For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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3answers
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Finding angles of triangles

I have what appears to be a $3$-sided triangle: it is two lines on a 180 degree line at the bottom. The bottom left angle is $4x-3$ the top angle is $6x + 3$ and the bottom right angle is not given, ...
3
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1answer
146 views

Projective Plane vs. Reference Plane

I was told that the Projective Plane was also known as the Reference Plane in Projective geometry, but when I told my professor this, he freaked and told me I was completely wrong. He said that the ...
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3answers
559 views

Distance from a point to a line in vector geometry - real world applications?

In vector geometry it is a standard example how to calculate the distance between a point and a line in space. However are there any interesting real world applications of this. I.e. where one wants ...
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4answers
1k views

Find control point on piecewise quadratic Bézier curve

I need to write an OpenGL program to generate and display a piecewise quadratic Bézier curve that interpolates each set of data points: $$(0.1, 0), (0, 0), (0, 5), (0.25, 5), (0.25, 0), (5, 0), (5, ...
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2answers
571 views

Simple way to measure or calculate the volume of clothing?

Quick disclaimer: I'm a StackOverflow regular and completely out of my element here...so go easy on me. Just wanted to ask if anyone know of a simple way to measure or calculate the volume of ...
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1answer
154 views

question about paralelepiped

please see this link question 20 http://www.naec.ge/images/doc/documents/GRE-2010-MAT.pdf problem is following on figure 1 as you see there is parallelipiped ...
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1answer
1k views

Finding the area of the intersection of two circles

The following is problem 8 from a GRE exam found here. The problem states that the two circles with radius $r=3$ intersect each other such that the area of the darkened region is equal to the sum ...
2
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1answer
231 views

Existence of $\pi$ [duplicate]

Possible Duplicates: Why is the ratio of the circumference of a circle to its diameter independent of the circle? Proof that Pi is constant (the same for all circles), without using limits ...
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3answers
281 views

given two lines in 2D, how to select the angle bisector related to the smallest angle between the lines

I have two lines: first line: $a_1x+b_1y=c_1 \qquad(1)$ second line: $a_2x+b_2y=c_2 \qquad(2)$ I know that the two angle bisectors are expressed by ...
5
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1answer
423 views

A way to see that $\int_{0}^{\infty}\exp(-x)dx=1$?

One can easily find the integral $\int_{0}^{\infty}\exp(-x)dx$. It is equal to 1. But is there a way to understand this geometrically without integration? If i rotate the picture i see that ...
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8answers
7k views

Is there an equation to describe regular polygons?

For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides ...
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2answers
122 views

Showing a (relatively simple) set of polynomial zeros in projective space is irreducible

I'm teaching myself a little algebraic geometry and I was hoping you could help me with an exercise. I have my head around affine spaces alright but I am having a little more trouble with projective ...
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1answer
63 views

right angle turn

i have following question suppose we have some AB length and have turned it by 90 angle about some arbitrary o point lies on the AB length.after turning AB maps some A'B' length.we should find ...
5
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1answer
403 views

Is there any (nontrivial) constructible rational angle?

Yesterday, I talked with a friend about a problem where the solution would be an angle of $2$ radians (about $114.6°$). Then somehow the question arose whether such an angle would be constructible ...
5
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1answer
246 views

Trying to find an unknown point just with angles

This is my model: What I do know: A, B, C, which form an equilateral triangle Mab, Mbc, Mac which are the middle points Angles x and y, which are the angles formed by the segment from the unknown ...
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4answers
1k views

Drawing a thickened Möbius strip in Mathematica

I would like to have Mathematica plot a "thickened Möbius strip", i.e. a torus with square cross section that is given a one-half twist. Ideally, I would like this thickened Möbius strip to be ...
17
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2answers
395 views

No hypersurface with odd Euler characteristic

Here is a classic problem which I encountered and could not solve: Prove that a simply connected closed smooth manifold has no closed sub-manifold of co-dimension $1$ with odd Euler ...
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2answers
258 views

Why can any affine transformaton be constructed from a sequence of rotations, translations, and scalings?

