrs. In our Python script, let's create some data to work with. Use the Draft Arc tool to create a circular arc by specifying the center, the radius, and the start and end angles. Depending on these states, the feature vector is subdivided into 3 subsets, , ,. This document provides usage examples for using the Python Shapefile Library. However, the center needs to be an object from a previously defined Point class. Here the Greek letter π represents a constant, approximately equal to 3. From the above we can find the coordinates of any point on the circle if we know the radius and the subtended angle. It is very simple Python function. I've applied circular hough tranform for identifing the circlular objects in an image and i got the results. 3 older comments. Through any three points that do not exist on the same line, there lies a unique circle. READ MORE READ MORE. 859834, which is almost straight line. A circle is the locus of all points equidistant from a static point, and the equation of a circle is a way to express the definition of a circle on the coordinate plane. This is how I do it: 1) Create a square (f. When dealing with a line segment it may be more efficient to first determine whether the line actually intersects the sphere or circle. The class needs two attributes: the center and the radius. Also, Scipy. Given 3 random points in a 2 dimensional space. 000000) / 3, or 3. The Canvas widget supplies graphics facilities for Tkinter. In this program, area of the triangle is calculated when three sides are given using Heron's formula. If we drop a line down from (x,y) to the x-axis, we'll have a right triangle (and an opportunity to therefore apply the Pythagorean Theorem to this circle):. This calculator allows you to work back from these fixed points and find the centre point and radius of the circle which passes through them. 0000 and Circle C2 with center (-0. Find the slopes of the lines through two pairs of points. Given three points p1, p2 and p3, find orientation. Which is great for taking down Pythons/Dropships. I'll be starting with the simplest kind of figure: a line plot, with points plotted on an X-Y Cartesian plane. Step by Step Math Worksheets Solvers New ! Find Points Of Intersection of Circle and Line - Calculator. The radius of a circle is the distance from the center to any point on the circle's edge. 7521), radius 2. plot a circle with pyplot. Computational Geometry Library in Python - Arc/Circle:. Computing the distance between objects is very similar to computing the size of objects in an image — it all starts with the reference object. Therefore, it is located at the spot in the equation where x. You can define a circle through 3 points (provided they are not coincident and not colinear), and you can define a circle through two points and a single tangent. find circle equation? Using Python and Numpy here is my initial code. e P, Q, or R) passing through the plane. 3: With G02 we will give the coordinates of the circle end point, as for a full circle the start-point and end-point remains the same, so we will use same X,Y coordinate G02 X10 Y25 4: Now to complete G02 G-code we will have to give the circle-center-point coordinates, I - Distance from circle start-point to circle-center-point in X-axis. If distance is less than or equal to radius. Given 3 random points in a 2 dimensional space. Each line of input to your program will contain the x and y coordinates of three points, in the order A(x),A(y),B(x),B(y),C(x),C(y). The corresponding problem in n-dimensional space, the smallest bounding sphere problem, is to compute the smallest n-sphere that. Three coordinates = three center points. The Web Machinist Software Program has a Circle Center 3 Known Points Calculator in it. Video transcript. Plug into the x input of a Point. The average of the three Python's workers estimates is (4. If you have just triangle (3 points) in plane, you can compute determinant from matrix, where lines are coordinates of points (3rd coordinate is 1). You will see updates in your activity feed. The class needs two attributes: the center and the radius. ax if not "label" in kwargs: kwargs["label"] = poly_id coords=[] for p in points: try: coords. READ MORE READ MORE. In a previous Hypotrochoid project, you drew. e a parabola). #N#def get_avg_bgr(in_img, in_cntrs. I suppose that making the circles in 3D, instead of 2D, and letting the 4th circle pass through the 3 points would be so crazy, that the effect you got there would not be so smooth. The Python is tougher and comes with 3 x class 3 weapons + 2 class 2. 5 circular regression was applied to the points collected in step 4 to minimize the deviations from the curve and overcome the simplistic assessment of a 3-point circle. If orientation of (p1, p2, p3) is clockwise, then orientation of (p3, p2, p1) is counterclockwise and vice versa is also true. Step by Step Math Worksheets Solvers New ! Find Points Of Intersection of Circle and Line - Calculator. A tangent intersects a circle in exactly one place. Some version of this problem often appears in the SAT Math 2 subject test! Problem. ProjectPoints (points, object_project, projection_type=3) ¶ Project a point or a list of points given its coordinates. 7 proceed with the next straight line segment or next tangent or next circular curve depending upon road layout. Active 1 year, 10 months ago. Circle by 3 points: Draws a circle from three points on the circumference: Ellipse by. Write a Python program which accepts the radius of a circle from the user and compute the area. Make $$y$$ the subject of the equation. I was working on a similar problem and found this. The circle should look tangent as soon as the tangent snap is turned on. Euclidean distance is probably harder to pronounce than it is to calculate. 3 Qt version: 4. If determinant is > 0, points are in CCW order. Two points form. For more see Parametric equation of a circle. Given 3 points in space (3D): A = (x1, y1, z1), B = (x2, y2, z2) C = (x3, y3, z3); then how to find the center and radius of the circle (arc) that passes through these three points, i. circle — draw a circle int or float)) -- center point of the circle as a sequence of 2 ints/floats, e. Using a Circle. One Python worker receives an input point inside and an input point outside the circle, estimating Pi as (4 * 1) / 2, or 2. The Web Machinist Software Program has a Circle Center 3 Known Points Calculator in it. Try this Drag the orange dots on each vertex to reshape the triangle. I've tried the geometric equations (finding midpoint, slope, etc. The points can be objects such as cubes, cylinder, then selected coordinates will be the centre of these forms. We demonstrate in our first example, how to draw a line. This is how I do it: 1) Create a square (f. You want the point on the line that is closest to B - lets call thay C. e P, Q, or R) passing through the plane. The three points will not be on a straight line. By applying this twice to two different chords, the center is established where the two bisectors intersect. I'll add this to the tinyCAD. Circles-4 | Equation of Tangent at a point | JEE Main and Advanced | Powered by IITians - Duration: 26:43. You can open/save every DXF file without losing any content (except comments), Unknown tags in the DXF file will be ignored but preserved for saving. Please check your connection and try running the trinket again. be/HBLoKLaDL4. The Calculator quickly and easily can calculate the Circle Center and Circle. There are various ways to handle this calculation problem. Sage provides extensive 2D plotting functionality. Given line AB, point P, and radius R (Figure 4. Computational Geometry Library in Python - Arc/Circle:. Applying the formula of the centroid, you will find the coordinate of the centroid to be [(1 + 3 + 5)/3, (2 + 4 + 0)/3] = (9/3, 6/3) = (3,2) Solution #2: Find the mid points of 2 of the sides of. If I have the x and y coordinates of 3 points - i. Description. The word "intercept" means crossing point, and the y-intercept of a graph refers to the point at which the equation crosses the y-axis of the coordinate plane. The following graphics primitives are supported: • arrow()- an arrow from a min point to a max point. If determinant is > 0, points are in CCW order. The Shapefile format is a popular Geographic Information System vector data format. create_arc (coord, start=0, extent=150, fill="blue") image − Creates an image item, which can be an instance of either the BitmapImage or the PhotoImage classes. I made a Point class (x, y) which we need to use, but I have no idea how to configure a Circle class. The Web Machinist Software Program has a Circle Center 3 Known Points Calculator in it. In this definition, the points 0 and 3 correspond to the end points (the knots). Reference for filled-, unfilled- and custom marker types with Matplotlib. markers documentation. The number of values in the list corresponds with the number of dimensions of the space it resides in. Posted by Karl Wecker on November 19, 2009 at 9:56am in VB, C# and Python Coding; In fact, this is not possible, you cannot define a circle with 3 tangents. Euclidean distance is probably harder to pronounce than it is to calculate. In a previous Hypotrochoid project, you drew. 83, when finding the actual circle passing through the 3 points. 0000 and Circle C2 with center (-0. Computational Geometry is a field of mathematics that seeks the development of efficient algorithms to solve problems described in terms of basic geometrical objects. 14 radius = float (input (' Please Enter the radius of a circle: ')) area. There are various ways to handle this calculation problem. One Python worker receives an input point inside and an input point outside the circle, estimating Pi as (4 * 1) / 2, or 2. The python is incredibly sluggish, so has trouble taking down more maneuverable enemies. This gives an immediate $O(N^3)$ algorithm - for all pairs of two points, find the line joining these two and find all other. # -*- coding: utf-8 -*- # Cercle sur 3 points. Euclidean distance refers to the distance between two points. plot a circle with pyplot. Now, create three equations in three cells, say C2:C4, with the formulas corr. I want to find a way to define the circle's center. For example, your problem is nearly trivial if you let the major axis go through the two most distant points of the 3. From P draw an arc with radius R, cutting line DE at C, the center of the required tangent arc. Writing the equation of a circle passing through three points on its circumference. We're currently learning Classes. 