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

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4
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2answers
152 views

Vectors transformation

Give a necessary and sufficient condition ("if and only if") for when three vectors $a, b, c, \in \mathbb{R^2}$ can be transformed to unit length vectors by a single affine transformation. This is ...
2
votes
3answers
3k views

How to find the rectangle center point (origin position)?

If I have a rectangle and I try to rotate it, how to find the rectangle center point (origin position)? I know these $3$ corners of the rectangle: left top: $(0,5)$ left bottom: $(0,0)$ right ...
3
votes
1answer
102 views

From (algebraic) topology to geometry

I am thinking about a "correct" didactic way of linking topology (algebraic topology) to geometry. Usually, we are taught introducing geometry first, then topology, almost as an abstraction of ...
0
votes
2answers
502 views

How to recognize ellipse/ellipsoid from random points? UN-weighted average?

Suppose we are getting random points in 2D (or 3D) which tend to be on ellipse (or ellipsoid). We can't guarantee points are uniformly distributed over ellipse (ellipsoid surface). The task is to ...
3
votes
2answers
94 views

Help with old Mu Alpha Theta Question. (Geometry)

I'm working on a question from an old Mu Alpha Theta exam. I'm stuck and would appreciate any hints. (Please no answers! I'm sure I'm just missing something stupid. Also, this is for fun. Not HW.) A ...
4
votes
1answer
162 views

Union of two self-intersecting planes is not a surface

I need to show that the union of xy-plane and xz-plane, i.e. the set $S:=\lbrace (x,y,z)\in\mathbb{R}^3 : z=0 \mbox{ or } y=0\rbrace$, is not a surface. Here is my claim, $\textbf{Claim :}$ Suppose ...
0
votes
0answers
143 views

geometry-internal angle sum of triangle

Problem: Let A, B, C be three non-collinear points. Let D, E, F be points on the respective interiors of segments BC, AC and AB. Let θ, φ and ψ be the measures of the respective angles ∠BFC, ∠CDA and ...
3
votes
2answers
95 views

Lie bracket on vector fields

I started reading about Lie derivative on vector fields and its properties, found an exercise, but i have doubts about my solution. Given are two vector fields $X_{1}=\frac{\partial }{\partial x_{1}} ...
3
votes
0answers
48 views

Prove the shortest path is closed (as in no intersections) [duplicate]

Picture of problem: I'm so stuck! Thank you!
2
votes
1answer
120 views

How to formally prove that this proof is (not) correct?

In lemma 2 in this article's section 5 there is a proof below and at the end it states that equation $\|(1-\alpha)p_j+\alpha p_i-p_k\|=\|p_i-p_k\|$ has only one solution $\alpha=1$. Examples can be ...
2
votes
2answers
194 views

Prove that $H$ is the orthocenter of $\Delta ABC$.

In acute $\Delta ABC$, $D$ and $E$ are points on $AB$ and $AC$ respectively such that $B, C, D, E$ are concyclic, $BE$ and $CD$ intersect at $H$, and $H$ is on the altitude of $\Delta ABC$ passing ...
1
vote
2answers
600 views

Which of the following are possible measures of the exterior angles of a polygon and how many sides does the polygon have: 90, 80, 75, 30, 46, 36, 2

Please help with these geometry question that have to due with polygons: 1.Which of the following are possible measures of the exterior angles of a polygon and how many sides does the polygon have: ...
1
vote
3answers
282 views
8
votes
2answers
177 views

Why does the term ${\frac{1}{n-1}} {2n-4\choose n-2}$ counts the number of possible triangulations in a polygon?

In the given picture bellow, it counts the number of different triangloations in a polygon, how do the get to this expression, why is it: $$ {2n-4\choose n-2} $$ and why do we multiply it by ...
1
vote
1answer
465 views

Interpretation of “three coplanar lines intersect”

How should "three coplanar lines intersect" be drawn? Will they form something that looks like an asterisk? Or three lines that somehow form a triangle? Or either interpretation is acceptable?
1
vote
2answers
321 views

How to workout the determinant of the matrix $D_n(\alpha, \beta, \gamma)$.

I am going through an example in my lecture notes. This is it: Let's introduce the matrix $D_n(\alpha, \beta, \gamma)$, which looks like this: $$\pmatrix{\beta & \gamma & 0 & 0 ...
3
votes
1answer
165 views

Intersection of two $n$-dimensional quadratic inequalities?

