Finding a point on an ellipsoid where the normal is perpendicular to a line segment.

If I have an ellipsoid in three dimensions (actually an oblate spheroid, the Earth).

$$\frac{x^2+y^2}{a^2} + \frac{z^2}{b^2} = 1$$

And a ray represented by the point $P = (x_p, y_p, z_p)$ with direction vector $D = (x_d, y_d, z_d)$

I can form an ellipse by intersecting the plane formed by the ray and the origin with the ellipsoid. How can I find the points on that ellipse (there should be two I believe), where the normal vector is perpendicular to the ray?

I'm effectively looking for the furthest points on the ellipse that are "visible" to the origin of the ray, $O$. Since there's two solutions, I'd like the one closer to the ray.

Here's a picture that I hope clarifies, after forming an ellipse by intersecting a plane with the ellipsoid, I want to find point A where the normal vector is perpendicular to the line containing the point and the ray origin.

I believe this is equivalent to finding the tangent line that contains the point P.

• Is the ray a line a through the origin? If that's the case, the plane you speak of is not well-defined. – Fimpellizieri May 11 '16 at 2:13
• Do you know how to find the tangent plane at a point? With this you can find the normal vector at a point. I think it's really ugly though – lordoftheshadows May 11 '16 at 2:19
• It is not clear what you are asking, a ray is described by an origin and a direction. Does $(x_l,y_l,z_l)$ stand for the origin or the direction? Also your ellipse looks strange, should the denominator be $a^2$ and $b^2$? Assume that is the case and you are looking for the locus of point $(x,y,z)$ on $\frac{x^2+y^2}{a^2} + \frac{z^2}{b^2} = 1$ whose normal is perpendicular to the ray joining $(x_l,y_l,z_l)$ and $(x,y,z)$, the locus will be the intersection of the ellipse and the plane $\frac{xx_l + yy_l}{a^2} + \frac{zz_l}{b^2} = 1$. – achille hui May 11 '16 at 4:09
• Does the plane "formed by the ray and the origin refer to the plane through the origin which has the normal vector $L$? – coffeemath May 11 '16 at 4:10
• @Fimpellizieri no in my case, I'll be doing a test for ray-ellipse intersection before finding these points, so any ray that goes through the origin will intersect the ellipsoid and not be considered. – gct May 11 '16 at 12:54

The plane through the ray and the origin intersects your ellipsoid to form a planar ellipse. To find this planar ellipse one must establish a basis vector on the plane such that any point on the plane can be represented by two parameters ($\alpha$ and $\beta$). $$\vec{r} = \alpha \vec{p} + \beta \vec{d}$$

The ellipse on the plane is now given by the equation

$$\alpha^2 C_{11} + 2 \alpha\beta C_{12} + \beta^2 C_{22} = 1$$

$$\pmatrix{\alpha \\ \beta \\ 1}^\top \pmatrix{C_{11} & C_{12} & \\ C_{12} & C_{22} & \\ & & -1} \pmatrix{\alpha \\ \beta \\ 1} =0$$

where \begin{align} C_{11} &= \frac{x_p^2}{a^2} + \frac{y_p^2}{a^2} + \frac{z_p^2}{b^2} \\ C_{22} &= \frac{x_d^2}{a^2} + \frac{y_d^2}{a^2} + \frac{z_d^2}{b^2} \\ C_{12} &= \frac{x_p x_d}{a^2} + \frac{y_p y_d}{a^2} + \frac{z_p z_d}{b^2} \end{align}

Now for the fun part. To be perpendicular to the ray the point $\vec{r}$ must obey the projection $$\vec{d}^\top (\vec{r}-\vec{p}) =0$$

From the expansion of $\vec{r}$ in terms of the planar coordinates $\alpha$ and $\beta$ this projection is used to solve for

$$\alpha = 1- \frac{\vec{d}^\top \vec{d}}{\vec{d}^\top \vec{p}} \beta = 1-\frac{x_d^2+y_d^2+z_d^2}{x_d x_p + y_d y_p + z_d z_p} \beta = 1- \lambda \, \beta$$

So given any distance along the ray $\beta$ the 3D position of the point is $\vec{r} = (1-\lambda \,\beta) \vec{p} + \beta \vec{d}$ where $\lambda$ is the fixed ratio $\lambda = \frac{x_d^2+y_d^2+z_d^2}{x_d x_p + y_d y_p + z_d z_p}$.