0e-10: is checking for collinearity. I'm trying to create a 3D heat map model out of data that I have from a csv file. Write PQ in general form. When working with mathematics and plotting graphs or drawing points, lines, and curves on images, Matplotlib is a good graphics library with much more powerful features than the plotting available in PIL. Let the co-ordinate of the given point be (x1, y1, z1) and equation of the plane be given by the equation a * x + b * y + c * z + d = 0, where a, b and c are real constants. $\endgroup$ - Yasmin Apr 20 at 1:41. The idea behind this python workbench for FreeCAD was to give users some "quick" access tiny tools. Intersection of a circle and a line. So every time you do a conditional or something, you'll be sending a request over the network (wifi apparently) to the robot, getting a response, running the if. These points can be in different dimensional space and are represented by different forms of coordinates. Python version: 2. Ok thank you this is very useful to the google! More posts from the code community. Each line of input to your program will contain the x and y coordinates of three points, in the order A(x),A(y),B(x),B(y),C(x),C(y). Thanks for contributing an answer to. Is there a way when importing the data to somehow join the points to create faces instead of generating thousands of. 000000) / 3, or 3. Shown below is the graph of the circle, the line and the two points of intersection. As detailed in our previous blog post, our reference object should have two important properties:. If you have points A, B, C, then your matrix looks like:. Sage provides extensive 2D plotting functionality. These coordinates will be real numbers less than. ), but I haven't found a way for Python to find both. This method relies on the fact that, for any chord of a circle, the perpendicular bisector of the chord always passes through the center of the circle. 13*3// 2 * ( 14 % 2 ) % 5 b. Polygon: polygon(xy, fill, The start and end points of the arc are connected by a straight line to the center of the circle. For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. The following tool visualize what the computer is doing step-by-step as it executes the said program: Customize visualization ( NEW!) There was a problem connecting to the server. Draw a circle on an iPad map from three points in Pythonista Jerry Stratton, December 1, 2015 I happened across a copy of 57 Practical Programs & Games in BASIC while traveling a month ago, and was once again fascinated by the simple toolchests we used back in the seventies and eighties. I am allowed to use modules already included with Python. The algorithm is non-iterative, so the computation time is bounded and small. Calculate Distance Between GPS Points in Python 09 Mar 2018. My implementation will work on either selecting two adjacent edges, or 3 vertices (not necessarily adjacent) edit This is now added to the github version as 'Circle Centers', prior to updating the addons/contrib version. A point is nothing more than a list of values called a coordinate. 0 OCC version: 6. (Mat, list of joints or Item) - final circle target. If you just want to draw some basic shapes to the screen you could use python-turtle or pygame. Substitute the given x and y-coordinates in the circle. X Research source To set the width, place the tip of the compass on the center of the circle, and open the compass to any one of your original points. I plotted the last one. (1) First we need to generate the actual points that will make up the surface plot. A circle, for instance, can be defined be a radius and a point, which is two variables. I have a few thousand data points with XYZ coordinates. m, returns points on an implicit circle in 2D. In this python program, we will find area of a circle using radius. I'm trying to create a 3D heat map model out of data that I have from a csv file. Can a circle always be drawn which goes through these 3 points? we assumed yes. Angle bisectors. Please check your connection and try running the trinket again. Any two points in a space will form a straight line. One Python worker receives an input point inside and an input point outside the circle, estimating Pi as (4 * 1) / 2, or 2. (10 points) Write a Python function that will calculate the volume of a cylinder (area function should accept two incoming parameters and return the volume. Space is usually denoted with an R and a superscript value indicating the number of dimensions. Approach: Let P, Q and R be the three points with coordinates (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) respectively. Calculate Distance Between GPS Points in Python 09 Mar 2018. This is how I do it: 1) Create a square (f. pdf and used with the kind permission of its author. 000000) / 3, or 3. The Arc 3Points tool creates a circular arc in the current work plane by entering three points that lie on the circumference; the center and radius are determined from these three points. It is very simple Python function. The Python is tougher and comes with 3 x class 3 weapons + 2 class 2. 100 x 100) 2) Create a volume PULL 100 3) Make it transparent, choose, COLOR - OPACITY 4) Put a point on the baseline of one side. 8013271108436504 Click me to see the sample solution. Posted by 3 days ago. The Canvas widget supplies graphics facilities for Tkinter. Writing Python code All workbenches at a glance. Shown below is the graph of the circle, the line and the two points of intersection. I'm semi-new to Python. The average of the three Python's workers estimates is (4. Python version: 2. These caught me by. Use the diameter form to find the circle with PQ as diameter. So three points in a plane also define a circle. Is there any function in matlab that could find the distance between two points. 5 bronze badges. So there is no practical circle that can pass through three collinear points. Note that the circumcircle always passes through all three points. 31,70) & (63. Instructions for manual positioning mode: Left-click in a blank space to add a new point. I plotted the last one. Left-click an existing point and drag to move it. You can vote up the examples you like or vote down the ones you don't like. Approach: As we know all three point lie on the circle so they will satisfy the equation of circle and putting them in general equation we get three equation with three variable g, f and c and by further solving we can get the values. The square of the length of tangent segment equals to the difference of the square of length of the radius and square of the distance between circle center and exterior point. the euclidean distance between the points and the circle is a common choice. Note that the circumcircle always passes through all three points. plot a circle with pyplot. I'm trying to create a 3D heat map model out of data that I have from a csv file. You should find the circle for three given points,. Designate a random point on the circle (x,y). Case 1) Two circles are possible. Draw a circle on an iPad map from three points in Pythonista Jerry Stratton, December 1, 2015 I happened across a copy of 57 Practical Programs & Games in BASIC while traveling a month ago, and was once again fascinated by the simple toolchests we used back in the seventies and eighties. (1) First we need to generate the actual points that will make up the surface plot. Dec 10, 2017 · When people say that 5 points are needed to define a unique ellipse, they are right. ByCoordinates node. Then the equation of plane is a * (x - x0) + b * (y - y0) + c * (z - z0) = 0, where a, b, c are direction ratios of normal to the plane and (x0, y0, z0) are co-ordinates of any point(i. It sometimes happens that you need 3 points to sit on a circle. There are infinitely many curves you can draw through three points. X Research source To set the width, place the tip of the compass on the center of the circle, and open the compass to any one of your original points. Calculate Distance Between GPS Points in Python 09 Mar 2018. Find the midpoints of those lines The perpendicular bisectors are the lines through the midpoints with slopes equal to the negative reciprocals of the slopes of the secant lines. So, i've been trying to find this out for such a long time. If not, you can swith for example last two points and you will get CCW order. Here the Greek letter π represents a constant, approximately equal to 3. Therefore, it is located at the spot in the equation where x. (x1,y1),(x2,y2),(x3,y3. I'm trying to use this point (and a given radius) to figure out the angle in radians for it, so that I can place the control knob on the circle at the same angle. Note that the circumcircle always passes through all three points. 14 radius = float (input (' Please Enter the radius of a circle: ')) area. A {\displaystyle A} Draw a another small arc intersecting the previous one. Try clicking Run and if you like the result, try sharing again. Plug into the x input of a Point. READ MORE READ MORE. Given three random points that do not lie on the same line, draw three circles centered at these points that are tangent to one another. I made a Point class (x, y) which we need to use, but I have no idea how to configure a Circle class. Find the center point and radius for the equation of a circle passing through (2,2), (2,4) and (5,5) x and y-coordinates. If you tried the construction, you would find that the two radii (JO, LO) would be parallel and so never meet at a center. Description. Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. The Canvas widget supplies graphics facilities for Tkinter. It can be used when the points are decimal. Place the compass tip. Three coordinates = three center points. Instructions for manual positioning mode: Left-click in a blank space to add a new point. Among these graphical objects are lines, circles, images, and even other widgets. Therefore, it is located at the spot in the equation where x. Maths Wallah - Vineet Loomba [IITR] 14,253 views 26:43. Find the equations of the perpendicular bisectors. py # create a circle from 3 points selected # combination of the two programs 'create a circle from 3 points selected' # with as options (orthogonal) circle can be built on a choice plans # to coordinate one of the three selected choice shapes # mode by default, creates the circle in 3D space # the. In our Python script, let's create some data to work with. The points of this circle are represented in a string with the coordinates like so: "(x1,y1),(x2,y2),(x3,y3)" Where x1,y1,x2,y2,x3,y3 are digits. A tangent to a circle is a straight line, in the plane of the circle, which touches the circle at only one point. We can derive the formula to obtain the value of g, f and c as: Putting coordinates in eqn of circle, we get: x1 2 + y1 2 + 2gx1 + 2fy1 + c = 0 - (1). I also try cvHoughCircles(), it give me a lot of circles but not the 3 circles I expected. The average of the three Python's workers estimates is (4. But when i used the code into an app designer, insted of ploting the circles into the image where the centers are marked its ploting the circle in a different figure. Ok thank you this is very useful to the google! More posts from the code community. Multiply this measurement by pi. One of the biggest difficulty for new users of FreeCAD, is to know in which workbench to find a specific tool. One Python worker receives an input point inside and an input point outside the circle, estimating Pi as (4 * 1) / 2, or 2. Finally, we could use the method of least-squares to fit a 2D circle into the planar points and then project the 2D fitting circle back to the 3D coordinations. This is the currently selected item. Description. 2 Coin version: 3. In a previous Hypotrochoid project, you drew. Write a Python program which accepts the radius of a circle from the user and compute the area. Please check your connection and try running the trinket again. and the closest distance depends on when and where the user clicks on the point. 