I have two quadratic inequalities of the form $$ a_1x^TAx + b_1^Tx + c_1 \le 0\\ a_2x^TAx + b_2^Tx + c_2 \le 0 $$ where $A\in\mathbb{R}^{n\times n}$ is positive semidefinite, $x\in\mathbb{R}^n$, ...
4
votes
3answers
230 views

Intuition about Hyperplane

I'm having a hard time understanding hyperplane ideas. So, can anyone explain to me how to easily understand what a Hyperplane is ?
4
votes
2answers
163 views

Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png An equilateral triangle can be divided into 3 congruent shapes. Questions: ...
3
votes
0answers
545 views

How to prove that a given curve is actually a straight line

Given a curve, I have to prove or disprove that it is a straight line. How do I do this? I tried by finding and comparing slopes but I can see that this will not be a very computationally efficient ...
4
votes
3answers
2k views

Why do we have 360 degrees in a circle and why we need radians? [duplicate]

I have two related questions: 1- Why do we have 360 degrees in a circle? 2- I have seen in most of the mathematical concepts, angle is expressed in radians not in degrees. Why was radian ...
2
votes
1answer
264 views

Contest Math Geometry

I'm currently prepping for some high school math competitions soon, and I was wondering if anyone knows any resources that are out there with an abundance of contest-math-related geometry problems. ...
0
votes
2answers
3k views

Difference between a Gradient and Tangent

I am unable to understand the fundamental difference between a Gradient vector and a Tangent vector. I need to understand the geometrical difference between the both. By Gradient I mean a vector ...
2
votes
3answers
156 views

Beer crate square shape versus rectangle shape

I am being asked this question by a consultant to test my logic and I was wondering whether one of you had an interesting mathematical perspective on it: Why are beer crates usually shaped in ...
0
votes
1answer
124 views

Given an angle at a point A and given another segment BC, construct a point D so that the angle DBC equals the given angle at A.

I'm not sure if I understand this question right. Q. Given an angle at a point A and given another segment BC, construct a point D so that the angle DBC equals the given angle at A. Figure - ...
0
votes
1answer
118 views

Why the solution of this brainteaser a linear function?

I have been asked the following brainteaser: Imagine that you have a grid of dots in 2D placed at regular interval, you draw a convex shape by joining dots. Let us call M the number of dots ...
2
votes
1answer
449 views

What does the definition of curvature mean?

First question, am I right in saying that curvature measure how quickly the direction of a cruve changes? Also, we have been given the "definition": $$T'(t) = \kappa(t) | \gamma'(t) | U(t)$$ where ...
3
votes
1answer
146 views

Edges of the convex hull of a finite point set

I'm looking for a formal proof of a statement that seems obvious (and yet may be wrong) to make sure one of my algorithms is correct. Given a set S of N points in $\mathbb{R}^3$, suppose we have a ...
2
votes
1answer
52 views

dimensions of isometries

I was studying something about isomoteries in space and read that the dimension of translation is 3, rotation is 5,screw is 6, reflection is 3 and etc. What does this mean? How to calculate this?
0
votes
1answer
30 views

Intersection of a 2-Dimensional body and a Line given west-most point and south-most point

I have a 2-Dimensional Closed Convex Compact Body $\mathbb{S}$(a set, for eg, a circular disc)). Assume, it has a non-zero intersection with the all-negative quadrant. Consider the following 2-D point ...
0
votes
1answer
253 views

How to extend rational parametrization of the circle to three dimensions?

I recently became aware of the rational parametrization of the circle in two dimensions: $$\left(\frac{1-m^2}{1+m^2}, \frac{2m}{1+m^2}\right)$$ for a unit circle centered on the origin. I'm ...
0
votes
0answers
89 views

A problem in differential geometry

How can we get $ \large w= \frac{1}{k(a)} + \frac{1}{k(a+pi)}$ by using those $4$ facts I got? Let $y (a)$ be a simple closed planar curve with curvature $k > 0$ parametrized by $a$, where $a$ is ...
1
vote
2answers
112 views

How do I calculate a point on each of three circles that have specific distance to each other?

I am trying to write code for a computer simulator. I need to simulate a complex mechanism where each link has a known length and the ends of the links are connected to a triangle. I would like help ...
1
vote
4answers
788 views

How can I prove that exterior angles of a pentagon add up to four right angles?

