Cone ellipse intersection

Question

Given the cone $K=\{(x,y,z)\in \mathbb R^3 :x^2+y^2-z^2=0\}$ and the equation of $E=\{(x,y,z)\in \mathbb R^3 :z=my+c\}$
Find an equation for the intersecion of the cone and the plane

The intersection is obviously $\{(x,y,z)\in \mathbb R : x^2+y^2-(my+c)^2,z=my+c\}$ I got a little confused by the problem. Is it possible to find an equation $f(x,y,z)=0$ that describes the given ellipse ? ($f \in \mathbb R$)
If thats not possible how would one parameterize the given equation ?

Would appreciate your help

It’s only an ellipse if $m^2\lt1$. When $m^2\gt1$ you get a hyperbola instead. — amd, Jan 15 '17 at 00:56

score 1 · Accepted Answer · answered Jan 14 '17 at 23:28

1

You can let

$$f(x,y,z)=\left\| x^2+y^2-z^2\right\|^2+\left\| z-my-c\right\|^2$$

answered Jan 14 '17 at 23:28

Siong Thye Goh

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Thanks.I Feel pretty dumb for not figuring that out xD – XPenguen Jan 14 '17 at 23:31
What does ||...||^2 mean in this case? – Michael Murray Mar 08 '21 at 22:25
The square of norm. – Siong Thye Goh Mar 09 '21 at 02:20

score 1 · Answer 2 · edited Apr 13 '17 at 12:21

Assuming that $m^2\lt1$ so that you actually have an ellipse, the answers to your previous question give you everything you need for a parameterization of it. If you have the center $\mathbf p$, the unit direction vectors of the major and minor axes $\mathbf u$ and $\mathbf v$ and the semi-axis lengths $a$ and $b$, then the ellipse can be parameterized as $\mathbf p + a\cos t\,\mathbf v+b\sin t\,\mathbf w$.

Cone ellipse intersection

2 Answers2