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

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1answer
52 views

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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1answer
372 views

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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1answer
390 views

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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1answer
47 views

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 ...
2
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2answers
207 views

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 ...
2
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2answers
382 views

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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1answer
176 views

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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2answers
66 views

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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2answers
107 views

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 ...
7
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1answer
193 views

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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1answer
188 views

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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1answer
1k views

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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1answer
28 views

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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2answers
147 views

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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0answers
158 views

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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3answers
2k views

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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2answers
311 views

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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0answers
57 views

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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1answer
50 views

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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2answers
37 views

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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2answers
240 views

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 ...
2
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1answer
430 views

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 ...
1
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1answer
92 views

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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1answer
53 views

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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2answers
250 views

“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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1answer
277 views

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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1answer
211 views

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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2answers
846 views

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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1answer
75 views

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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2answers
226 views

Find the absolute value of the difference between the area of these triangles.

Let $\triangle ABC$ and $\triangle ABC'$ be two non congruent triangles with side $AB=4$, $AC=AC'=2$$\sqrt{2}$ and $\displaystyle\angle B=30^\circ$. Find the absolute value of the difference between ...
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1answer
1k views

How many n square can fit into a square of side N

Suppose we have n small squares of equal sizes that has area w. Suppose we have a fix square S of area A such that for area A, one area w < area A. If square S's area A, length, and width are ...
21
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7answers
61k views

Finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates ...
4
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1answer
93 views

Find the $3$ angles of triangle $ABC$

We have a non obtuse triangle $ABC$. With $$\bf\dfrac{1}{2}\cos(2A)+\sqrt{2}\cos(B)+\sqrt{2}\cos(C)=\dfrac{3}{2}$$ Find the $3$ angles $A,B,C$.
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1answer
692 views

Find hypotenuse given acute angle bisectors

In a right triangle $ABC$ (right-angled at $B$), $D$ and $E$ are points of $\overline{AB}$ and $\overline{BC}$ respectively such that $\overline{CD}$ and $\overline{AE}$ are the angle bisectors of the ...
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2answers
631 views

Prove that a rotation and a translation never commute

How to prove that a rotation and a translation never commute, unless one of them is the identity map i
2
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1answer
121 views

Proving that $\frac{6\tan\phi}{\tan^2\phi-9}=\tan A$

In a triange $ABC$, points $D$ and $E$ are taken on side $BC$ such that $BD=DE=EC$. If $\angle ADE=\angle AED=\phi$, how can we prove that $$\frac{6\tan\phi}{\tan^2\phi-9}=\tan A$$
0
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1answer
41 views

Transformations and Isometries

Suppose an isometry, $\alpha$, of the Euclidean plane $\Bbb R^2$ fixes different points $A$ and $B$, and does not fix a third point $C$. Describe $α$, with justication. Since $\alpha$ fixes $A$ and $...
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2answers
1k views

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 ...
8
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1answer
185 views

Is this curve the circumference of a circle?

Let $\Gamma$ be a single closed curve with no self-intersections on a plane which satisfies the following condition : Condition : For any distinct four points $P, Q, R, S$ on $\Gamma$, if the line $...
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3answers
286 views

Point P(x,y) inside a square, differences of distances to corners known

My problem is the following: A square wooden plate is hit with a bullet. There are sensors at the four corners. Length of the sides are known. Speed of sound in wood is known. The moments where the ...
9
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2answers
294 views

Partition of plane into parabolas

The plane is partitioned into parabolas (each point belongs to exactly one parabola). Does it follow that their axes have the same direction?
0
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1answer
44 views

Problem form Geometry

In the square $ABCD$ with side $AB = 2$ a straight line $MN$ is drawn perpendicular to $AC$. Denoting the distance from the vertex $A$ to the line $MN$ as $x$, express through $x$ the area $S$ of the ...
0
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2answers
102 views

Why is it said that every point in hyperbolic space is a saddle point?

I have read that since hyperbolic space has a constant negative curvature (a concept that I think I understand), every point is a saddle point. I am trying to understand what that means. Can we say ...
2
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2answers
66 views

Hyperbox, partitioned into hyperboxes with $k$ integer side lengths each, must have at least $k$ integer side lengths?

In Intuitive/direct proof that a rectangle partitioned into rectangles each with at least one integer side must itself have an integer side I asked a question about intuitively proving that a ...
1
vote
1answer
329 views

Perpendicular unit vectors

I have a known unit vector $p (a,b,c)$. First I want a unit vector $q$ which is perpendicular to $p$ and passing through a known point $V(X_0,Y_0,Z_0)$. Then a another unit vector $r$ which ...
0
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1answer
62 views

Solutions of triangles, homework

I have 2 questions in which I have doubt :- Q1. prove that:- a cosBcosC + b cosAcosC + c cosBcosA = ar(ABC)/R A1. I have used cosine rule and have put the values of all cosines here and after adding ...
2
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1answer
58 views

Solutions of triangles

$AA_1$, $BB_1$, $CC_1$ are the medians of triangle $ABC$ whose centroid is $G$. If points $A, C_1, G, B_1$ are concylic then prove that $2a^2= b^2 + c^2$. Thanks My try:- $ar(GBC)=1/3ar(ABC)$ $\...
2
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1answer
112 views

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 ...
0
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1answer
933 views

What is the volume of a hyper cylinder in d - dimension?

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?
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1answer
928 views

How To Scale A Rectangle

I know this is probably pretty easy for people on this site but I am no mathematician so I thought I would ask here. I am looking for a formula for scaling a rectangle from an arbitrary point as in ...