How can I prove that exterior angles of a pentagon add up to four right angles I have thought about dividing the pentagon into 3 triangles, then maybe using the exterior angle sum equal to two ...
6
votes
2answers
488 views

How to Construct orthogonal circles?

Let $C_{1}$ be a circle of unit radius. Let A and B be two points inside $C_{1}$. Now I want to construct another circle $C_{2}$ such that A and B lie on $C_{2}$ and $C_{2}$ is orthogonal to $C_{1}$ ...
1
vote
1answer
73 views

Minimising Length and Energy for Finsler Manifold

Is it true that a minimiser of Finsler energy is automatically parameterised by arc length? As in the Riemannian case. Is there a reference for this fact?
1
vote
3answers
71 views

Composition of Reflections

If R1 and R2 are two planar reflections corresponding to the lines d1 and d2. What is the necessary and sufficient condition on the lines d1 and d2 such that R1oR2=R2oR1 ?
0
votes
1answer
55 views

$\frac{AM}{MB}=\frac{AN}{NC}$ iff $\frac{MQ}{QN}=1$

I have a triangle ABC and point $M \in AB$ and $N \in AC$. Defining the points $O= BN\cap CM$ and $Q=AO\cap MN$. I want to show that $\Large\frac {AM}{MB}=\frac {AN}{NC}$ iff $\Large\frac{MQ}{QN}=1$. ...
5
votes
3answers
2k views

Car racing: How to calculate the radius of the racing line through a turn of varying length

I am in the process of designing a board game involving car chases, and I am stumped by the following problem: A car will have a maximum speed through a constant radius speed turn, giving a maximum ...
26
votes
2answers
582 views

New twist on a Putnam problem

A recent Putnam problem: Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f$ is identically zero? The ...
1
vote
1answer
136 views

For any simple polygon in 2d and for any point in 2d there is always an edge of the polygon that is entirely visible from the point.

prove For any simple polygon in 2d and for any point in 2d there is always an edge of the polygon that is entirely visible from the point.
1
vote
0answers
229 views

Proof of Pappus' Theorem using Affine geometry

I want to prove Pappus' Theorem using affine combinations. The theorem states that given two lines $l_1$ and $l_2$ in the plane and six points $A_i,B_i,C_i \in l_i (i=1,2)$ show that the points $A_3= ...
8
votes
6answers
6k views

How do I find the area of this region?

A square with edge length 2 cm has semicircles drawn on each side. Find the total area of the shaded region. Here is an image of the diagram shown : Please show your work in pictures, numbers, ...
1
vote
3answers
449 views

Euclidean Isometry

This is part of my homework problems, it is not for assignment nothing to hand in first of all, i just dont get how to go about this proof. If i can see it it would be good. Question is Show that a ...
1
vote
1answer
137 views

How to map points in a unit square to a regular polygon?

I have a set of points in a unit square $x = [-1,1]$ and $y = [-1,1]$ and I want to remap them to their equivalent points in a regular polygon ($n \geq 3$; Triangle and so on). I've found a really ...
2
votes
1answer
202 views

Unitary matrices and rotation of n-dimensional torus

I am reading a paper on quantum automata, and within the proof of theorem $6$, there is the following statement (I have added a definition for completeness of the statement). Let $U$ be a unitary ...
0
votes
1answer
44 views

Conditions for the existence of a conical combination of some given vectors such that it lies in a cone?

Let $v_1,v_2,\dots,v_n,u_1,u_2,\dots,u_r\in\mathbb{R}^n$. Can one find analytical conditions (not write the problem up as a convex optomisation problem and argue it can be solved this way) under which ...
6
votes
2answers
224 views

Embedded surface in $\mathbb{R}^3$

Let $U \subseteq \mathbb{R}^2$ be an open set and let $\sigma : U \rightarrow \mathbb{R}^3$ be a parametrization of an oriented surface $S$ embedded in $\mathbb{R}^3$ whose unit normal in $\sigma ...
3
votes
2answers
956 views

Using equations to draw out complex objects

How do people come up with equations of curves to draw out complex objects? Some popular examples would include: batman curve & PSY curve. This stackexchange link explains the rationale for the ...
1
vote
1answer
334 views

Finding the volume of a cube based off the mass and density of a sphere contained inside it.

This is an extension to my last question, but since people didn't like that the problem involved atoms, I will change them to small spheres of cheese. The mass of a sphere of cheese is $1.37\cdot ...