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

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16 views

What are some techniques of specifying a molecules structure using the least amount of information?

For instance say I have a water molecule I can describe it's structure by two bond lengths and a bond angle. Are there any neat math tricks or representations of objects that I could use to describe ...
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1answer
29 views

Curvature of plane curves

What is the neatest way to derive the formula for the curvature $\kappa =\frac{||y'x''-y''x'||}{(x'^2+y'^2)^{\frac{3}{2}}} $?
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12 views

Signed unit normal

I'm trying to study for one of my exams and the past papers have no solutions. I had to define the signed unit normal and the signed curvature. The signed curvature, $\kappa_s$ being such that ...
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1answer
52 views

Probability of triangle to be acute?

Suppose that someone randomly picks $3$ points $A, B$ and $C$ on a fixed circle. What is the probability of triangle $ABC$ to be acute?
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0answers
12 views

How do I find all lines of $\mathbb{P^{3}}$ meeting $L, M$ and $N$?

Let $L, M$ and $N$ be $3$ lines of $\mathbb{P^{3}_{R}}$ such that $$L \cap N = \emptyset, \quad M \cap N = \emptyset \quad \text{and} \quad L \cap M = \{ G \} \ \ \text{(one point)}.$$ I'm asked to ...
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1answer
13 views

Differential length of a logarithmic spiral

I am working on a problem that asks to find the magnetic field at the origin of a logarithmic spiral $r = e^\theta$ from $\theta = 0$ to $\theta = 2\pi$, where the angle $\theta$ is measured ...
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2answers
37 views

finding angle value inside this triangle

I need a method to calculate the angle $X$ in the image below, I know the its value ($30^\circ$) but how ?!! thank you.
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0answers
26 views

Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$

Let $P_1 P_2 \dots P_n$ be a convex polygon in the plane. Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$ Show ...
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1answer
21 views

Why do lines in the poincare model meet the infinite edge at right angles?

I know the lines are generated by projecting geodesics on a hyperboloid to a plane and the boundary of the disk comes from the asymptotic cone around the hyperboloid, but I just don't see why the ...
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0answers
13 views

What are the images of a point in a multi-layer mirror systems?

We all know that there are infinite images of a point which is located between two parallel mirrors. Also, the locations of the images can be easily obtained. Generally, how to locate the images of a ...
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1answer
20 views

Let the diameter of a subset S of the plane be defined as the maximum of the distance between arbitrary of points of S

Let the diameter of a subset S of the plane be defined as the maximum of the distance between arbitrary of points of S. Let$$S=\{(x,y):(y-x)\le0,(x+y)\ge0,x^2+y^2\le2\}$$ Then the diameter of S is ...
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1answer
22 views

Given sides and a bisection, find angles in a triangle

Consider a triangle $ABC$ where the angle $A$ is $60^{\circ}$. Draw its bisection intersecting $BC$ at $D$. Let $AB = x$, $BD = y$ and $AC=x+y$, $\angle ABC = \alpha$ and $\angle ACB = \beta$. Find ...
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0answers
10 views

Boundedness condition of Minkowski's Theorem

Statement: "Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if ...
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1answer
31 views

“Reverse engineering” of a geometric illustration

The following enigmatic illustration can be found here, unfortunately without any accompanied comment or short description: Can you deduce its meaning? What was the way it was constructed?
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2answers
40 views

Complicated Planar Geometry [duplicate]

A regular polygon $\mathcal{P}$ is inscribed in a circle $\Gamma$. Let $A$, $B$, and $C$ be three consecutive vertices of the polygon $\mathcal{P}$, and let $M$ be a point on the arc $AC$ of $\Gamma$ ...
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0answers
13 views

Find out rotation and angle given a square

Consider I have a camera and a square, the projection of the square on the camera would give me a "distorted" square. Can I possibly find out the angle of the camera relative to the square and the ...
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1answer
13 views

what is the difference between an elliptical and circular paraboloid? (3D)

My textbook uses the terms interchangably, and they look the same in graphs, so I was wondering if there a difference between the two? Thanks!
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2answers
34 views

How to reverse this mathematical formula

I am using the following function to compute a "Y" value from a Latitude ...
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1answer
29 views

Relative distance to rotated object

An object (with center $O_{2}$) has been rotated by an angle $\alpha$. There are two images of the object taken by a camera (centered at $O_{1}$), and two points $x_{1}$ and $x_{2}$ that are actually ...
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0answers
25 views

How can one show that the vector $AK=\frac{1}{3}AI$?