0000) and radius 1. Intersection methods allow this higher dimensional geometry to generate lower dimensional objects, while the trim and select trim commands allow script to heavily modify geometric forms after they. This is very easy with a simple 3-points spline, but I would need a real circle-arc. Don't worry about trying to draw the straight line so it's in the center -- anywhere on the circle will do. A tutorial on finding the points of intersection of a circle with a line; general solution. Project: Lazy-Cleaner-9000 Author: TheOriginalBDM File: clean_sweep_vision. Circle through three points This online calculator will find and plot the equation of the circle that passes through three given points. Constructing a surface plot in Matplotlib is a 3-step process. In other words if P is the closest point on the line then. This is achieved by noting that the closest point on the line through P 1 P 2 to the point P 3 is along a perpendicular from P 3 to the line. e P, Q, or R) passing through the plane. A tangent to a circle is a straight line, in the plane of the circle, which touches the circle at only one point. These are on different branches so compile time is longer. Reference for filled-, unfilled- and custom marker types with Matplotlib. This page shows how to construct (draw) a circle through 3 given points with compass and straightedge or ruler. From the above we can find the coordinates of any point on the circle if we know the radius and the subtended angle. A circle, for instance, can be defined be a radius and a point, which is two variables. r/dataisbeautiful: A place for visual representations of data: Graphs, charts, maps, etc. A Python package to create and modify DXF drawings, independent from the DXF version. There are infinitely many curves you can draw through three points. The table below will give you an overview of the most important workbenches and their tools. Sage provides extensive 2D plotting functionality. Math, Alison. To do this, open the compass a little wider. ByCoordinates node. As detailed in our previous blog post, our reference object should have two important properties:. Find the midpoints of those lines The perpendicular bisectors are the lines through the midpoints with slopes equal to the negative reciprocals of the slopes of the secant lines. There's a bigger issue with 3_point_circle because tangency auto-snaps don't modify the circle preview. The formula comes from cercle_3pts. Given the Cartesian coordinates of three points on a plane, find the equation of the circle through them all. When working with mathematics and plotting graphs or drawing points, lines, and curves on images, Matplotlib is a good graphics library with much more powerful features than the plotting available in PIL. The ceter point is 1 foot from a line drawn between the other 2 points. The points of this circle are represented in a string with the coordinates like so: "(x1,y1),(x2,y2),(x3,y3)" Where x1,y1,x2,y2,x3,y3 are digits. Three points of a circle on the x-y coordinate plane have coordinates (-5,-6); (-5, -2) and (1, -2) respectively. A tangent intersects a circle in exactly one place. So if I were to take three random points here, so let's call that point A, point B, and then let's say this is point C right. boundingRect (). Radius, Center of Circle Given 3 Points What is an easy formula to calculate the center point and radius of a circle given three points on the circumference? Finding the Radius and Center of a Circle How do I find the center and radius for the circle defined by the equation x^2 + 8x + y^2 - 6y = 11?. The three points will not be on a straight line. find circle equation? Using Python and Numpy here is my initial code. It then checks to see if the point is inside the circle (it knows it's inside the circle if x^2 + y^2 R^2, where x and y are the coordinates of the point and R is the radius of the circle). If extent is not given, draw the entire circle. Examples of lines, circle, rectangle, and path. Calculate Distance Between GPS Points in Python 09 Mar 2018. The Web Machinist Software Program has a Circle Center 3 Known Points Calculator in it. Well, I'll jump in with what I've got so far. The circumcircle always passes through all three vertices of a triangle. Is there any function in matlab that could find the distance between two points. Intersection. A related topic is regression analysis, which. What Kind of Data are we talking…. The formula comes from cercle_3pts. Space is usually denoted with an R and a superscript value indicating the number of dimensions. You should find the circle for three given points,. Give 3 points and a plot circle. Smooth curve generator implemented in python Given a list of ordered points as in the bottom of this image, my program produces the top part of the image: For more about the theory behind it see my last blog post. Therefore, it is located at the spot in the equation where x. This document provides usage examples for using the Python Shapefile Library. This gives an immediate $O(N^3)$ algorithm - for all pairs of two points, find the line joining these two and find all other. One Python worker receives an input point inside and an input point outside the circle, estimating Pi as (4 * 1) / 2, or 2. For example, I have a sequence of 3 points, but after linking all of them with a line, I want to insert a new point on that line between each original point, making it a line with 5 points (on that case). Draw a small arc opposite the given point. It sometimes happens that you need 3 points to sit on a circle. I am allowed to use modules already included with Python. Approach: The perpendicular distance (i. If orientation of (p1, p2, p3) is clockwise, then orientation of (p3, p2, p1) is counterclockwise and vice versa is also true. (10 points) Write a Python function that will calculate the volume of a cylinder (area function should accept two incoming parameters and return the volume. We like to find one of the circles in. (x1,y1),(x2,y2),(x3,y3. I'm semi-new to Python. The point is called the point of tangency or the point of contact. For example, your problem is nearly trivial if you let the major axis go through the two most distant points of the 3. The first line above makes all the types of object of Zelle's module accessible, as if they were already defined like built-in types str or list. e P, Q, or R) passing through the plane. He not only wants us to use the 'main' function, but he wants us to call the circle function afterwards. When working with GPS, it is sometimes helpful to calculate distances between points. It uses the Draft Linestyle set on the Draft Tray. Date: 2019-05-14 / tags: Python, specifying 3 points or more with xy will make the connection look unattractive. The idea behind this python workbench for FreeCAD was to give users some "quick" access tiny tools. The word "intercept" means crossing point, and the y-intercept of a graph refers to the point at which the equation crosses the y-axis of the coordinate plane. blocking (bool) - True if the instruction. One Python worker receives an input point inside and an input point outside the circle, estimating Pi as (4 * 1) / 2, or 2. This page shows how to construct (draw) a circle through 3 given points with compass and straightedge or ruler. Suppose origin O(0,0) is the reference point and P1( a,b) and P2(c,d) are the two points given (. Draw a small arc opposite the given point. Place the point of the compass at point B and the marking tip at C, setting the radius of the compass to be equal to the radius of the circle. where r is the radius of the circle, and h,k are the coordinates of the center. Marker Reference¶. We have use multiple dimentional data like 1D, 2D, 3D and…. e P, Q, or R) passing through the plane. The program keeps track of how many points it's picked so far (N) and how many of those points fell. What Kind of Data are we talking…. In other words if P is the closest point on the line then. READ MORE READ MORE. Make $$y$$ the subject of the equation. A {\displaystyle A} Draw a another small arc intersecting the previous one. The following are code examples for showing how to use cv2. This document provides usage examples for using the Python Shapefile Library. 6 the exit segment (see 3) was determined. Draw the arc in counterclockwise direction if radius is positive, otherwise in clockwise direction. As detailed in our previous blog post, our reference object should have two important properties:. } C = (X_{C},Y_{C},Z_{C}) = \\mbox{Centre of the circle. Consider 3 points - O (the origin), A (the end point of the line), and B (the point you want to get closest to along the line). def add_polygon_2d(self, points=[], poly_id="undefined", ax=None, **kwargs): """Add a polygon specified by list of input points :param list points: list with :class:GeoPoint objects :param str poly_id: string ID of this object (e. As you can see, calculating a circle center based on 3 known points can be a bit complicated. E {\displaystyle E} , and draw an arc that sweeps directly across from point. 5 bronze badges. This is the first circle we'll create. If I run the code with the second set of points (uncommented) the resulting radius is R = 108576. Property #1: We know the dimensions of the object in some measurable unit (such as inches. Go to the editor Sample Output : r = 1. Writing the equation of a circle passing through three points on its circumference. # -*- coding: utf-8 -*- # Cercle sur 3 points. You may receive emails, depending on your notification preferences. Step by Step Math Worksheets Solvers New ! Find Points Of Intersection of Circle and Line - Calculator. They are from open source Python projects. Well, I'll jump in with what I've got so far. HSrdelic: yes, the first one. Looking for your reply. Our answer is slightly different to the answers obtained by using a parabolic model and linear approximations. Also Find Equation of Parabola Passing Through three Points - Step by Step Solver. It works by joining two pairs of points to create two chords. Given the Cartesian coordinates of three points on a plane, find the equation of the circle through them all. So three points in a plane also define a circle. I wanted find the approximate center of the circle passing through more than three points. Count 5 units up, down, left, and right from the center at (3, -1). Now, generating all the points of the 3D surface is impossible since there are an infinite number of them! So instead, we'll generate just enough to be able to estimate the surface and then. They will make you ♥ Physics. If distance is less than or equal to radius. # Python Program to find Area Of Circle using Radius PI = 3. Through any three points that do not exist on the same line, there lies a unique circle. It sometimes happens that you need 3 points to sit on a circle. the point is inside, else outside. Using code block, designate a value of 50; for the radius. Just write out 3 equations using the values of x & y: y 1 = a x 2 1 + b x 1 + c y 2 = a x 2 2 + b x 2 + c y 3 = a x 2 3 + b x 3 + c This is 3 equations in 3 unknowns (a,b,c) that can be solved by substitution. Three points defining a circle. In this tutorial I show you how you can find the equation of a circle passing through three points on the circumference. Three Circles can have none/one/or an area as its solution. Circles-4 | Equation of Tangent at a point | JEE Main and Advanced | Powered by IITians - Duration: 26:43. Matplotlib's PyLab interface is the set of functions that allows the user to create plots. Smooth curve generator implemented in python Given a list of ordered points as in the bottom of this image, my program produces the top part of the image: For more about the theory behind it see my last blog post. There was a problem connecting to the server. 