Eigenvalues of Representations of $SO(3)$

The set of homogeneous polynomials of degree $d$ are functions of the form

For this post $\vec{x} \in \mathbb{R}^3$, but one can also consider homogeneous functions on $\mathbb{R}^3$. $SO(3)$ the group of $3$ dimensional rotations acts on $P_d$ by rotating coordinates of the domain $\mathbb{R}^3$. Suppose $p \in P_d$ then the action of $A \in SO(3)$ is

The result of the action is just another homogeneous polynomial of degree $d$, because rotations preserve distance.

... read more

Area Forms & Normals

To compute the area form of $S^3$, I used a general formula for $S^n \subset \mathbb{R}^{n+1}$ that I found in Lee’s Introduction to Smooth Manifolds.

is the area form of $S^n$, in terms of its coordinate functions. While trying to understand this formula, I was reminded of the cross product formula, which is an alternating sum of determinants. Well, sure, alternating forms like are really determinants in disguise. Suppose $v_1, \ldots,v_n$ are tangent vectors at some point on $S^n$. Individually, 1-forms $dx_{j}(v)$ are linear functionals and act on tangent vectors $v \in T_{p}S^n$. Essentially, $dx_{j}(v_k )$ is a portion of the derivative $Dx_j$ in the $v_k$ direction; the directional derivative. Alternatively, it is the action of $v_k$ on coordinate $x_j$. I summarize these two descriptions below. ... read more

Parameterizing the Space of 3D Rotations

3D medical images in the NifTI format store data from MRI acquisitions of some anatomy, like a human brain or heart. When I was a postdoc @Penn my lab was primarily interested in studying brains. Each image associates grayscale values ( in the simplest case ) to discrete coordinates $(i,j,k)$, which describe voxel locations. And for reasons best explained by NifTI FAQ it’s useful/important to align the acquired image to some other coordinate system. This alignment is stored in the image header, as a rigid motion plus an offset. Here’s a bit of NifTI documentation motivating the need to keep the alignment.

This method can also be used to represent "aligned"
coordinates, which would typically result from some post-acquisition
alignment of the volume to a standard orientation (e.g., the same
subject on another day, or a rigid rotation to true anatomical
orientation from the tilted position of the subject in the scanner).

The NifTI standard allows for orientation reversing transforms, but in this post I focus on proper rotations. These rigid motions must preserve volume and orientation; rigidity necessitates linearity. Being a geometric property, volume is preserved when this linear transformation is an isometry; which is to say, for some matrix $A$, $A A^T = I$. Moreover, orientation is preserved when $\det(A) > 0$. The intersection of all these requirements is the group of 3D rotations, usually referred to as the special orthogonal group, ... read more

3 Points Make a Circle

While learning about Voronoi diagrams from Computational Geometry, I had trouble justifying a single statement from a proof regarding the complexity ( as a function of $n$ sites ) of Voronoi diagrams. For completeness I’ll restate the theorem here.

Theorem. For $n \geq 3$, the number of vertices in the Voronoi diagram of a set of $n$ point sites in the plane is at most $2n-5$ and the number of edges is at most $3n-6$.

To establish the stated bounds, the proof hinges on the constancy of the Euler’s characteristic of graphs (that embed in $S^2$) and every vertex in the Voronoi diagram has degree at least three. Really? At the point in the proof when this fact is invoked we know that all the $n$ sites are not colinear; so given any pair of sites there’s a third site that does not lie on the line passing through them. Three points induce a vertex. But why?? Voronoi vertices have a specific meaning/interpretation; they are equidistant from sites, which means that a Voronoi vertex is the center of circle that passes through at least three sites. ... read more

Functional Prime Number Sieve

In a post a few years ago I discussed primality testing. For some reason I thought I described the basic prime number sieve (I’m pretty sure there is only one) there, but apparently I didn’t. No worries. The prime number sieve solves a different problem; that of generating a list of the first $N$ primes. One could use a primality testing method to do this as well; namely filter a list of numbers with the primality test as a predicate; but sieve based methods are faster and conceptually much easier.

The prime number sieve iteratively filters out all the composite numbers, with the caveat that the leading number in the input is a prime. For , we put aside $2$, the smallest number, and filter out all multiples of $2$; then put aside $3$ and filter out multiples of $3$, and so on. This process, shown below as a decreasing sequence, terminates with a set of primes. ... read more

Losing My Orientation

While trying to grok $SO(3) \cong \mathbb{RP}^3$ and understand various parametrizations of $SO(3)$, I wandered a bit (okay maybe more that litte:) ) and started to think about projective space itself. It is well know that real projective spaces alternate between being orientable and non-orientable, as dimension increases. Specifically odd dimensional projective spaces are orientable but even ones are not. For example, $\mathbb{RP}^1$ and $\mathbb{RP}^3$ are orientable but $\mathbb{RP}^2$ is not.

Jeffery Week’s Shape of Space gives a Flatland inspired explanation of orientations on the projective plane and projective 3-space. It’s a fantastic book and a wonderful example of storytelling as a teaching method. In this note, I want to invoke Week’s style and describe orientations on real projective spaces of any dimension. ... read more

Sum of the First $N$ Natural Numbers

Lately I’ve been thinking about the content of my high school mathematics courses. In my college algebra and trigonmetry class we were introduced to proofs by induction–of all things–somewhere towards the end of my junior year. An important proof technique, for sure. However, the combinatorial expressions they were applied to, like the sum of first $N$ natural numbers, seemed miraculous. Proof by induction is fairly straight forward. But how does one even guess at a closed formula for such expressions? I remember the teacher saying something to the effect “you need to be smart”. I suppose so, but we can actually construct the closed form of the sum of the first $N$ natural numbers. ... read more

Average Length of the Longest Arc in $S^1$

Suppose that we draw $n$ points $ a_i \sim Uniform(S^1)$ for $i = 1 \ldots n $. These points determine a partition or set of disjoint arcs of $S^1$. What is the average length of the longest arc? To even measure the $n$ arc lengths, we need an ordering of the $\{ a_i \}$ , $a_{(1)}, \ldots, a_{(N)} $.

the problem

The arc lengths are: ... read more