$ABC$ is a triangle then $I$ is the medium of $[CB]$ and $J$ the medium of $[AI]$ and $K$ the intersection of $(BJ)$ and $(AI)$. Then how can one show that $AK=\frac{1}{3}AC$ Do we have to add ...
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0answers
14 views

Cover the entire sphere with a moving cone

Assume you have a cone with half-angle theta in a 3D cartesian space, with the vertex of the cone in the origin, and that you want to rotate the cone along a curve so that it covers the entire sphere. ...
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0answers
10 views

Maximize the intersection over union of oriented rectangles

I have an oriented rectangle in the form region=(x1,y1, ..., x4, y4) I want to know which is the axis-aligned rectangle with the same center that maximize the intersection over union of the areas of ...
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0answers
22 views

How to calculate the solid angle subtended by the torus?

The diagram above shows a torus, with circular cross section, having inner radius $r=19 cm\space$ & outer radius $R=33 cm\space$. How to evaluate the solid angle subtended by the torus at a ...
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1answer
33 views

Volume of paraboloid

A paraboloid is formed by revolving a parabola, $y=kx^2$, about its axis of symmetry. The paraboloid is bounded by a plane cutting the axis of symmetry perpendicularly at the point (0,20). The ...
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2answers
33 views

Finding the max area for a rectangle inside of a circle

I know how to find the max area, inside of a normal circle. This circle however is the inside of an aircraft and the floor must be there. How can this be done? Here's a sketch of the inside of the ...
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0answers
19 views

Conditions for point lying inside triangle formed by three complex numbers.

The question states $z_1,z_2,z_3$ are three non-collinear complex numbers such that $$z=\frac{lz_1+mz_2+nz_3}{l+m+n}$$ lies inside the triangle formed by $z_1,z_2,z_3$. If $l,m,n$ are the ...
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1answer
37 views

Area of triangle, variable sides. High school level. [on hold]

The triangle $PQM$ is inside the triangle $AOB$. a) Show that the area of the triangle $PQM$ can be expressed using the function $T$ given by $T(x) = -\frac{1}{2}x^2 + 3x \quad (0\leq x \leq ...
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0answers
23 views

Finding position of point (in 3D space ) which are at x,y offset from corner of a rectangle in 3D world

So I am writing a 3D graphic software. And I am stuck at mathematical problem. Mathematically speaking: There's a rectangle (plane) of finite size in 3D space. It can be of any orientation and ...
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0answers
37 views

Find maximum width of a rectangle contained with another (diagonally)

It appears that the question of figuring out if a rectangle will fit inside of another, specifically at a diagonal, has been asked before. As such, utilizing the equation linked here, I was able to ...
2
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1answer
12 views

Reflection combined with a glide reflection

Suppose that $A, B, C,$ and $D$ are the vertices $(0, 0), (2, 0), (2, 2),$ and $(0, 2)$ of a square. The transformation $ρ_{AC} ◦ γ_{DA}$ can be decomposed as the combination of reflections across ...
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1answer
47 views

How to prove that I can trisect any angle? [on hold]

Given that I can trisect any angle including a 60 degree angle with a compass and straight edge how can I go about providing proof. For instance I can construct a nonagon inscribed in a circle
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1answer
11 views

Given a known line segment and two known horizontal lines, how do I find the line subsegment between the two parallels?

I am trying to clip a line segment between two parallel horizontal lines. I know the location of the two vertices on the larger segment, but need to know the points at which it intersects the ...
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0answers
20 views

Has anyone proven how to trisect an angle with a compass and ruler? [duplicate]

I was curious if anyone has proven how to trisect any angle with compass and straight edge. I keep seeing that it is not possible.
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1answer
59 views

How can we prove that a three legged chair will never be wobbly?

I am taking the geometry approach. We know from intuition that more than three legs on a chair will make it unstable if any of the legs have a different length than the others. So by "wobble" I mean ...
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0answers
13 views

How can one show that (DI) , (JB) and (AC) concurrents on G?