0 OCC version: 6. 14 radius = float (input (' Please Enter the radius of a circle: ')) area. Two Python workers receive an input point within the circle, each estimates Pi as (4 * 1) / 1, or 4. create_arc (coord, start=0, extent=150, fill="blue") image − Creates an image item, which can be an instance of either the BitmapImage or the PhotoImage classes. Consider 3 points - O (the origin), A (the end point of the line), and B (the point you want to get closest to along the line). The underlying rendering is done using the matplotlib Python library. In this program, area of the triangle is calculated when three sides are given using Heron's formula. Its center is at the point where all the perpendicular bisectors of the triangle's sides meet. The Arc 3Points tool creates a circular arc in the current work plane by entering three points that lie on the circumference; the center and radius are determined from these three points. def add_polygon_2d(self, points=[], poly_id="undefined", ax=None, **kwargs): """Add a polygon specified by list of input points :param list points: list with :class:GeoPoint` objects :param str poly_id: string ID of this object (e. The Web Machinist Software Program has a Circle Center 3 Known Points Calculator in it. Computational Geometry Library in Python - Arc/Circle:. Is there a simple formula which gives you the radius of this circle?. An Arc or Circle is another important entity in Computational Geometry. Video transcript. So every time you do a conditional or something, you'll be sending a request over the network (wifi apparently) to the robot, getting a response, running the if. P = (X_{P},Y_{P},Z_{P}) = \\mbox{General point on diameter of circle circle. $\endgroup$ - Yasmin Apr 20 at 1:41. 1 - Enter the x and y coordinates of three points A, B and C and press "enter". This calculator finds the equation of parabola with vertical axis given three points on the graph of the parabola. Given line AB, point P, and radius R (Figure 4. I'm able to import the data into blender as a point cloud with this python script, however I would like to create a model with solid faces. Given a circle (coordinates of centre and radius) and a point (coordinate), find if the point lies inside or on the circle, or not. Reference for filled-, unfilled- and custom marker types with Matplotlib. The idea behind this python workbench for FreeCAD was to give users some "quick" access tiny tools. Each line of input to your program will contain the x and y coordinates of three points, in the order A(x),A(y),B(x),B(y),C(x),C(y). The radius of a circle is the distance from the center to any point on the circle's edge. 859834, which is almost straight line. By polling three points on every circle I can predict the co-ordinates of the center. When a point is on the y-axis, it is neither to the left nor the right of the origin. Find the midpoints of those lines The perpendicular bisectors are the lines through the midpoints with slopes equal to the negative reciprocals of the slopes of the secant lines. For the Python interpreter to find Zelle's module, it must be imported. (x1,y1),(x2,y2),(x3,y3. Everything begins with points. Given 3 Nulls, A, B, C, I would like to have an ARC continuously interpolated between them (no matter you move any of nulls on XYZ space) in a way that every time the ARC is updated and starts on A, ends on B and pass through C (looking at image on my first post). Computational Geometry Library in Python - Arc/Circle:. We like to find one of the circles in. A point is nothing more than a list of values called a coordinate. This utility is up next in the combo view with "Work Features" label. You can vote up the examples you like or vote down the ones you don't like. The parametric equation of a circle. The Degree is a positive whole number. Let the co-ordinate of the given point be (x1, y1, z1) and equation of the plane be given by the equation a * x + b * y + c * z + d = 0, where a, b and c are real constants. This calculator finds the equation of parabola with vertical axis given three points on the graph of the parabola. 14 radius = float (input (' Please Enter the radius of a circle: ')) area. If the three points are collinear (all lying on a straight line) then the circle passing through all three will have a radius of infinity. It sometimes happens that you need 3 points to sit on a circle. For more see Parametric equation of a circle. for deletion, default: "undefined") """ if ax is None: ax = self. The points of this circle are represented in a string with the coordinates like so: "(x1,y1),(x2,y2),(x3,y3)" Where x1,y1,x2,y2,x3,y3 are digits. Given three points p1, p2 and p3, find orientation. Solar Eclipse Animation. coord = 10, 50, 240, 210 arc = canvas. Go to the editor. Scroll down the page for more examples and solutions. A related topic is regression analysis, which. The following tool visualize what the computer is doing step-by-step as it executes the said program: Customize visualization ( NEW!) There was a problem connecting to the server. 9742), radius 2. Run your code first! It looks like you haven't tried running your new code. Approach: As we know all three point lie on the circle so they will satisfy the equation of circle and putting them in general equation we get three equation with three variable g, f and c and by further solving we can get the values. The center is radius units left of the turtle; extent - an angle - determines which part of the circle is drawn. In a previous Hypotrochoid project, you drew. Each with X and Y values. circle_imp_point_dist_2d. The average of the three Python's workers estimates is (4. Example of arc (arc()),. So if I were to take three random points here, so let's call that point A, point B, and then let's say this is point C right. m, finds the distance from a point to an implicit circle in 2D; circle_imp_point_dist_signed_2d. The Circle tool creates a circle in the current work plane by entering two points, the center and the radius, or by picking tangents, or any combination of those. Please circle your answer and show work for partial credit: a. The simplest method it to use a 2nd order polynomial (i. Measuring distance between objects in an image with OpenCV. Find the equations of the perpendicular bisectors. 31,70) & (63. Run your code first! It looks like you haven't tried running your new code. This JavaScript program computes the smallest circle that encloses an arbitrary set of points in the plane. 3-Points Circle Developing 3-Points Circle is even more complicated. A tangent to a circle is a straight line, in the plane of the circle, which touches the circle at only one point. One Python worker receives an input point inside and an input point outside the circle, estimating Pi as (4 * 1) / 2, or 2. The points of this circle are represented in a string with the coordinates like so: "(x1,y1),(x2,y2),(x3,y3)" Where x1,y1,x2,y2,x3,y3 are digits. I plotted the last one. Because a center point can be anywhere depends on the positions of three points. Suppose origin O(0,0) is the reference point and P1( a,b) and P2(c,d) are the two points given (. Use MathJax to format equations. For the Python interpreter to find Zelle's module, it must be imported. Description. 0 release, some three-dimensional plotting utilities were built on top of Matplotlib's two-dimensional display, and the result is a convenient (if somewhat limited) set of tools for three-dimensional data visualization. The number of values in the list corresponds with the number of dimensions of the space it resides in. 000000) / 3, or 3. It uses the Draft Linestyle set on the Draft Tray. This is achieved by noting that the closest point on the line through P 1 P 2 to the point P 3 is along a perpendicular from P 3 to the line. Matplotlib produces high-quality figures like many of the illustrations used in this book. The simplest method it to use a 2nd order polynomial (i. Marker Reference¶. Find the slopes of the lines through two pairs of points. Make $$y$$ the subject of the equation. So, you can't really find an angle between the two points unless you are using a reference point. The three points will not be on a straight line. From P draw an arc with radius R, cutting line DE at C, the center of the required tangent arc. The corresponding problem in n-dimensional space, the smallest bounding sphere problem, is to compute the smallest n-sphere that. The parametric equation of a circle. Because a center point can be anywhere depends on the positions of three points. This is the currently selected item. The ceter point is 1 foot from a line drawn between the other 2 points. To do this, keep the compass set to the same width. There's a bigger issue with 3_point_circle because tangency auto-snaps don't modify the circle preview. They should. The smallest-circle problem (also known as minimum covering circle problem, bounding circle problem, smallest enclosing circle problem) is a mathematical problem of computing the smallest circle that contains all of a given set of points in the Euclidean plane. Next lesson. I'm able to import the data into blender as a point cloud with this python script, however I would like to create a model with solid faces. Two circles that intersect, can have a possibility of none/one/two solutions. Posted by 3 days ago. 31,70) & (63. I plotted the last one. Unless you have some prior information as to what the form of the mathematical function "should be", there is no way of distinguishing which of those infinite number of functions is the "right" one. The task is to find the equation of the circle and then print the center and the radius of the circle. Draw a small arc opposite the given point. You will see updates in your activity feed. With this behavior it is possible to open also DXF drawings that contains data from 3rd party applications. One of the biggest difficulty for new users of FreeCAD, is to know in which workbench to find a specific tool. The following diagrams show the Radius Tangent Theorem and the Two-Tangent Theorem. Equation of a circle passing through 3 points (x1, y1) (x2, y2) and (x3, y3) #N#The equation of the circle is described by the equation: #N#After substituting the three given points which lies on the circle we get the set of equations that can be described by the determinant: #N#The coefficienta A, B, C and D can be found by solving the. If determinant is > 0, points are in CCW order. coord = 10, 50, 240, 210 arc = canvas. Three Circles can have none/one/or an area as its solution. I want to give the points [0,1],[1,0] and [0,-1] to python and plot the circle that passes over them. Because a center point can be anywhere depends on the positions of three points. Mark the points of intersection "D" and "E. The Overflow Blog Introducing Collections on Stack Overflow for Teams Give 3 points and a plot circle. Find the center point and radius for the equation of a circle passing through (2,2), (2,4) and (5,5) x and y-coordinates. } C = (X_{C},Y_{C},Z_{C}) = \\mbox{Centre of the circle. Plug into the x input of a Point. We go through the prerequisite skills (knowing how to write the equation, distance, midpoint, equation of a line.

