# Tagged Questions

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

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### Question about property of circle

We know that equal chords are equidistant from the center. However, I was curious if the lengths involved are proportional as well since the circle is a pretty symmetrical shape. Here's what I mean: ...
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### 3D Convex Hull and The Gift Wrapping Principle

I am currently trying to implement a 3D convex hull algorithm that is based on the paper Convex Hulls of Finite Sets of Points in Two and Three Dimensions by F.P. Preparata and S.J. Hong, but I’ve run ...
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### Prove the product of two distinct, opposite rotations is a translation

My homework question is what is the product of rotations through opposite angles α,−α about two distinct points. The answer is clearly a translation, but I'm not sure how to prove it. My idea on how ...
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### Least number of circles that can fill a box

My application is this: I have a robot with a laser scanner on top, and I am trying to program it in such a way that it moves through rectangular areas so that it has maximum view of everything in the ...
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### Product of two opposite, distinct rotations

My homework question is what is the product of rotations through opposite angles $\alpha, -\alpha$ about two distinct points. The answer is clearly a translation, but I'm not sure how to prove it. My ...
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### Rate of Change of Cylindrical Roll's Volume as it Unrolls

This is purely a "for-fun" question. I was minding my own business in the washroom this morning when I began to unroll some toilet paper from the roll, and in typical Breaking Bad fashion (sorry if ...
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### Catmull-Rom blending functions

I have a non-uniform Catmull-Rom spline (so the $t_i$ parameter values are not uniformly distributed). Is there a simple way to calculate the blending functions of the control points? So the spline ...
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### A quadrilateral and circularity of vertices

So we have a convex quadrilateral ABCD, which satisfies these conditions: $m(\widehat{DAC})=m(\widehat{BDC})=36°$ and $m(\widehat{BAC})=72°$. If $P=AC\cap BD, \,m(\widehat{APD})=\,?$ I did a quick ...
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### Intersection of two congruent spirals

Let $S_1$ and $S_2$ be two congruent circular spirals in $\mathbb{R}^3$, both with their axes passing through the origin. They are congruent in that their radii are equal, as are their winding ...
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### Generalized Heron's formula for n-dimensional “n-angle” instead of “triangle”

Is there a generalized version of Heron's formula for calculating the equivalent of a "volume" of an n-dimensional "n-angle" based on the length of it's sides? I've seen the equivalent formula for a ...
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### Determining a distance from a hexagonal close packing system

My friend and I have been feverishly arguing about something and I really want to know who's right. The question is as follows (bear with me, it's a little hard to define the problem) An HCP ...
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### Calculating individual wheel velocities from a desired angle in a differential wheeled robot

I am working on a simulation of a two-wheeled robot, and at present am driving it by setting each individual wheel's velocity. The robot is similar to an ePuck: What I would like to do is set an ...
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### any intersection point (if it exists) of H ∩ M is also constructible?

call a real number constructible if it can be obtained using whole numbers and a finite number of applications of operations. Given the equation of the circle H with centre $(h, k)$ and radius $r$ is ...
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### Lattice property of coprime integers

I was reading on the Wikipedia page for coprime numbers that (for $a \gt b$), gcd($a,b$)$=1$ if and only if the diagonal connecting $(0,0)$ and $(a,b)$ does not cross through any lattice points ...
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### Find vertex of a parallelogram/parallelepiped/parallelotope with minimum distance to a point

Suppose you have a parallelogram and a point. It's easy to tell which of the parallelogram's vertices is closest to the point (Euclidean distance) by checking the distance for every vertex - but this ...
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### Integral points on a circle

Given radius $r$ which is an integer and center $(0,0)$, find the number of integral points on the circumference of the circle.
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### 3D geometry cube; find a distance

Let A1B1C1D1A2B2C2D2 be a cube with A1B1C1D1 being the bottom face and A2B2C2D2 the top face. Given that A1A2 is of length 1 what's the distance between D2A1 and A2B1.
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### Linejoin for fat lines?

I draw a figure with 2 fat lines. I need to draw a join between these lines correctly. Long red lines are in a middle of each fat line. What I know: coordinates of white points. the angle between ...
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### Cube - Plane cut

I have a unit cube and plane, given by its normal vector and variable d. How can I found d value, so plane-cube intersection ...
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### Constructing a triangle

Is it possible to construct a triangle $\triangle ABC$ with $\angle BAC = 24$ $AB=\frac{1+\sqrt{5}}{2}$ and $AC=\sqrt{1+\sqrt{2}}$? I am really lost how to solve this.
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### How to calculate where a line through the earth will exit

If we assume that it is possible to dig a hole through the earth, how can we calculate exactly where the hole would exit the earth if we know .... 1) The point of entry (gps coords) 2) The angle of ...
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### Determine if a set of points on a sphere come from a uniform distribution?

I have a large distribution of points on the unit sphere $S^2$ and I want to determine if those points came from a uniform distribution on the surface. Essentially, I'm looking for a two dimensional ...
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### Common direction between two vectors

I have two vectors with the same origin and I need to find the common direction between them, that is the vector perpendicular to the line that join them. For instance, referring to this image I need ...
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### Geometry, inscribed quadrilateral and angles

How to find angle e? I tried using sum of angles and sum of interior angles. But does not work. Thanks in advance.
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### “Point P lies on the sphere described a cube.”

Point P lies on the sphere described a cube. Show that the sum of squared distances of the point P of the vertices of the cube does not depend on the choice of P. I cannot found any logical ...
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### What figure does one obtain from a Möbius band if one shrinks the boundary circle to a point?

'Im trying to solve the following problem: What figure does one obtain from a Möbius band if one shrinks the boundary circle to a point? I don't really quite understand the problem. What does it ...
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### Determining matrices for an affine transformation

Determine the matrices A and b for the affine transformation t(x) = Ax + b, where A and b are $2 \times 2$ and $2 \times 1$ matrices, respectively, given that t maps each point of the line $y = 0$ ...
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### Reflection on a circle [duplicate]

Given two points "A" and "B" outside of a given circle of center "O". Where is the point X on the circle, such that AX + XB is the shortest possible? For the problem "Given two points "A" and "B" on ...
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### Find coordinates of intersections

I have a coordinate system, shown in black below, in which a point is situated along the $x$-axis. There is a different coordinate system rotated along the $z$-axis by $a$ degrees, shown in red in the ...
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### Find an angle of an isosceles triangle

$\triangle ABC$ is an isosceles triangle such that $AB=AC$ and $\angle BAC$=$20^\circ$. And a point D is on $\overline{AC}$ so that AD=BC, , How to find $\angle{DBC}$? I could not get how to use ...
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### Getting an angle

I have a unit circle, and two angles: $\alpha=\angle{JON}\in[0,\pi]$ and $\beta=\angle{IOM}\in[0,\frac{\pi}{2}]$. Using angles, we can get points $N$, $M$ as on the image. Then, dropping a ...
Is it $L \times S^{d-1}$ where $S$ is the hyper sphere of $d-1$ dimension and $L$ is length in usual 1 dimension?