ABCD is a square , We add outside it two equilateral triangles ADJ and ABI How can I show that (DI) and (BJ) and (AC) occur in the same point ? Here can we demontrate that saying that IGB and JGD are ...
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0answers
8 views

How to evaluate solid angle subtended by composite plane figure?

How to calculate the solid angle subtended by the shaded portion of the circular plane at a point $P(0,0,h)$. This composite plane figure is obtained by removing the right $\Delta ABC$ $(AB=BC)$ ...
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1answer
30 views

Distribution of an angle between a random and fixed unit-length $n$-vectors

Suppose I have a random unit-length $n$-element vector $\mathbf{x}$ that is uniformly distributed on an $n$-dimensional sphere, and let vector $\mathbf{a}$ be some other unit-length $n$-element vector ...
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0answers
14 views

Cone with 3 mutually perpendicular generators [on hold]

what is general equation of cone touching coordinate planes and please show it diagrammatically
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1answer
11 views

How to evaluate volume of oblique frustum of right circular cone with elliptical section?

Let there be an oblique frustum, with an elliptical section, of a right circular cone with apex point O & cone angle $2\alpha=60^{o}$. It is obtained by cutting the cone by a plane at a normal ...
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4answers
42 views

Find the type of triangle from equation.

In triangle $ABC$, the angle($BAC$) is a root of the equation $$\sqrt{3}\cos x + \sin x = \frac{1}{2}.$$ Then the triangle $ABC$ is a) obtuse angled b) right angled c) acute angled but not ...
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1answer
54 views

Volume of a hole through cylinder (from the side)

I need to calculate the volume of a circular hole in a cylinder and I've come across a problem. The problem is finding the "cap-volume", which is needed to complete the volume of the hole. I created a ...
3
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1answer
92 views

Rectangle randomly thrown on chessboard

) I'm an electrical engineer and having a tough problem with... math :) geometry and probability... Here's the problem : We have an infinite chessboard. Each square of the chessboard is of known ...
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0answers
18 views

Find points which form angle X from a triangle formed by 2 other points

Here's a diagram Given $\angle{ACB}$, $\angle {CAD}$, position C, and position E, and given that $\angle{ADB}$ equals $\angle{ACB}$, find position D.
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4answers
62 views

2010 unit circles

$C$ is a unit circle with radius $r$. $C_1,C_2,\ldots, C_{2010}$ are unit circles along the circumference of $C$ touching $C$ externally. Also the pairs $C_1C_2;C_2C_3,\ldots;C_{2010}C_1$ touch. Then ...
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0answers
15 views

Packing spheres into a rectangular prism

So, this was a problem in the new standardized high school tests California has started using(CAASP). These new tests are completely done on the computer, and feature what they call Computer Adaptive ...
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2answers
32 views

Geometry, a triangle with all sides and no angles [on hold]

So, there is an undefined triangle which has the side lengths of $ [3,4,5] $, can I have any hints about how to find the acute angles?
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1answer
20 views

A rectangle has a perimeter of 8s x 1/4 units. Which of the following could not be the dimensions of the rectangle?

A rectangle has a perimeter of $8s \times \frac{1}{4}$ units. Which of the following could not be the dimensions of the rectangle? A. $3s$ units long and $s \times 1/8$ units wide B. ...
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1answer
30 views

A few questions regarding Pappus' Theorem

I am trying to understand a proof of Pappus' Theorem. This is taken from the book Geometry and Topology by Miles Reid and Balázs Szendröi. Definition 1 Let $PQ$ be the line through two points $P=(x_0 ...
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1answer
25 views

What is the significance of $SL(2, \mathbb{R} / SL(2, \mathbb{Z}))$ in studying lattices in geometry of numbers?

I was listening to a talk about lattices and the geometry of numbers and at one point they jumped from discussing a 2d lattice into discussing $SL(2, \mathbb{R})\ /\ SL(2, \mathbb{Z}))$ and it was not ...
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2answers
34 views

Why are Euclid axioms of geometry considered 'not sound'?

The five postulates (axioms) are: "To draw a straight line from any point to any point." "To produce [extend] a finite straight line continuously in a straight line." "To describe a circle with ...