The points on the ellipse that are perpendicular to the ray solve the equation

$$(1-\lambda\,\beta)^2 C_{11} + 2 (1-\lambda\,\beta)\beta C_{12} + \beta^2 C_{22} = 1 \\ (C_{11} \lambda^2 -2 C_{12} \lambda + C_{22}) \beta^2 + 2 (C_{12}-C_{11} \lambda) \beta + (C_{11-1}) = 0$$

This has two solutions (as expected)

$$\beta = \frac{C_{11} \lambda-C_{12} \pm \sqrt{ C_{11} (\lambda^2-C_{22}) + C_{12}^2 -2 C_{12} \lambda + C_{22}}}{C_11 \lambda^2 -2 C_{12} \lambda + C_{22}}$$

And the back substitution

$$\alpha = 1 - \lambda\, \beta \\ \vec{r} = \alpha \vec{p} + \beta \vec{d}$$

Edit 1 To find the lines tangent to the planar ellipse, and passing through P you do the following:

1. The planar coordinates of P are $(\alpha=1,\beta=0)$, or in homogeneous coordinates $$P=\pmatrix{1,& 0,& 1}$$
2. The homogeneous coordinates of the planar ellipse are $$C=\left| \matrix{C_{11} & C_{12} & \\ C_{12} & C_{22} & \\ & & -1} \right|$$
3. The polar line of P is $L=C\,P$ $$L = \left[ \matrix{C_{11}, & C_{12},& -1} \right]$$
4. The polar line intersects the ellipse at the tangent points $Q_1$ and $Q_2$. To find these points form the equation of the line $C_{11} \alpha + C_{12} \beta - 1 =0$ and plug it into the ellipse equation to find $$Q_1 = \pmatrix{ 1 + \tfrac{C_{12} \sqrt{C_{11}-1}}{K}, & - \tfrac{C_{11} \sqrt{ C_{11}-1 }}{K}, & C_{11} }$$ $$Q_2 = \pmatrix{ 1 - \tfrac{C_{12} \sqrt{C_{11}-1}}{K}, & \tfrac{C_{11} \sqrt{ C_{11}-1 }}{K}, & C_{11} }$$ where $K=\sqrt{C_{11} C_{22}-C_{12}^2}$.

5. The tangent line coordinates are $T_1 = C \,Q_1$ and $T_2 = C \, Q_2$ $$T_1 = \left[ \matrix{ C_{11}, & C_{12} -K \sqrt{C_{11}-1} , & -C_{11} } \right]$$ $$T_2 = \left[ \matrix{ C_{11}, & C_{12} +K \sqrt{C_{11}-1} , & -C_{11} } \right]$$

6. In equation form the tangent lines are $$(C_{11}) \alpha + (C_{12} - K \sqrt{C_{11}-1}) \beta - C_{11} = 0$$ $$(C_{11}) \alpha + (C_{12} + K \sqrt{C_{11}-1}) \beta - C_{11} = 0$$

• Brilliant answer! I've seen some of your other answers and was hoping you'd see my question. – gct May 16 '16 at 13:01
• Actually, I'm not sure this is quite what I need. I don't necessarily want the points on the ellipse perpendicular to the ray itself, rather I want the points on the ellipse whose tangent lines contain the point P. This is for a ray caster, and in particular when a ray misses the ellipsoid, I wish to find the point furthest away which is "visible" to the origin of the ray (the viewpoint), which are the aforementioned tangent points. – gct May 16 '16 at 13:15
• Also I'm curious how you got the initial equation for the ellipse, is that just what falls out when you intersect the ellipsoid and plane in homogeneous coordinates? – gct May 16 '16 at 13:30
• @gct I took the ellipsoid equation (in terms of xyz) and plugged in $\vec{r} = (x,y,z)$ to get an equation in terms of $\alpha$, $\beta$. – ja72 May 16 '16 at 14:16
• that makes perfect sense, thanks. What do you think about finding the tangent points I mentioned above? My thought is that $C*l$ gives you the tangent line for a point $l$ on the conic. But actually finding the tangent point $l$ given that the tangent line must contain $\vec{p}$ eludes me. – gct May 16 '16 at 15:34