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• Sarat Chandra Babu N

Articles written in Resonance – Journal of Science Education

• Virtual Chemistry Lab to Virtual Reality Chemistry Lab

The recent Covid-19 pandemic has brought about drastic changes in the teaching-learning process. We witnessed a paradigm shift from conventional classrooms to virtual online learning. Although online learning is not a new concept, the covid-19 pandemic situation came with subsequent restrictions that shook the entire school educational system. Education has become highly uncertain for the time being, especially in India. This has been more evident in rural and village areas where the teaching-learning process does not include technology-enabled methods. Over one year, we have observed that conducting theory classes through online conferencing and virtual classroom software systems has become very common, and students are coping with it. However, in an online environment, students miss out on the practical aspects, i.e., laboratory experiments which are an essential part of science and engineering education. Software-based virtual labs are available, but they lack the real feel of performing an experiment, especially getting familiar with the laboratory apparatus. In this context, the present article proposes an innovative teaching-learning system that helps conduct chemistry volumetric experiments through a hardwareenabled platform named avatar-shell. The proposed system helps students develop titration skills using the actual apparatus. However, while performing experiments with the $avatar-shell$, we don't need to use real chemicals; instead, we use ordinary water. The proposed virtual reality chemistry laboratory system comprises two units-the administrator module for teachers to configure the experiment with appropriate molarity associated with a solution and the simulator module for students to perform the volumetric analysis. The simulator module $avatar-shell$ comprises a high sensitive weight sensor, which communicates with the administrator module and verifies the correctness during every instance of titration. $avatar-shell$ can connect to the administrator module via the internet, and teachers can monitor the experiment through a mobile-based application.

• # Resonance – Journal of Science Education | News

• Posted on 13 July 2021
• # Workshop in Algebra, 7–9 September 2021

Posted on 9 August 2021


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# Exercise 1 - The Mandelbrot Set

The mandelbrot set is a beautiful fractal (Wikipedia). More precisely, it contains all numbers from the complex plane where the complex quadratic polynomial

\begin{align} z_{n+1} = z_{n}^2 + c \end{align}

remains bounded. A complex number $c$ is part of the Mandelbrot set when starting with $z_0=0$ and applying the iteration repeatedly, the absolute value of $z_n$ remains smaller or equal than 2 regardless how large $n$ becomes.

Moreover, one can draw very beautiful images by looking at the escape time of a numerical computation: We will numerically determine whether a complex number $c$ belongs to the set, and if not we will keep track how many iterations are needed until $|z_n| > 2$. Plotting the escape times of a sampled grid of complex numbers as a 2D image will yield the famous Apfelmaennchen you might have seen before.

• Write a function mandelbrot(relim, imlim, resteps, imsteps, maxiterations) that computes and returns the escape times for the mandelbrot set.

• relim and imlim are tuples that define the boundary of the complex plane (e.g. relim=(-2,1) and imlim=(-1,1)), for convenience imlim should contain real numbers although it represents the complex axis.
• resteps and imsteps are integers and define the sampling resolution along each axis (e.g. 300 and 200 steps).
• maxiterations is an integer defining the maximum number of iterations (e.g. 50).

• The function should sample complex numbers from the plane as defined by the parameters relim, imlim, resteps, imsteps and repeatedly apply the quadratic polynomial from above until the absolute value of a complex number is larger than 2. In case a maximum number of iterations is reached (maxiterations) the number is believed to be part of the set.

• The function should return a 2D array containing the number of iterations needed for every sampled complex number and two 1D arrays containing the sampled values along each axis. Escape times in the 2D array of exactly maxiterations indicate complex numbers that are believed to be part of the set.
• Make a 2D color plot of the escape times, you might want to look at matplotlib's imshow function.

### Hints how the function mandelbrot may work:

• Create a 2D numpy array containing the escape times, in the beginning filled with zeros.

• Secondly, create two 1D arrays sampling each dimension of the complex plane. You might want to take a look at the np.linspace function. You can multiply one of the arrays with 1j to create the imaginary axis. Later on you can calculate each starting complex number simply from summing individual elements of both arrays.

• The easy brute-force solution involves three nested for loops. Be aware that you can use python's break statement to leave a for loop and spare unnecessary computations.

• Iterate at most maxiterations times for any value in your complex plane and apply the quadratic polynomial from above. If the absolute value gets larger than 2 you can stop iterating and store the number of iterations you have needed in the 2D escape array. Thus, for complex numbers that are actually part of the Mandelbrot set, the 2D array of escape times should contain the value maxiterations.

### Hints for optimization:

• You can aim for an optimized version that uses vectorized computation. Here you should only need a single for loop.

• Create a second and third 2D array containing the starting complex numbers and the intermediate values, checkout the meshgrid function to create these.

• Use the second 2D array to iteratively update all values in the third array at once (vectorized computation!). Mask all values that already escaped the boundary via boolean indexing to speed up the computation.

• Remember that boolean indexing creates copies, not views! So you need to find a smart way to only apply the polynomial to values that haven't escaped without always keeping the full 2D array, otherwise internally numpy has to reiterate the full array every timestep. If you also create mesh grids of matrix indices you can reduce the 2D arrays each iteration to 1D arrays containing only the currently not escaped values. This is tricky!

• Now you can use finer resolutions than above!

# Brute-Force Solution

In [1]:

import numpy as np

def mandelbrot(relim=(-2,1), imlim=(-1, 1), resteps=300, imsteps=200, maxiterations=50):
"""Computes the *escape times* of the numeric approximation of the Mandelbrot set.

Values of maxiterations refer to complex numbers that are believed to be part of the set.

:param relim: The limits of the real axis
:param imlim: The limits of the imaginary axis
:param resteps: Sampling resolution along real axis
:param imsteps: Sampling resolution along the imaginary axis
:param maxiterations: Maximum number of iterations

:return:

2D Array of escape times
1D Array of real samples
1D Array of imaginary samples

"""
realaxis = np.linspace(relim[0], relim[1], resteps) # Sample the real axis
imaxis = np.linspace(imlim[0], imlim[1], imsteps) * 1j # Sample the imaginary axis
escape = np.zeros((imsteps, resteps), dtype=int) # 2D array of escape times

for irun in range(resteps): # Iterate over real axis
for jrun in range(imsteps): # Iterate over imaginary axis
c = realaxis[irun] + imaxis[jrun] # Starting value
z = 0.0 # Helper variable
for krun in range(maxiterations): # Compute the escape time
z = z**2 + c
if np.abs(z) >= 2:
break # c is not part of the Mandelbrot set
else:
krun = maxiterations # If the for loop did not break c is part of the set
escape[jrun, irun] = krun

return escape, realaxis, imaxis

In [2]:

xlim = (-2,1)
ylim = (-1,1)
xsteps = 300
ysteps = 200
maxiterations = 50

In [3]:

escape, realaxis, imaxis = mandelbrot(xlim, ylim, xsteps, ysteps, maxiterations)

In [4]:

import matplotlib.pyplot as plt
%matplotlib inline

plt.imshow(escape, aspect='auto', origin='lower')
plt.xticks(np.arange(xsteps)[::50], ['%.2f' % x for x in realaxis[::50]])
plt.yticks(np.arange(ysteps)[::50], ['%.2f' % x.imag for x in imaxis[::50]])
plt.xlabel('real axis')
plt.ylabel('imaginary axis')

Out[4]:

<matplotlib.text.Text at 0x7f5801461f90>

# Optimized Version

In [5]:

def mandelbrot_pro(relim=(-2,1), imlim=(-1, 1), resteps=1500, imsteps=1000, maxiterations=50):
"""As function mandelbrot, but faster if the number of maxiterations is rather low.

It uses a vectorized approach.