A book on CG says: ... we can construct any affine transformation from a sequence of rotations, translations, and scalings. But I don't know how to prove it. Even in a particular case, I found ...
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1answer
252 views

Similar - perspective triangles implies corresponding sides are parallel?

In a general homothetic transformation, if two triangles have corresponding sides parallel then the lines joining respective vertices are concurrent at the homothetic center. I was wondering if the ...
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0answers
77 views

Questions about boundary faces of simplices and triangulations

Let $S$ be a simplex in $\mathbf R^n$ and let $\{S_i\}$ be a triangulation of $S$. The boundary of $S$ is defined as the union of the boundary faces of $S$. Is this union equal to the topological ...
4
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3answers
457 views

Dissecting a square into congruent pieces that all touch the centre

Edited for clarity: I thought I had a complete set of solutions to this: Cut a square into identical pieces so that they all touch the center point. It became clear, after some discussions, ...
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3answers
644 views

Find opposite vertices of a rhombus, given the other 2

I am stuck with this problem. I posted an earlier problem with a square, where rotation with i of 90 degrees was possible. This one is a rhombus, how should I proceed? Given ABCD is a rhombus with ...
3
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1answer
430 views

Reflecting a point over a line created by two other points

This problem came up while discussing using a simplex to solve systems of equations. (By the way, yes, this is very similar to this one.) Given three points, how do I find the location of the point ...
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1answer
125 views

Distance between two ranges

I'm working on a clustering algorithm to group similar objects that are represented by ranges of real numbers. Let's say that I have a group of people who are buying sugar. Each of them defines ...
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2answers
207 views

Dimension of the manifold O(n)

$O(n)$ is the manifold of the orthogonal $n \times n$ matrices. How do I prove that its dimension is $\displaystyle \frac{n(n-1)}{2}$? Edit: Thanks to all your answers. I appreciate your help.
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3answers
4k views

How do I explain why $dA/dr = 2 \pi r$ geometrically?

There's this question in my calculus book that goes something like this: The derivative of the area of a circle with respect to its radius is equal to the circle's circumference ($dA/dr = 2 \pi r$). ...
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1answer
182 views

Can you go backwards from a Morley triangle?

Definition 1. If T is a triangle, let E(T) be the triangle formed by the intersections of the adjacent trisectors of the (interior) angles of T. Synonymously, E(T) will be called the Morley triangle ...
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0answers
116 views

Is there an interpretation for this classic identity? [duplicate]

Possible Duplicate: Different methods to compute $\sum_{n=1}^\infty \frac{1}{n^2}$. $\sum \frac{1}{n^2}= \frac{\pi^2}{6}$ There are a few proofs for that fact but can anybody see why is ...
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3answers
735 views

What is the area of the portion of 1/8 of an sphere cut off by two parallel planes?

So the problem that I'm trying to solve is as follows: Assume 1/8 of a sphere with radius $r$ whose center is at the origin (for example the 1/8 which is in $R^{+}$). Now two parallel planes are ...
4
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1answer
512 views

How to project the surface of a hypersphere into the full volume of a sphere?

The game I mentioned in "Navigating though the surface of a hypersphere in a computer game" is taking shape in here. The world is a 3-sphere where everything belongs. In Euclidean coordinates, for ...
6
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1answer
403 views

Torus: Circle cut

Given a Torus $T$ with major and minor radius $R$ and $r$, respectively, I can obtain a circle lying in $T$ by cutting $T$ with a bi-tangential plane. Now I don't want circles, but Tori with major ...
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2answers
2k views

Counting number of distinct regions with intersecting circles

Given $n$ circles of possibly different radii, how many distinct regions can there be? For small $n$, I can work it out with pictures. (I'm pretty sure $n=4$ can yield 13 distinct regions, but not ...
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3answers
111 views

Formal proof for detection of intersections for constrained segments

They told me it was off-topic at stackoverflow. So I am trying my luck here. Yes, it's a homework, but I'm looking for some guidance (or related literature) instead of complete solutions. Please see ...
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0answers
50 views

Formal proof for detection of intersections for constrained segments [duplicate]

