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

learn more… | top users | synonyms

0
votes
1answer
1k views

How can I find equivalent Euler angles?

I have a rotation over time represented as a series of Euler angles (heading, pitch and bank). I'd like to represent the individual rotation curves as continuously as possible. An anyone help me ...
-1
votes
2answers
501 views

In △ ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees?

In △ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees? I know already that angle A and angle D are congruent because ...
14
votes
3answers
431 views

Is there a geometric realization of Quaternion group?

Is there a geometric realization of the Quaternion group: $$Q = \langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle$$ I dont think it can be realized as the symmetries/rotations of a 3D shape so could ...
12
votes
3answers
836 views

Equation of Cone vs Elliptic Paraboloid

I can't understand why $$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2} \tag{*}$$ corresponds to an elliptic paraboloid and $$\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2} \tag{**}$$ to a cone, ...
10
votes
5answers
2k views

Calculating the area of an irregular polygon

Given the length of the sides of an irregular polygon (no coordinates provided) how do you compute the area of the maximum area of the polygon? Thanks in advance
8
votes
1answer
264 views

How to parameterize an orange peel

I'm trying to parametrize the space curve determined by the boundary of a standard orange peel: for example, the one on this photo: For example, the ideal curve would be inside the unit cube; have ...
7
votes
1answer
164 views

Rotation of $\mathbb{R}^3$ by using quaternion

Express the rotation of $\mathbb{R}^3$ by $\frac{\pi}{3}$ about the $x=y=z$ axis by using quaternions and identifying $\mathbb{R}^3$ with $(i,j,k)$-space. Thoughts: From my point of view, every ...
7
votes
3answers
6k views

Finding the intersecting points on two circles

Given 2 circles on a plane, how do you calculate the intersecting points? In this example I can do the calculation using the equilateral triangles that are described by the intersection and centres ...
6
votes
1answer
131 views

Trajectories on the $k$-dimensional torus

Let $r_1,\dots,r_k$ be irrational and linearly independent over $\mathbb Q$. My intuition clearly tells me that the set $$\{(nr_1,\dots,nr_k)+\mathbb Z^k:n\in\mathbb N\}$$ is dense in $\mathbb ...
5
votes
2answers
620 views

Aren't asteroids contradicting Euler's rotation theorem?

I am totally confused about Euler's rotation theorem. Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that: In geometry, ...
5
votes
1answer
196 views

For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero

Consider an $n$-sided convex polygon $P$ that contains the origin in the complex plane. Let the $j$-th vertex be denoted $z_j = r_j e^{i\theta_j}$ ($0 \leq \theta_j < 2 \pi$) for $j= 1 \dots n$. ...
5
votes
2answers
562 views

What is the name for a shape that is like a capsule, but with two different radii?

I'm looking for the name of a shape that is like a capsule, but where each circle can have different radii. The shape could be described using two circles (two centers and two radii). Something like ...
4
votes
1answer
234 views

Shortest path on unit sphere under $\|\cdot\|_\infty$

Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y$ define $d(x,y)$ to be the arc length of shortest paths on $Y$ joining $x,y$. (It ...
4
votes
3answers
349 views

Tetrahedral torus

Is it possible to form a closed loop by joining regular (platonic) tetrahedrons together side-to-side, with each tetrahedron having two neighbours? It should be a loop with a hole in, as can be done ...
4
votes
3answers
799 views

Determinants and volume of parallelotopes

The absolute value of a 2 by 2 matrix determinant is the area of a corresponding parallelogram with the 2 row vectors as sides. The absolute value of a 3 by 3 matrix determinant is the volume of a ...
4
votes
3answers
1k views

Calculating an Angle from $2$ points in space

Given two points, around an origin $(0,0)$, in $2$D space how would you calculate an angle from $p_1$ to $p_2$. How would this change in $3$D space?
3
votes
1answer
72 views

If $\gamma$ is spherical, the equation $\frac{\tau}{\kappa}=\frac{d}{ds}(\frac{\dot{\kappa}}{\tau \kappa^2})$ holds.