"""
realaxis = np.linspace(relim[0], relim[1], resteps)
imaxis = np.linspace(imlim[0], imlim[1], imsteps) * 1j
escape = np.ones((imsteps, resteps), dtype=int) * maxiterations

realarray, imarray = np.meshgrid(realaxis, imaxis)
complexarray = realarray + imarray

indices_real = np.arange(len(realaxis)) # index array
indices_im = np.arange(len(imaxis)) # index array
mesh_indices_real, mesh_indices_im = np.meshgrid(indices_real, indices_im) # Create a mesh version

zarray = complexarray.copy()

for irun in range(maxiterations):

if len(zarray) == 0:
break

zarray *= zarray # Application of the polinomial
zarray += complexarray

mesh_indices_real = mesh_indices_real[inv_mask] # Filter the indeices
zarray = zarray[inv_mask] # Filter the z values
complexarray = complexarray[inv_mask] # Filter the starting values

return escape, realaxis, imaxis

In [6]:

xlim = (-2,1)
ylim = (-1,1)
xsteps = 1500
ysteps = 1000
maxiterations = 50

In [7]:

escape, realaxis, imaxis = mandelbrot_pro(xlim, ylim, xsteps, ysteps, maxiterations)

In [8]:

plt.imshow(escape, aspect='auto', origin='lower')
plt.xticks(np.arange(xsteps)[::100], ['%.2f' % x for x in realaxis[::100]])
plt.yticks(np.arange(ysteps)[::100], ['%.2f' % x.imag for x in imaxis[::100]])
plt.xlabel('real axis')
plt.ylabel('imaginary axis')

Out[8]:

<matplotlib.text.Text at 0x7f5801349b10>

In [9]:

"""Short timing test"""
%timeit mandelbrot((-2,1), (-1,1), 45, 30, 10)
%timeit mandelbrot_pro((-2,1), (-1,1), 45, 30, 10)

10 loops, best of 3: 41.9 ms per loop
1000 loops, best of 3: 673 µs per loop

# Exercise 2 - Numerical Integration of a Neuron Model

We are going to simulate our first neuron model with Euler integration (Wikipedia). Basically, our neuron model consists of a single differential equation that describes the development of the membrane voltage over time. For now let's assume this equation is an arbitrary function $f$: \begin{align} \frac{dV}{dt} = f(V) \end{align} To obtain the voltage as a function of time, i.e. $V(t)$, we have to solve the differential equation. Lucky for you, we are in the computer practical and not the analytical tutorial. We are going to solve it numerically, so there's no need for a complicated Ansatz :-)

As said before, we will use simple Euler integration. Accordingly, if we assume discretized time we can easily compute $V(t+1)$, the membrane voltage of the next time step, in case we now the previous voltage $V(t)$: \begin{align} V(t+1) = V(t) + f(V) * dt \end{align} with $dt$ the size of the discretized timesteps.

If we start with a chosen initial value of $V(0)$ we can iteratively solve the differential equation.

By this method we can simulate very complex neuron models. Let's simulate a rescaled version of the exponential integrate and fire neuron (Scholarpedia): \begin{align} \frac{dV}{dt} = -V + \exp(V) + I \end{align}

$I$ describes a fixed input current. We will simulate several neurons fed with different current values. For some values of $I$ the membrane potential $V$ will rise to infinity, this corresponds to the upstroke of an action potential (you remember action potentials from a neurobio course, right?). However, our neuron model cannot recover from this upstroke by itself. For a smooth recovery we would need a second differential equation. However, we will keep our model simple and add a so called reset rule: whenever $V$ crosses a particular threshold $V(t)\geq V_t$ then we set it back to a reset value at the next timestep $V(t+1)=V_r$.

• Write a function expIF_neuron(V, I, Vt, Vr, duration, dt) that simulates one or more exponetial integrate-and-fire neurons.

• The parameter V can be a scalar or numpy array describing the initial conditions
• The parameter I can be a scalar or numpy array describing the input currents
• The parameter Vt is a scalar value defining the spiking threshold
• The parameter Vr is a scalar defining the reset value after threshold crossing
• The parameter duration gives the length of the simulation
• The parameter dt describes the stepsize of the Euler integration
• The function should return
• A 2D array of voltage traces, i.e. the simulated development of the membrane potential
• First dimension is the number of neurons, second dimension the voltage trace over time
• First entries in second dimension should contain the initial values
• A 1D array containing the discretized timesteps
• In case V and I are arrays, the function should be vectorized, i.e. there should only be a single loop over all timesteps, but no loop over all neurons!
• Simulate 5 neurons at once (do NOT call the function 5 times!) with 5 different input currents $I\in\{-3.0, -2.0, -1.0, 0.0, 1.0\}$.

• Choose the other parameters as
• $V_r=-1.0$
• $V_t=5.0$
• $duration=10.0$
• $dt = 0.01$
• Set the initial $V$ values to $V_r$
• Plot all 5 voltage traces in a single plot, add a legend and label the axis.

### Hints

• You can use the template provided below. This exercise can be solved in just a handful of lines ;-)
• To incorporate the reset rule you may try boolean indexing.

In [10]:

## The template: ##

def expIF_neuron(V=-1.0, I=0.0, Vt=5.0, Vr=-1.0, duration=10.0, dt=0.01):
"""Numerically integrates the expIF neuron membrane equation with the Euler-Method.

The neurons obey a reset rule, when the membrane potential crosses Vt
it is set back to Vr.

:param V: array of initial membrane values (or a scalar)
:param VT: spiking threshold (scalar)
:param I: array of input currents (or scalar)
:param duration: duration of experiment (scalar)
:param dt: stepsize of Euler integration (scalar)

:return:

2D array of voltage time series, first dimension the neurons,
second dimension the voltage trace.
First entries contain the initial values.

1D array of simulation times

"""

steps = int(duration/dt) # Calculate the number of simulation steps

if isinstance(V, np.ndarray): # V can be scalar or an array, we need to check first
nneurons = len(V) # Infer the number of neurons from the length of the initial conditions
else:
nneurons = 1

V_series = np.zeros((nneurons, steps+1)) # Array that will contain the voltage traces
# 1st dim neurons, 2nd dim voltage traces
# i.e. V[2,10] would return the voltage of neuron #2 at the 10th timestep!
# Wee need steps+1 since the 0th entry should contain the initial conditions

V_series[:, 0] = V # Set initial conditions

times = np.zeros(steps+1) # Array of timesteps

for step in range(1, steps+1): # Loop starting from step 1 (0th contains initial conditions)

# Manipulate V_series here to simulate the neuron model
# Iteratively compute f(V(t)) = -V(t) + exp(V(t)) + I and V(t+1) = V(t) + f(V(t)) * dt
# Do not introduce another for loop, try to think vectorized
# Try using boolean indexing to implement the threshold crossing and voltage reset

######### End of your code ###########

times[step] = times[step-1] + dt # You actually don't need the times explicitly, but returning them
# will make plotting easier

return V_series, times

In [11]:

def expIF_neuron(V=-1.0, I=0.0, Vt=5.0, Vr=-1.0, duration=10.0, dt=0.01):
"""Numerically integrates the expIF neuron membrane equation with the Euler-Method.

The neurons obey a reset rule, when the membrane potential crosses Vt
it is set back to Vr.

:param V: array of initial membrane values (or a scalar)
:param VT: spiking threshold (scalar)
:param I: array of input currents (or scalar)
:param duration: duration of experiment (scalar)
:param dt: stepsize of Euler integration (scalar)

:return:

2D array of voltage time series, first dimension the neurons,
second dimension the voltage trace.
First entry contains the initial values.

1D array of simulation times

"""

steps = int(duration/dt) # Calculate the number of simulation steps

if isinstance(V, np.ndarray): # V can be scalar or array, we need to check first
nneurons = len(V)
else:
nneurons = 1

V_series = np.zeros((nneurons, steps+1)) # Array that will contain the voltage traces
# 1st dim neurons, 2nd dim voltage traces
# i.e. V[2,10] would return the voltage of neuron 3 at the 11th timestep!
# Wee need steps+1 since the 0th entry should contain the initial conditions
V_series[:, 0] = V # Set initial conditions

times = np.zeros(steps+1) # Array of timesteps

for step in range(1, steps+1): # Loop starting from step 1 (0th contains initial conditions)

prev_V = V_series[:, step-1]
dV = -prev_V + np.exp(prev_V) + I

next_V = prev_V + dt * dV # Euler step
next_V[prev_V>=Vt] = Vr # Voltage reset
prev_V[prev_V>=Vt] = Vt # For better plotting we bound also the previous step

V_series[:, step] = next_V

times[step] = times[step-1] + dt # You actually don't need the times explicitly, but returning them
# will make plotting easier

return V_series, times

In [12]:

nneurons = 5
Vt = 5.0
Vr = -1.0
dt = 0.01
duration = 10.0
I = np.linspace(-3.,1.0, nneurons)
V = np.ones(nneurons)*-1.0

In [13]:

V_series, times = expIF_neuron(V, I, Vt, Vr, duration, dt)

In [14]:

for neuron in range(nneurons):
plt.plot(times, V_series[neuron,:], linewidth=2, label='I=%.1f' % (I[neuron]))

ax = plt.gca()
ax.ticklabel_format(useOffset=False) # prevents strange units
plt.xlabel('time')
plt.ylabel('V')
plt.legend()

Out[14]:

<matplotlib.legend.Legend at 0x7f5801411250>

In [ ]:


text:
# Not understanding the proof of ramification groups are $p-$groups in Taylor Frohlich

This is the part I struggled for the whole afternoon and I could not figure out obvious reason.

Let $L/K$ be a finite galois extension with $K=Frac(O)$ where $K$ has a finite residue field.($O$ is dedekind domain. Then $p\in Spec(O)-0$, $|\frac{O}{p}|<\infty$.) Assume $K$ is complete. Since $K$ is complete, $O$ has a single prime and $O_L$ has a single prime $p$ corresponding to previous prime. Let $\Delta_0$ be the inertia group of $L/K$ extension. Let $\Delta_1$ be the 1st ramification group.

$\Delta_1=\{\sigma\in\Delta_0\vert \forall a\in O_L, \sigma(a)-a\in p^2\}$.

