Articles of geometry

Distance between two points on the Clifford torus

How can I obtain the distance between two points $\mathbf{x}=(x_1,x_2,x_3,x_4)$ and $\mathbf{y}=(y_1,y_2,y_3,y_4)$ that belong to the $2$-torus $\mathbb{S}^1\times \mathbb{S}^1$? This is, I want to measure the distance (I do not require the geodesic) of $\mathbf{x}$ to $\mathbf{y}$ along the manifold $$\mathbb{S}^1\times\mathbb{S}^1=\big\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4:x_1^2+x_2^2=1,x_3^2+x_4^2=1\big\}.$$ I guess that there should be a way of doing this considering an isomorphism(?) […]

Correspondence between eigenvalues and eigenvectors in ellipsoids

think of an ellipsoid in the n-dimensional space defined by $$(x-\mu)’A(x-\mu)=1.$$ I was calculating the volumes of n-dimensional ellipsoids like the one from above for a while, which is straightforward once the eigenvalues of matrix $A$ are retrieved. The volume $V$ is then given by (using the log scale so that it will not create […]

What are some general strategies to build measure preserving real-analytic diffeomorphisms?

One could prove the following theorem in the smooth setting: Theorem Let $(M,m)$ be a $d$ dimensional $C^\infty$ manifold with smooth volume $m$. Let $\{F_i\}_{i=1}^k$ and $\{G_i\}_{i=1}^k$ be two systems of disjoint open subsets, satisfying $m(F_i)=m(G_i)$. Assume $\bar{F}_i$ and $\bar{G}_i$ are diffeomorphic to the $d$ dimensional closed ball in $\mathbb{R}^m$ for $i=1,\ldots k$. Then, given […]

Find the Closest and Farthest Points of a Cube

This has to do with collision detection between a ray and a cube. I have a camera position that I am looking from and a ray is being shot into the scene which contains a cube. I have the ray starting position and direction in 3D space. The cube is defined by its center position […]

Math Behind Creating a “Perfect” Star

I am busy looking to create star paths in my app, and I was wondering how to determine the ratio between the inner radius and the outer radius of the points of a star so that the star has “straight” lines across. I have a function that takes 3 parameters: pointCount = 5 outerRadius = […]

On constructing a triangle given the circumradius, inradius, and altitude .

I was recently pondering about constructing triangles given different attributes of it. I am wondering whether we could construct a triangle given its Circumradius $R$ , Inradius $r$, and length of one altitude.

What is a vector?

What is a vector? As the question says what is a vector and what are its uses or, I mean, when should we use vectors? Is this a branch of geometry or algebra or trigonometry?

Number of ways to separate $n$ points in the plane

Say you are given $n$ points such that no three are colinear. Show the number of ways to separate them into two subsets by drawing a straight line depends on $n$ but not the position of the points.

What is the equation describing a three dimensional, 14 point Star?

I need to model a 14 point star. This is a three dimensional surface where there is a point at each of the eight corners of a cube and each of the six sides. The object is uniform (i.e. planar symmetry). This is a real object. It has a volume of 5000 $cm^3$ per side. […]

Decompose rotation matrix to plane of rotation and angle

I would like to decompose an $n$-dimensional orthogonal rotation matrix (restricting to simple rotation with a single plane of rotation) to the two basis vectors of the plane of rotation, and an angle of rotation. The common method is decomposing the rotation matrix to an axis and angle, but this doesn’t work in higher dimensions. […]