Question: Please help me doing this question. In fact, there is its solution as I posted below. But I don't understand the answer. Please explain this more clearly. Thank you for helping. ...
3
votes
1answer
737 views

Calculate Camera Pitch & Yaw To Face Point

How do you calculate pitch & yaw for a camera so that it faces a certain 3D point? Variables Camera X, Y, Z Point X, Y, Z Current Half Solution Currently I know how to calculate the pitch, ...
3
votes
1answer
435 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 ...
3
votes
1answer
176 views

Expectation of $QQ^T$ where $Q^TQ=I$

It's exercise 1.1 on p.2 of this book. The goal is to is to show that, for some random matrix $Q \in \mathbb{R}^{n\times k}$ where $k<n$ and the columns of $Q$ are orthogonal (i.e. $Q^T Q = I$; ...
3
votes
3answers
4k views

Convert coordinates from Cartesian system to non-orthogonal axes

I have a 2D coordinate system defined by two non-perpendicular axes. I wish to convert from a standard Cartesian (rectangular) coordinate system into mine. Any tips on how to go about it?
3
votes
1answer
1k views

How to prove the midpoint of a chord is also the midpoint of the line segment defined by the points of intersection of other two chords with it?

Bernhard Elsner, alias MathOMan, posted this exercise in plane Geometry, Theorem about a circle, three chords and a midpoint on January 29th, 2010. "Let $\mathcal{C}$ be a circle, $A,B$ two distinct ...
2
votes
1answer
24 views

How do I calculate the number of times angle X should be added to obtain angle Y?

Suppose I have a angle X=100 and angle Y=60. when we add X 15 times it gives ...
2
votes
2answers
113 views

When is $Ar(APD)=Ar(ABCD)$?

This question arose while I was answering this question, (we need to show $Ar(\Delta APD)=Ar(ABCD)$). First the original question: $ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets ...
2
votes
1answer
61 views

An isosceles trapezoid $ABCD$ function of $AB$

Ok so i have an isosceles trapezoid $ABCD(AD=BC)$ with bigger base $AB$ and let $O$ be the point of the crossed diagonals and let $OH$ be perpendicular to $AB$. I want to find a function for $AB$ with ...
2
votes
3answers
59 views

How to find a center/axis of rotation?

I have a 3d model (M1) consisted of several points. I know all their coordinates. I also have another model (M2). M2 and M1 are the same, but M2 is a model after rigid transformation. I don't know the ...
2
votes
3answers
106 views

finding a topological group with specific conditions

I have a question, it sounds difficult. The question is the following: Let $X$ be a topological group such that the binary operation defined on it is $*$. For any two points $a$ and $b$ in $X$ ...
2
votes
2answers
62 views

Help me conclude a statement about two points given the following…

Two points on the graph of $y=kx^p$ are labeled $A$ and $C$. Point $A$ has coordinates $(a,b)$, where $0<a<1$ and point $C$ has coordinates $(c,d)$, where $1<c$. If we are told that that the ...
2
votes
0answers
52 views

Derive the centroid of an area from a limiting procedure

I wondered if the centroid of an area could be derived as a limit from the centroid of the solid of revolution built by the same area revolving around an axis of revolution. e.g.: the volume ...
2
votes
2answers
75 views

Perpendiculars on a line segment

Two points A and B are given. Find the set of feet of the perpendiculars dropped from the point A onto all possivle straight lines passing through the point B.
2
votes
2answers
388 views

Offseting a Bezier curve

I searched this site and I read that in general it is not possible to calculate offset of a Bezier curve. But is it possible to calculate the offset in some special cases? Obviously, if the Bezier ...
2
votes
1answer
194 views

Area of a portion of an arbitrarily-placed circle?