Now $\frac{\Delta_0}{\Delta_1}\to k_L^\star$ is embedding where $k_L$ is the residue field associated to $L$. This yields $p\nmid |\frac{\Delta_0}{\Delta_1}|$. The book starts trying to show $|\Delta_1|=p^s$ for some $s\in Z_{\geq 0}$. Prove by contradiction.

"Since $\Delta_1$ is normal in $\Delta_0$ by $\Delta_1$ kernel of $\Delta_0\to k_L^\star$, it suffices to argue $\Delta_1$ is a $p-$group. Assume $r\mid|\Delta_1|$ s.t. $(r,p)=1$. Then pick out the $p-$group in $r$. Pick out an element $\sigma\in\Delta_1$ s.t. $\sigma^p\neq 1$ and $\sigma^r=1$. Take uniformizer $\pi\in L$ as $L$ is local. Then $\sigma(\pi)-\pi\in p^2$. Hence $\sigma(\pi)=\pi(1+a)$ where $a\in p^x-p^{x+1}$ for some $x\geq 1$. Consider $\sigma^p(\pi)=\pi\prod_{0\leq i\leq p-1}(1+\sigma^i(a))$. This reduction by $p^{x+1}$ yields $\sigma^p(\pi)\equiv\pi(\mod p^{x+1})$ via $\sigma^i(a)\equiv a(\mod p^{x+1})$.

Then the book concludes for any integer $s$, $\sigma^{ps}(\pi)=\pi(1+\beta)$ s.t. $\beta\in p^{x+1}$."

$\textbf{Q:}$ How did the book deduce $\beta\in p^{x+1}$?

$\textbf{Q':}$ I guess $\sum_{0\leq i\leq p-1}$ does not annhilate everything. Is this correct?

Ref. Algebraic Number Theory by Taylor, Frohlich Chpt 3 Section 4.

Although I did not read the book you mentioned, I modified the proof a bit so that it works.

Assume $\sigma \in \Delta_1$ with order $r>1, p\nmid r$. Choose $\lambda$ a uniformizer of $\mathcal{O}_L$. (I guess this is what meant by $\lambda$ in the image). Then $\sigma(\lambda) = \lambda(1+\alpha)$ for some $\alpha \in \mathfrak{P}^t$, $\alpha \notin \mathfrak{P}^{t+1}$. Then it can be deduced that $\sigma^i(\alpha) \equiv \alpha \pmod{\mathfrak{P}^{t+1}}$. I will show that $$\tag{1} \sigma^p(\lambda) \equiv \lambda \pmod{\mathfrak{P}^{\color{red}{t+2}}}$$

Firstly, we have $$\prod_{i=0}^{p-1}(1+\sigma^i(\alpha)) \equiv (1+\alpha)^p \pmod{\mathfrak{P}^{t+1}}$$ therefore $$\sigma^p(\lambda) = \lambda \prod_{i=0}^{p-1}(1+\sigma^i(\alpha)) \equiv \lambda(1+\alpha)^p \pmod{\mathfrak{P}^{t+2}}$$ Now $$\lambda(1+\alpha)^p = \lambda + \binom{p}{1}\lambda \alpha + \cdots + \lambda \alpha^p$$ since $p\geq 2$, we have $\lambda\alpha^p \in \mathfrak{P}^{2t+1} \subset \mathfrak{P}^{t+2}$. Also, since $p\mid \binom{p}{i}$ for $0<i<p$ and $p\in \mathfrak{P}$. Hence $\lambda(1+\alpha)^p - \lambda \in \mathfrak{P}^{t+2}$, this established $(1)$.

Thus $$\sigma^{sp}(\lambda) \equiv \lambda \pmod{\mathfrak{P}^{t+2}}$$ so $\sigma^{sp}(\lambda) = \lambda(1+\beta_s)$ for some $\beta_s \in \mathfrak{P}^{t+1}$. Then the proceed as in the picture.

In fact, for $i\geq 1$, quotients of higher ramification groups $\Delta_i / \Delta_{i+1}$ can be embedded into the additive group of $\mathcal{O}_L/\mathfrak{P}$. Hence $\Delta_1$ must be a $p$-group.

• Thanks. I think the important observation is $p\in B$ due to $char(O/B)=p$. And may be this is a dumb question. I run into the proof in Cassels' local fields that one of step requires showing $\cap_i\Delta_i=\{1\}$ in conjunction with $\Delta_i/\Delta_{i+1}$ embedding into $k$ for $i>1$. Cassels said one can find $A$ s.t. $\sigma(A)\neq A$(which is true by fixed fied.) Then he said $\sigma\not\in\Delta_i$ if $\sigma(A)-A\not\in B^{i+1}$. How to achieve the last step directly picking $A$ here? I did by contradiction if $\sigma\in\Delta_i$ for all $i$, then $\sigma$ fixes everything. – user45765 Jul 28 '18 at 12:33
• Hence $\sigma=Id_K$ – user45765 Jul 28 '18 at 12:33


text:
# Milne on Using “And”

At a used bookstore today, I picked up the 1893 text Elements of Arithmetic: For Primary and Intermediate Classes in Public and Private Schools by Dr. William J. Milne. One thing that I noticed was that he is adamant that “and” is never to be used when naming integers: “In reading numbers expressed by three figures, the tens are read after the hundreds without the word and. Thus, 478 is read four hundred seventy-eight, instead of four hundred and seventy-eight” (p. 78). That Milne feels the need to emphasize this suggests that “and” was in standard use here, so the he feels the need to “correct” it, at least in rigorous mathematical contexts.

This stood out to me because I don’t recall hearing about this convention when I was in school, but I understand it’s prevalent in elementary school these days. My students have corrected my using “and” in that context, but they weren’t clear on why. It had something to do with “and” only being used in decimals. However, that doesn’t make sense, because the standard way of reading decimals is to use the word “point” (e.g., “pi is approximately three point one four one five nine”). Given the age of the book, I was curious if Dr. Milne had anything else to say on the matter.

He defines mixed numbers on p. 150, but doesn’t explicitly tell how they’re supposed to be read. However, prior to introducing mixed numbers formally, he gives examples such as “1 and 5 sixths” and “2 and a half” (p. 147). This strongly suggests that mixed numbers are to be read with “and” between the integer portion and the fraction.

In the chapter called “Decimal Fractions”, Milne makes this explicit. He describes decimals as being shorthand for fractions: 0.9 represents $$\frac{9}{10}$$, while 0.48 represents $$\frac{48}{100}$$. “Since decimals have the same law of increase and decrease as integers,” he writes, “the denominator of the fraction may be indicated by the position of the figures” (p. 188).

When reading the number 796.584694, for instance, Milne feels we should say, “Seven hundred ninety-six and five hundred eighty-four thousand six hundred ninety-four millionths”. Milne writes, “Hence it is evident that: In reading a decimal, the decimal should be read as an integer, and the denomination of the right-hand figure should be added. In reading an integer and a decimal, use the word and only between the integral and decimal parts of the number” (p. 189).

This approach is naturally problematic when it comes to irrational numbers, which Milne doesn’t address in the book regardless. I reckon he may have gotten around this by describing the functions on algebraic numbers, using the names for transcendentals, and describing the decimal approximations as just that (e.g., “$$\sqrt{2}\pi \approx 4.443$$” would be read aloud as “the square root of two times pi about equals four and four hundred forty-three thousandths”). However, it seems needlessly confining to require someone to decide how many decimal values of π they’re going to give before they’ve started speaking (“three and fourteen-hundredths” vs. “three and one thousand four hundred sixteen ten-thousandths”). And, at any rate, this is confusing to the audience, compared to the standard convention of “three point one four one six”.

While I’m intrigued by Milne’s suggestion that decimal forms are abbreviations for fractions with denominators which are powers of ten, I don’t agree with his method of reading such numbers aloud, nor do I agree that “and” is necessarily wrong in the integer portion. He uses “and” elsewhere for addition (“three and one” etc. on p. 15, for instance). It is true that 400 + 78 = 478, which he would read as “four hundred and seventy-eight are four hundred seventy-eight” (he uses “are” rather than “is” for equality, which is worth a separate post).

Overall, though, I was intrigued to find a convention that I have been identifying as a relatively recent shift in such an old book.

Follow-up: It’s been pointed out to me that, in many applied fields, people already know the level of precision they’re using from the outset, so it wouldn’t be as much of a burden, nor would it necessarily be confusing, to always refer to decimals in this fractional form. If everyone involved in a conversation knows, for instance, that all decimal values are being given in thousandths, then it may actually be clearer to say “four thousand thirty and twenty-five thousandths” than “four thousand and thirty point zero two five”, both of which are clearer than the casual speech “forty thirty point zero twenty-five”. H/T Tyler Bartunek and others.

Clio Corvid

1. I went to elementary school (USA) in the early 1970s, and the convention about omitting “and” was definitely a standard part of the curriculum.

2. The basic idea, and many texts old and new assert this, is that we are reading units AND pieces, just as we give fourteen dollars and fifty-two cents, or five feet and three inches. It is a conjunction joining two different things.

1. Paul Hartzer says:

Fair enough, but the claim is that “and” in “five hundred and three” will lead to confusion. It doesn’t. Everyone knows what “five hundred and three” means. At worst, it will lead to the question “three WHAT?”, the correct answer to which should be “three UNITS” — thus being an opportunity to repeat a clarification that bears repeating regardless.