Possible Duplicate: Formal proof for detection of intersections for constrained segments Hi I need to come up with a formal proof for the following statement: Given an arbitrary count of ...
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2answers
137 views

Determining the position of a vertex of a triangle

A triangle has a vertex $A$ at $(0,3)$, vertex $B$ at $(4,0)$, and vertex $C$ at $(x,5)$. If the area of the triangle is $8$, what is the value of $x$? I did this problem out by using the formula ...
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2answers
2k views

Intersecting Circles Circumference Problem

If two equal circles intersect so that each circle's centre lies on the other circle's circumference, what is the ratio of the part of a circle's circumference that overlaps, to its whole ...
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2answers
1k views

Finding the measure of an angle

Let E be a point outside of the square $ABCD$ such $\triangle ABE$ is an equilateral triangle. What is the measure of $\angle CED$, in degrees? I need help with this problem. I made a diagram ...
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2answers
142 views

In the area of sphere, which angle with $0\leq \alpha<2\pi$ and which angle with $0\leq \beta < \pi$?

I am lacking the skill of visualizing the problem, picture here, to decide the right intervals. The way I do it currently is to try things but the technique fails with anything more complicated to 2D. ...
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2answers
1k views

What is the name of the quadrilateral shape described by two radii and two arcs?

What is the name of the four sided shape described by two radii and two concentric arcs? Like each black and white section taken individually on this image Or the double score / triple score areas ...
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2answers
7k views

Finding the intersecting points on two circles

Given 2 circles on a plane, how do you calculate the intersecting points? In this example I can do the calculation using the equilateral triangles that are described by the intersection and centres ...
5
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1answer
513 views

Circle packing representation of a given graph

Based on the Circle packing theorem: "For every connected simple planar graph G there is a circle packing in the plane whose intersection graph is (isomorphic to) G." I would like to draw the circle ...
5
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1answer
156 views

Origin of a problem in graph theory/planar geometry

Does anyone remember where does the following problem comes from: Let $P_n$ be a set of $n$ points on the plane, and denote by $d$ the minimal distance between any two points of $P_n$ (i.e. ...
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2answers
533 views

In △ ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees?

In △ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees? I know already that angle A and angle D are congruent because ...
3
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0answers
90 views

Computing the proportion of vectors with the same sign

Let $s \propto (1, \ldots, 1)$ be a d-dimensional unit vector, so that $s_i = 1 / \sqrt{d}$. Let ${\cal V} = \{ u \in {\mathbb S}^{d-1}:u \cdot s = \cos \theta \}$ be a collection of unit vectors that ...
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0answers
290 views

Number of closed paths formed by arcs of one fifth of a circle

**I was trying to solve the following issue: Find the number of possible closed paths using one fifth of an arc (72 degrees), where at each time step we can move either clockwise or anti-clockwise. ...
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2answers
992 views

Determining end coordinates of line with the specified length and angle?

I have the point $(x_1, y_1)$, the angle $0 \leq a < 360$ in degrees and the length $l > 0$. How do I determine the end point $(x_2, y_2)$ if there is a line between $(x_1, y_1)$ and $(x_2, ...
3
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0answers
42 views

What are the techniques one can used for rule based plane generation?

I've asked the question here at gamedev SE, but the response wasn't too encouraging. So I try to reask again, from a slightly difference perspective. I have a terrain, which is defined by mesh. And ...
3
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1answer
232 views

How to define a topological tunnel?

I would like to define a notion of a topological tunnel, but I don't know how (or even if it is possible) to capture it topologically. I am interested in closed 2-manifolds in $\mathbb{R^3}$. Suppose ...
7
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2answers
333 views

Are rotations of $(0,1)$ by $n \arccos(\frac{1}{3})$ dense in the unit circle?

Under which conditions will successive rotations of $(0,1)$ by an angle $\theta$ guarantee that given $\delta > 0$ and some point $p$ on the unit circle, there exists some $n$ such that rotating ...
6
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2answers
1k views

How do I calculate the equation of a circle given 3 complex numbers?

Given three complex values (for example, $2i, 4, i+3$), how would you calculate the equation of the circle that contains those three points? I know it has something to do with the cross ratio of the ...