I have a circle that's off-center, but I want to find out the area of the part of the circle in the positive x and y region. Not sure how to do this because of the multiple variables involved.
2
votes
1answer
2k views

calculating the Fermat point of a triangle

Is there any algorithm by which one can calculate the fermat's point for a set of 3 points in a triangle? a fermat's point is such a point that the sum of distances of the vertices of the triangle to ...
2
votes
3answers
108 views

Formal proof for detection of intersections for constrained segments

They told me it was off-topic at stackoverflow. So I am trying my luck here. Yes, it's a homework, but I'm looking for some guidance (or related literature) instead of complete solutions. Please see ...
2
votes
3answers
238 views

How to obtain Line Equation of the form $ax + by + c = 0$

I'm trying to check if a line hits a rectangle, and for that, I found this nice solution: Line triangle intersection The problem is that, having forgot almost all I ever knew about math, I don't ...
1
vote
2answers
65 views

If $ABCD$ is a cyclic quadrilateral, then $AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD)$

If $ABCD$ is a cyclic quadrilateral, then $$ AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD) $$ I tried using many approaches, but I could not find a proper solution. Can anyone please ...
1
vote
0answers
74 views

Possible combinations for a cube

When drawing one of two possible diagonals on each side of a cube, how many unique patterns are possible with regard to all sides of the cube and all possible diagonal orientations. I am stuck on ...
1
vote
1answer
86 views

A weak version of Markov-Kakutani fixed point theorem

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a commuting family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element ...
1
vote
4answers
1k views

Angle between two 3D vectors is not what I expected.

Using the definition of the dot product, you can find the angle between two vectors. I am experiencing an unexpected result, so my question is where did I go wrong. I have two unit vectors in 3 ...
1
vote
1answer
441 views

How to find the intersection of the area of multiple triangles

I have a couple of questions regarding finding the intersection of triangles. I have a system of 16 projectors that all have slightly different color gamuts. The color gamuts are represented by a ...
1
vote
0answers
76 views

Count Exclusive Partitionings of Points in Circle, Closing Double Recurrence?

I am studying a problem that I have worked out is equivalent to the following: Problem Description Given N distinct points on the border of a circle, there are $B_N$ ways to partition them - where ...
1
vote
3answers
1k views

Rotating two vectors to point in the same direction

I have two vectors, $v$ and $u$. How do I rotate $u$ around the x-, y-, and z-axes (or one axis) so that it points in the same direction as $v$?
1
vote
0answers
130 views

Frustrating geometry question

I have been battling with this problem for some time now but I am still stuck. Please would some kind soul help me out? So I have a Riemann sphere with 2 point $X,Y$ on it. (The Riemann sphere is ...
1
vote
1answer
2k views

Transforming from one spherical coordinate system to another

I have a set of points on the surface of a sphere specified in one coordinate system (specifically, the equatorial coordinate system), and for each point I need to work on all its neighbouring points ...
1
vote
2answers
596 views

How to get the cardinal direction from one location to another?

Given are two geo locations, each with latitude and longitude. One is the current location, the other is a target location. is there a formula for calculating the target's cardinal direction for 0 ...
1
vote
1answer
344 views

Foci of a general conic equation

The general equation of a conic is $A x^2 + B x y + C y^2 + D x + E y + F = 0$. At Wikipedia, there is an equation for the eccentricity, based on ABCDEF. Is there a similar equation for getting ...
1
vote
2answers
2k views

How to calculate the middle of a line?

My question is following. I have a line with a given (X1, Y1) and (X2, Y2) coordinates (see figure below). I need to calculate ...
1
vote
2answers
1k views

Finding the mean distance between n points evenly distributed in a disc of radius r

In reading this article about updated estimates for the number of exoplanets in the Milky Way, I am curious how to get an estimate of the mean distance between them. The Milky Way is ~50,000 light ...
1
vote
3answers
148 views

Approximation For Difference Of Two Sides Of A Triangle

I have been trying to derive this approximation but have been unsuccessful in doing so. Any help would be greatly appreciated.
0
votes
2answers
69 views

Axis and Angle of Rotation of 3x3 matrix

\begin{pmatrix} √ 3/2 & -1/4 & √3/4\\ 1/2 & √3/4 & -3/4\\ 0 & √3/2 & 1/2 \end{pmatrix} How do I find the axis and angle of rotation of this matrix?