3. The 1890s was a time when Gauss’ 1801 book “discussions on arithmetic” was provided as background to elementary arithmetic books. By 1910 an international convention further placed modular arithmetic as a foundation for college arithmetic courses.

Of course, WW I ended the math fad experiment, with all things German were removed from US, British and French math books. Classical Greek geometry was re-introduced in ways that need to be recalled.

Math fads continue to muddle arithmetic education.

My 1950s high school math education used set theory as background. By the 1960s set theory was moved to the front of the class in another failed math fad, one os several in vogue over thte last 50 years.

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text:
# Completely-integrable differential equation

An equation of the form

$$\tag{* } \omega \equiv \ \sum _ {i = 1 } ^ { n } P _ {i} ( x) dx ^ {i} = 0,\ \ P _ {i} \in C ^ {1} ,$$

for which an $( n - 1 )$- dimensional integral manifold passes through each point of a certain domain in the space $\mathbf R ^ {n}$. A necessary and sufficient condition for complete integrability of the differential equation (*) is the Frobenius condition $\omega \wedge d \omega = 0$, where $\wedge$ is the symbol of the exterior product [1]. If $n = 3$, this condition has the form:

$$P _ {1} \left ( \frac{\partial P _ {3} }{\partial x ^ {2} } - \frac{\partial P _ {2} }{\partial x ^ {3} } \right ) + P _ {2} \left ( \frac{\partial P _ {1} }{\partial x ^ {3} } - \frac{\partial P _ {3} }{\partial x ^ {1} } \right ) + P _ {3} \left ( \frac{\partial P _ {2} }{\partial x ^ {1} } - \frac{\partial P _ {1} }{\partial x ^ {2} } \right ) =$$

$$= 0.$$

Instead of equation (*) the following system of equations is sometimes considered [2]:

$$dx ^ {i} = \ \sum _ {j = 1 } ^ { {n } - 1 } a _ {j} ^ {i} ( x, t) dt ^ {j} ,\ \ i = 1 \dots n.$$

In this case the conditions of complete integrability assume the form:

$$\sum _ {l = 1 } ^ { n } \frac{\partial a _ {j} ^ {i} }{\partial x ^ {l} } ( x, t) a _ {k} ^ {l} ( x, t) + \frac{\partial a _ {j} ^ {i} }{\partial t ^ {k} } ( x, t) =$$

$$= \ \sum _ {l = 1 } ^ { n } \frac{\partial a _ {k} ^ {i} }{\partial x ^ {l} } ( x, t) a _ {j} ^ {l} ( x, t) + \frac{\partial a _ {k} ^ {i} }{\partial t ^ {j} } ( x, t),$$

$$i = 1 \dots n ; \ j , k = 1 \dots n .$$

The family of integral manifolds of a completely-integrable differential equation is a foliation [3].

#### References

 [1] G. Frobenius, "Ueber das Pfaffsche Problem" J. Reine Angew. Math. , 82 (1877) pp. 230–315 [2] V.V. Nemytskii, "On the orbit theory of general dynamic systems" Mat. Sb. , 23 (65) : 2 (1948) pp. 161–186 (In Russian) [3] S.P. Novikov, "Topology of foliations" Trans. Moscow Math. Soc. , 14 (1965) pp. 268–304 Trudy Moskov. Mat. Obshch. , 14 (1965) pp. 248–278 MR0200938 Zbl 0247.57006

The exterior product $\wedge$ is also called the outer product.
An $( n - 1 )$- dimensional submanifold $M$ of $\mathbf R ^ {n}$ is an integral manifold of (*) if the restriction of $\omega$ to $M$ is zero; cf. also Pfaffian equation. Another (dual) way to formulate this is as follows. Let $U$ be an open subset where $\omega \neq 0$. For each $x \in U$ let $D _ {x}$ be the set of all (tangent) vectors $\xi$ at $x \in \mathbf R ^ {n}$ such that $\omega ( \xi ) = 0$. Then $D _ {x} \subset T _ {x} ( \mathbf R ^ {n} )$ is an $( n - 1 )$- dimensional subspace and the $D _ {x} \in U$ define a distribution on $U$. An integral manifold $M$ of $D$( or of the equation $\omega = 0$) is now an $( n - 1 )$- dimensional submanifold of $U$ such that $T _ {x} M = D _ {x}$ for all $x \in M$. A distribution $D$ on $U$ is called involutive if for all vector fields $\xi , \eta$ on $U$ such that $\xi ( x) , \eta ( x) \in D _ {x}$ for all $x$ also $[ \xi , \eta ] ( x) \in D _ {x}$ for all $x$. The Frobenius integrability condition $\omega \wedge d \omega = 0$ is equivalent in these terms to the condition that the distribution defined by $D$ be involutive. All this generalizes to systems of equations $\omega ^ {i} = 0$, $i = 1 \dots r$; cf. Integrable system.
The phase completely-integrable system (completely-integrable Hamiltonian system), completely-integrable Hamiltonian equation on an $n$- dimensional manifold refers to a rather different property, viz. that of having $n$( including the Hamiltonian (function) itself) integrals in involution; cf. Hamiltonian system.


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# Maintain log documents size dealt with without logrotate

Exists any kind of means to maintain documents size of log documents dealt with without revolving it by a new vacant documents and also deleting (or archiving) the old documents. As an example, if I set log documents maximum size to 1MB, after the documents size increase past that restriction it will certainly be instantly secured, the message is added 'tail' and also the earliest component of message is bulged to maintain the documents size 1MB.

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2019-05-06 01:21:57
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 *      prune - clips off the tops of a list of files described in the
*              file /etc/default/prune. Designed to be run periodically
*              from cron, the idea is to automatically shorten log
*              files that grow forever. This file contains a list of
*              files (full pathname), and the number of blocks that
*              should be retained.  To retain some sanity, prune will
*              clip the file after the next newline. The file
*              /etc/default/prune should look something like:
*
*
*              The crontab entry on infoswx for prune looks like:
*
*                      0 5 * * * /etc/prune >/usr/adm/prune.log 2>&1
*
*              Compile with:  cc -O -o prune prune.c
*
*              and your system block size.
*
*      Ray Davis  infoswx!bees  09/25/85

Many (most? all?) daemons will certainly re - open their log files if sent out a HUP signal (as an example by very same cron work that runs trim)

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2019-12-03 05:34:10
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You can write a little celebration manuscript to do this. Simply tail the documents to a particular byte matter making use of tail -c and also overwrite the documents.

from man tail :

-c, --bytes=N
output the last N bytes; alternatively, use +N to  output  bytes
starting with the Nth of each file

If  the  first  character of N (the number of bytes or lines) is a +',
print beginning with the Nth item from the start of each  file,  other‐
wise, print the last N items in the file.  N may have a multiplier suf‐
fix:  b  512,  kB  1000,  K  1024,  MB  1000*1000,  M   1024*1024,   GB
1000*1000*1000, G 1024*1024*1024, and so on for T, P, E, Z, Y.

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2019-05-08 13:52:29
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Your only remedy might be to write your very own userspace documents system or add to an existing one. Consider the partial checklist at Filesystem in Userspace

If you do not have the abilities to add, supply a task attention or \$ or both, to add it for you.

I desire I had the moment to do it, I have constantly needed something specifically similar to this.

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2019-05-08 13:37:41
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You can do something comparable making use of a FIFO, which is type of like an absolutely no - byte sized documents.

Nonetheless, note that if absolutely nothing reads from this documents, after that the syslog procedure might come to be obstructed, and also will certainly stop contacting ALL of your logfiles. I'm not exactly sure if this actions has actually been transformed with more recent variations of Ubuntu/CentOS.

One instance here

For an additional instance, attempt something similar to this.

sudo mkfifo /var/log/everything.fifo

And also add this to (r) syslog.conf, after that reactivate syslog :

*.*     |/var/log/everything.fifo

After that watch the FIFO from one window :

cat /var/log/everything.fifo

And also in an additional window, send some things to syslog :

logger Test1
logger Test2
logger Test3

You need to see the "Test *" lines in the result of cat over.

This attribute can be wonderful for debugging, specifically if you uncommitted to maintain the information around for longer. As an example, if you just intend to see every little thing with the exception of thefirewall spam, you can do something similar to this :

grep -vi "kernel: .* on wan" /var/log/everything.fifo

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2019-05-08 13:36:29
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Here's my 2nd solution. This is a rather hackish.

Usage watch (1) to repetitively execute tail --bytes=1024 (the last 1024 bytes of the logfile, many thanks to @jjclarkson for that solution).

watch --no-title tail --bytes=1024 /var/log/messages >/tmp/messages.watch

And afterwards watch the documents with :

less --raw-control-chars /tmp/messages.watch

The distinction in between watch and also a while loop is that watch will just upgrade/ tmp/messages. watch if there were adjustments to/ var/log/messages.

while true; do
tail --bytes=1024 /var/log/messages > /tmp/messages.watch
sleep 1
done

And also well, I presume you can place a test` in the while loop to make sure that tail is just implemented if/ var/log/messages was upgraded, yet I will not figure that out currently.

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2019-05-08 13:29:52
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