A Finding Minimal Mean Distance Curves on the Unit Sphere

AI Thread Summary
The discussion focuses on finding parametric equations for a simple closed curve of length 4π on the unit sphere that minimizes the mean spherical distance from the curve to the sphere. The solution involves using calculus of variations and Lagrange multipliers to derive a system of differential equations for the parameters of the curve. It is established that the mean distance can be minimized by applying specific transformations to the parametric equations, allowing for arbitrary lengths greater than 2π. The thread also highlights that combinations of great circles can achieve the required length and minimize distance, with suggestions for using step functions for angles. The approach emphasizes the need for numerical solutions to verify minimization.
iPuzzled
Messages
1
Reaction score
0
TL;DR Summary
problem and solution
in attachements there is more, there is a question map and corresponding answers.
**Problem:**

Find parametric equations for a simple closed curve of length 4π on the unit sphere which minimizes the mean spherical distance from the curve to the sphere; the solution must include proof of minimization. Can you solve this problem with arbitrary L > 2π instead of 4π?

There seems to be little precedent for this problem. A few recently studied spherical curves (which probably do not minimize the mean distance) can be viewed at Gallery of Space Curves Made from Circles and Gallery of Bishop Curves and Other Spherical Curves.

**SOLUTION**
To find the parametric equations for a simple closed curve of length 4π on the unit sphere that minimizes the mean spherical distance from the curve to the sphere, we can use the calculus of variations. Let the curve be given by the parametric equations ##\mathbf{r}(t) = (\sin\theta(t)\cos\phi(t), \sin\theta(t)\sin\phi(t), \cos\theta(t))##, where ##0 \leq t \leq 2\pi## and ##\theta(t)## and ##\phi(t)## are differentiable functions. The length of the curve is given by
$$L = \int_{0}^{2\pi} \|\mathbf{r}'(t)\| dt = \int_{0}^{2\pi} \sqrt{\theta'(t)^2\cos^2\phi(t) + \theta'(t)^2\sin^2\phi(t) + \phi'(t)^2\sin^2\theta(t)}\ dt.$$
We want to minimize the mean distance between points on the curve and points on the sphere, which is given by
$$D = \frac{1}{4\pi}\iint_{\text{sphere}} \mathrm{dist}(\mathbf{r}(t), \mathbf{x})\ d\mathbf{x}.$$
Here, ##\mathrm{dist}(\mathbf{r}(t), \mathbf{x})## is the distance between the point on the curve at parameter value ##t## and the point ##\mathbf{x}## on the sphere. Using the spherical law of cosines, we can express this distance as
$$\mathrm{dist}(\mathbf{r}(t), \mathbf{x}) = \arccos(\mathbf{r}(t) \cdot \mathbf{x}).$$
Substituting in the parametric equations for ##\mathbf{r}(t)## and using the fact that the sphere has radius 1, we have
$$\mathrm{dist}(\mathbf{r}(t), \mathbf{x}) = \arccos(\sin\theta(t)\cos\phi(t)x_1 + \sin\theta(t)\sin\phi(t)x_2 + \cos\theta(t)x_3).$$
We want to minimize ##D## subject to the constraint that ##L = 4\pi##. Using Lagrange multipliers, we consider the function
$$F = D + \lambda(L - 4\pi).$$
Taking the partial derivatives of ##F## with respect to ##\theta##, ##\phi##, and ##\lambda##, and setting them to zero, we obtain the following system of differential equations:
\begin{align*}
&\frac{d}{dt}\left(\frac{\cos\theta(t)}{\sqrt{\theta'(t)^2\cos^2\phi(t) + \theta'(t)^2\sin^2\phi(t) + \phi'(t)^2\sin^2\theta(t)}}\right) = \lambda\theta'(t)\sin\theta(t), \\
&\frac{d}{dt}\left(\frac{-\sin\phi(t)}{\sqrt{\theta'(t)^2\cos^2\phi(t) + \theta'(t)^2\sin^2\phi(t) + \phi'(t)^2\sin^2\theta(t)}}\right) = \lambda\phi'(t)\sin\theta(t), \\
&\int_{0}^{2\pi} \sqrt{\theta'(t)^2\cos^2\phi(t) + \theta'(t)^2\sin^2\phi(t) + \phi'(t)^2\sin^2\theta(t)}\ dt = 4\pi,
\end{align*}
where the second equation follows from the fact that the vector ##(\cos\theta(t)\cos\phi(t), \cos\theta(t)\sin\phi(t), -\sin\theta(t))## is normal to the curve at ##\mathbf{r}(t)##.

We can simplify the first two equations by multiplying them by ##\sqrt{\theta'(t)^2\cos^2\phi(t) + \theta'(t)^2\sin^2\phi(t) + \phi'(t)^2\sin^2\theta(t)}## and then taking the derivative of the resulting expressions with respect to ##t##. After some algebraic manipulation, we obtain the following system of differential equations:
\begin{align*}
&\frac{d^2\theta}{dt^2} + \frac{\cos\theta}{\sin\theta}\left(\frac{d\theta}{dt}\right)^2 + \frac{\sin\phi}{\sin\theta}\frac{d\theta}{dt}\frac{d\phi}{dt} - \lambda\sin\theta = 0, \\
&\frac{d^2\phi}{dt^2} + \frac{2\cos\theta}{\sin\theta}\frac{d\theta}{dt}\frac{d\phi}{dt} - \lambda\sin\theta\sin\phi = 0.
\end{align*}
This system of equations can be solved numerically using a suitable initial condition. We can then verify that the resulting curve indeed minimizes the mean spherical distance by computing the mean distance over a fine grid of points on the sphere.

To solve the problem for arbitrary ##L > 2\pi##, we can simply scale the parametric equations by a factor of ##L/(4\pi)##, since the mean distance is a scale-invariant quantity. That is, we can use the parametric equations
$$\mathbf{r}(t) = \left(\sin\theta(t)\cos\left(\frac{L}{4\pi}\phi(t)\right), \sin\theta(t)\sin\left(\frac{L}{4\pi}\phi(t)\right), \cos\theta(t)\right),$$
where ##0 \leq t \leq 2\pi## and ##\theta(t)## and ##\phi(t)## are differentiable functions. We can then use the same approach as above to find the solution that minimizes the mean distance subject to the constraint that the length of the curve is ##L##.
 

Attachments

Last edited by a moderator:
Mathematics news on Phys.org
Some observations:
  1. Every great circle on the unit sphere has length 2π
  2. Thus any combination of two great circles has length 4π
  3. All combinations of two great circles must intersect in at least two points
  4. The simplest combination is to run the same great circle twice
  5. If that is not allowed, use two great circles with an angle between them (θ) a step function the angle around (φ) modulo 4π - i.e. 0 for 0≤ φ<2π, 1 for 2π≤φ<4π.
  6. From that point on, you can try other functions for θ
 
Thread is now open again.
 
  • Like
Likes renormalize
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Thread 'Imaginary Pythagorus'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
Back
Top