Having an spheroid $S$ of semi-major axis $a$ in the equatorial plane in direction $x-axis$ and $y-axis$, and of semi-minor axis $b$ in direction $z-axis$.

$(S): (\frac{x}{a})^2+(\frac{y}{a})^2+(\frac{z}{b})^2=1$

A plane $(P): ux+vy+wz=d$ intersects $S$ That yields an ellipse $E$ in 3D.

I Intersected $S$ and $P$, I got a quadratic equation of quadratic conics, in the form: $(E): Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

After equating and solving, I got the coefficients of the ellipse $E$

$A=(wb)^2+(ua)^2$ $B=2uv(a)^2$ $C=(wb)^2+(va)^2$ $D=-2du(a)^2$ $E=-2dv(a)^2$ $F=a^2(d^2-(bw)^2)$

My question is as follow:

I have a point $M(i,j,k)$ lying in the plane of intersection, and it is outside the ellipse $E$, how to find the coordinates of the two tangent points $B$ and $C$ from the point $M$ to the ellipse $E$?

Please see figure from geogebra: enter image description here

Add enter image description here

  • $\begingroup$ You’ve already reduced this to a two-dimensional problem. Find the points of tangency there and map back to the three-dimensional space. Or, find the intersections of the quadric, the “slicing” plane, and the polar plane to $M$. $\endgroup$ – amd Jul 13 '18 at 19:14
  • $\begingroup$ I searched in internet for such problems, not found, that is why I ask here, I left the school since 10 years, I would get a help, or links helping me, e.g. polar plane, I have already asked similar thing, today otherway asked, I think you were there $\endgroup$ – Khaled Jul 13 '18 at 19:18
  • $\begingroup$ Do you remember that? $\endgroup$ – Khaled Jul 13 '18 at 19:19
  • $\begingroup$ I have an approach in javascript, but it didnt work, should I post it here? $\endgroup$ – Khaled Jul 13 '18 at 19:20
  • $\begingroup$ This is not a site for debugging your code. A description of the mathematics that you’re using in that program would be more appropriate. $\endgroup$ – amd Jul 13 '18 at 19:40

First of all, the equation that you’ve derived ostensibly by solving equation (P) for $z$ and then substituting into (S) isn’t the equation of the intersection curve. Indeed, a curve in a 3-D space can’t be represented by a single implicit Cartesian equation. You’ve got the equation of an elliptical cylinder that happens to have the same intersection with the cutting plane. The intersection of this cylinder with the $x$-$y$ plane is the orthogonal projection of the ellipse, so is in most cases foreshortened.

There are several ways to solve this problem, and the best/easiest really depends on what else you want to know about this configuration. If all you want are the two points of tangency, then a straightforward way to go is to find the intersection of $P$ and the polar plane to $M$, then computing the intersection of this line with the spheroid. If you use a standard parametric representation for the line, this is a straightforward matter of solving a single-variable quadratic equation.

Specifically, the polar plane to $M$ is ${ix\over a^2}+{jy\over a^2}+{kz\over b^2}=1$, so its intersection with $P$ has direction vector $$\mathbf v = \left(\frac i{a^2},\frac j{a^2},\frac k{b^2}\right)\times(u,v,w)$$ and with a bit of work you can find a point on this line, such as $$\mathbf p = \left({dj-a^2v \over ju-iv} , {di-a^2u \over iv-ju}, 0\right).$$ Plug $\mathbf p+t\mathbf v$ into the equation of the spheriod and solve for $t$.

A related approach is to transform the problem into a simpler one that involves intersection with and tangents to the unit sphere, solve that problem, then transform back. The required transformations are simple scaling operations that preserve intersections and tangency. Mapping the spheroid to the unit sphere takes $(i,j,k)$ to $(i/a,j/a,k/b)$ and takes $P$ to the plane $P'$ with equation $aux+avy+bwz=d$. The points of tangency can be found using the method described above, or you could further rotate or reflect to make $P'$ parallel to the $x$-$y$ plane and then solve the simple problem of finding the ends of the chord of contact to a circle. This additional rotation/reflection also makes it fairly easy to compute a parameterization of the intersection curve: it’s just the unit circle rotated/reflected and scaled back into position. Once you have this parameterization of the intersection curve, you can use it to generate test points for the search algorithm that you’ve described.

Finally, there’s a version of the tried-and-true method that you likely learned in planar analytic geometry. Generate a one-parameter family of lines through $M$ that lie in $P$ and work out their intersection with the spheriod. For tangency, you want the resulting equation to have a double root, which will then generate a simple quadratic equation in the line parameter. Once you have that, finding the points of tangency will be a matter of back-substitution.

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  • $\begingroup$ Solving for $t$... wolfram gave $t = ((-(2 b^4 j w (d j - a^2 v))/(j u - i v) - (2 b^2 (d i - a^2 u) (a^2 k u - b^2 i w))/(i v - j u) + (2 a^2 b^2 k v (d j - a^2 v))/(j u - i v)) ± sqrt(((2 b^4 j w (d j - a^2 v))/(j u - i v) + (2 b^2 (d i - a^2 u) (a^2 k u - b^2 i w))/(i v - j u) - (2 a^2 b^2 k v (d j - a^2 v))/(j u - i v))^2 - 4 ((b^2 (d j - a^2 v)^2)/(j u - i v)^2 + (b^2 (d i - a^2 u)^2)/(i v - j u)^2 - a^2 b^2) (a^4 b^2 k^2 v^2 - 2 a^2 b^4 j k v w + b^2 (a^2 k u - b^2 i w)^2 + a^2 + b^6 j^2 w^2)))/(2 (a^4 b^2 k^2 v^2 - 2 a^2 b^4 j k v w + b^2 (a^2 k u - b^2 i w)^2 + a^2 + b^6 j^2 w^2))$ So? $\endgroup$ – Khaled Jul 14 '18 at 15:11
  • $\begingroup$ Is that normal that is so long, what would be then when I put these $t$ into the the equation of the line $\mathbf p+t\mathbf v$ to get the coordinates of $B$ and $C$ ?!! $\endgroup$ – Khaled Jul 14 '18 at 15:23
  • $\begingroup$ @Khaled Why would you expect a fully-expanded general solution to be neat and tidy? If I were coding it, I’d do it step by step, with plenty of local variable to hold the intermediate results. Much less likely to introduce an error that way. $\endgroup$ – amd Jul 14 '18 at 19:16
  • $\begingroup$ Bcz I would write a code in programming language, for example, a function take parameters then it gives these intersections points $\endgroup$ – Khaled Jul 14 '18 at 19:19
  • $\begingroup$ I didnt find such thing in google $\endgroup$ – Khaled Jul 14 '18 at 19:20

Equation of the ellipsoid:

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

Equation of the polar plane w.r.t. $M$:

$$\frac{ix+jy}{a^2}+\frac{kz}{b^2}=1 \tag{2}$$

Equation of the plane $P$ containing $M$:

$$ux+vy+wz=d \tag{3}$$

under the constraint $ui+vj+wk=d$.

In principle, $(1) \cap (2) \cap (3)$, which is the required intersections, is solvable with discriminant $$\frac{i^2+j^2}{a^2}+\frac{k^2}{b^2}-1>0$$

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