How to solve this equations for alfa and beta ?

clc;
clear;

syms r alfa beta x0 y0 z0 x1 y1 z1

dist(alfa,beta) = norm(([x0 - r*(sin(alfa))*(cos(beta)),y0 - r*(sin(alfa))*(sin(beta)),z0 - r*(cos(alfa))] - [x1,y1,z1]));

D(alfa,beta) = diff(dist,alfa,2)*diff(dist,beta,2) - (diff(diff(dist,alfa),beta))^2;
[ALFA_MAX,BETA_MAX] = solve(D > 0, dist(alfa,beta) < 0,alfa,beta);
[ALFA_MIN,BETA_MIN] = solve(D > 0, dist(alfa,beta) > 0,alfa,beta);
[ALFA_SAD,BETA_SAD] = solve(D < 0,alfa,beta);
[ALFA_NAN,BETA_NAN] = solve(D == 0,alfa,beta);

Hi, I want solve distance equation of between any points and any point on sphere for alfa and beta. I have get distance equation and apply second derivation test but i can not any solution.

(x1,y1,z1) : coordinates of any points.

(x0,y0,z0) : center point of sphere.

alfa : angle between z axis or parallel line passes through (x0,y0,z0) and radius.

beta : angle between radius and xy-plane.

r : radius.

Answer 1

[Shishir Reddy]

Hi Tarik

As per my understanding, you would like to find the angles 'alpha' and 'beta' that minimize or maximize the distance between a point (x1, y1, z1) and any point on the surface of a sphere centered at (x0, y0, z0) with radius r.

Kindly refer the following steps to solve this problem -

1. Define the Distance Function - We'll start by defining the distance function symbolically.

syms r alpha beta x0 y0 z0 x1 y1 z1
x = x0 - r * sin(alpha) * cos(beta);
y = y0 - r * sin(alpha) * sin(beta);
z = z0 - r * cos(alpha);
dist = sqrt((x - x1)^2 + (y - y1)^2 + (z - z1)^2);

2. Find Critical Points - Compute the first derivatives and set them to zero to find critical points.

d_dist_alpha = diff(dist, alpha);
d_dist_beta = diff(dist, beta);
critical_points = solve([d_dist_alpha == 0, d_dist_beta == 0], [alpha, beta]);

3. Second Derivative Test - Compute the second derivatives and form the Hessian matrix to classify the critical points.

d2_dist_alpha2 = diff(d_dist_alpha, alpha);
d2_dist_beta2 = diff(d_dist_beta, beta);
d2_dist_alphabeta = diff(d_dist_alpha, beta);

Hessian = [d2_dist_alpha2, d2_dist_alphabeta; d2_dist_alphabeta, d2_dist_beta2];

Hessian_det = det(Hessian);

Numerical Optimization - If symbolic solutions are complex or not feasible, numerical methods like fminunc can be used.

distanceFunc = @(vars) sqrt((x0 - r*sin(vars(1))*cos(vars(2)) - x1)^2 + ...
                            (y0 - r*sin(vars(1))*sin(vars(2)) - y1)^2 + ...
                            (z0 - r*cos(vars(1)) - z1)^2);

initial_guess = [pi/4, pi/4];

options = optimoptions('fminunc', 'Algorithm', 'quasi-newton', 'Display', 'iter');
[optimal_vars, min_distance] = fminunc(distanceFunc, initial_guess, options);

optimal_alpha = optimal_vars(1);
optimal_beta = optimal_vars(2);
fprintf('Optimal alpha: %.4f\n', optimal_alpha);
fprintf('Optimal beta: %.4f\n', optimal_beta);
fprintf('Minimum distance: %.4f\n', min_distance);

Adjust the initial guess and options in fminunc based on your specific problem setup for best results.

For more information regarding fminunc function, kindly refer the following documentation - https://www.mathworks.com/help/optim/ug/fminunc.html

I hope this helps.