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

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2
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1answer
407 views

Finding a cone from 3D data

I have data in form of many 3D coordinates (say $(x_1,y_1,z_1)...(x_n,y_n,z_n)$). EDIT - The points are known to shape something similar to the top of a lemon. Assuming we know which point is the ...
3
votes
1answer
106 views

Why does $b^2 = c^2 + a^2 - 2ca\cos(B)$ in trigonometry?

http://i.stack.imgur.com/l0Dw7.png I have a (what I believe to be an isosceles) triangle and the formula $b^2 = c^2 + a^2 - 2ca \cos(B)$ and I just have to "prove it". Now this really confused me as ...
2
votes
1answer
105 views

Find points on a triangle

In the diagram below, I have all the points working except c,d,e,f. I need to find these points. cp and ep are known points. I dynamically calculate a and b with cp -> ep vector's perpendicular ...
2
votes
1answer
92 views

Geometry problem

In triangle ABC, $\measuredangle$ ABC is bisected by $\overline{BE}$ and $\measuredangle$ ACB is bisected by $\overline{CD}$. $\overline{BE} = \overline{CD} = x $. Show that $\overline{BD} = ...
2
votes
1answer
656 views

Cones' radiuses ratio

Volume of the upper cone is equal to half volume of the whole cone. What is the ratio of the base radius of the whole cone to base radius of the upper cone? I got: $\sqrt{2h \over H}$ where $h$ is ...
8
votes
2answers
1k views

Construction of three tangent circles in a triangle

Given a triangle I want to construct three tangent circles inscribed in the triangle (every two sides of the triangle are tangent lines of one of the circles). For better understanding of the problem ...
0
votes
1answer
231 views

Stereographic projection of ellipsoid

I am really new in geometry and especially in working with stereographic projection, so excuse me, please, if my question is too easy. Given is the ellipsoid: $E = \left \{ (x,y,z)\in \mathbb{R}^{3}: ...
3
votes
2answers
684 views

Resolving vectors into components

The problem is to determine the components of $F_2$. Problem image Method 1 Method 2 My question is why do I receive different answers?
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0answers
38 views

Proving vertex coloring of lines

I'm attempting to prove the following proposition: For n>2 lines arranged in the plane, in general position, it is possible to color the vertices of the lines with 3 colors such that no 2 adjacent ...
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vote
0answers
186 views

How to generalise a result regarding intersections of cones and other convex sets?

To test for a particular property of positive LTI systems using feasibility problems I've come across the following claim which, intuitively, I believe can be generalised. I think I've (rather ...
10
votes
1answer
238 views

How much of a cylinder can you see from a single point of view?

I'm creating wraps for trash cans and recycling bins to label them appropriately. They'll wrap 360 degrees around the bins. I want the text on the bins to be readable from a single point-of-view (so ...
3
votes
0answers
430 views

Geometric interpretations of matrix inverses

Let $A$ be an invertible $n \times n$ matrix. Suppose we interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through ...
-2
votes
1answer
499 views

Area enclosed by a semi-circular fence

A contractor has installed a silt fence around an area that is semi-circular and level to prevent soil from the construction site entering nearby streams. The diameter of the semi-circle is 900 feet. ...
0
votes
1answer
211 views

Perimeter and area of a hexagon

The floor of a theater was built in the shape of a hexagon. The interior angles of the hexagon are the same and one side is 8 meters long. How long is the perimeter of the floor in feet? What is the ...
0
votes
1answer
601 views

How to find the matrix of reflection operator

I got a 3D space. In got a canonical equation of a plane: $ax + by + cz+ dt = 0$ How I can find the matrix of symmetric transformation transforming a point to its reflection?
2
votes
1answer
422 views

Does there exist a constructible (by unmarked straightedge and compass) angle that cannot be quintsected?

I know that for example an angle of 20 degrees cannot be quintsected because an angle of 4 degree cannot be constructed (I'm thinking in terms of (unmarked) straightedge and compass. But an angle of ...
4
votes
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
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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
605 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
178 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
232 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
552 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
451 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
254 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
790 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 ...