# Equations

The equation for an ellipsoid with axises $$e_1, e_2$$ and $$e_3$$: $E(x) := \frac{x_1^2}{e_1^2} + \frac{x_2^2}{e_2^2} + \frac{x_3^2}{e_3^2} - 1 = 0$ Let $$y = \begin{bmatrix} y_1 & y_2 & y_3 \end{bmatrix}^T$$ be a point outside the ellipsoid. For any point $$x = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}^{T}$$ on the ellipsoid we have1: $y - x = \frac{1}{2} t \nabla E(x),$ or equivalently in component form, $y_k = \left( 1 + \frac{t}{e_k^2} \right) x_k, \qquad k=1,2,3.$ The values $$x_k$$ can be expressed as a function of $$t$$: $x_k(t):=\frac{e_k^2}{t+e_k^2} y_k, \qquad k=1,2,3,$ and their derivatives are given by, $x_k'(t) = -\frac{e_k^2}{(t+e_k^2)^2} y_k = -\frac{x_k(t)}{t+e_k^2}, \qquad k=1,2,3.$ The equation of the ellipsoid is reduced to a scalar equation in $$t$$: $E(t)=\frac{x_1^2(t)}{e_1^2} + \frac{x_2^2(t)}{e_2^2} + \frac{x_3^2(t)}{e_3^2} - 1.$ The derivative $$E'(t)$$ is simply: $E'(t)=\frac{2 x_1(t) x_1'(t)}{e_1^2} + \frac{2 x_2(t) x_2'(t)}{e_2^2} + \frac{2 x_3(t) x_3'(t)}{e_3^2}.$

1. http://www.geometrictools.com/Documentation/DistancePointEllipseEllipsoid.pdf

# Analytical solution

When $$e_1=e_2=e_3=R$$ we can solve $$E(t)=0$$ analytically. $y_1^2+y_2^2+y_3^2-(t+R^2)^2=0$ $t = - R^2 + \sqrt{y_1^2+y_2^2+y_3^2}$ Note: We consider the positive $$t$$. Negative $$t$$ corresponds to the point $$\tilde{x}$$ on the other side of the ellipsoid.

# Numerical algorithm

1. Find a starting guess $$t_0$$.

2. Iterate a few Newton Raphson steps: $t_{n+1} = t_n - \frac{E(t_n)}{E'(t_n)}, \qquad n=0,1,\dots.$

3. Until convergence $$|\Delta t| < TOL$$ and $$|E(t)| < RTOL$$.

# Code

int EllipsoidSolver(Vec3 E, Vec3 y,
float& error, float& resid, Vec3& x, Vec3& n,
float tol=1e-4, float rtol=1e-4,
int maxiter=10, bool useX=false)
{
const float Rmin = 1e-4;
const Vec3 One = Vec3(1.0f);

const Vec3 y2 = y*y;
const Vec3 E2 = E*E;
const float yn = norm(y);
const float R = min(E);

float t;
int iter = 0;

// Check if y is too close to center of the ellipsoid.
if (yn < Rmin)
return -maxiter;

// Start guess.
if (useX)
t = norm(y-x)/norm(x/E2);
else
t = -R + norm(y);

while (iter < maxiter)
{
x = (E2/(Vec3(t)+E2))*y;
Vec3 xp = -x/(Vec3(t)+E2);
float F = dot(One, x*x/E2) - 1.0f;
float Fp = dot(One, 2.0f*x*xp/E2);
float dt = F/Fp;
t = t - dt;
iter = iter + 1;
error = std::fabs(dt);
resid = std::fabs(F);
if (error < tol && resid < rtol)
{
n = x/E2;
float rn = 1.0f/norm(n);
n = rn*n;
return iter;
}
}

return -maxiter;
}