I am working in a project of calibration of a laser set using a geometry with know dimensions, I am using a cone to do my calibration, the cone in 2D section gives an ellipse , my problem right now I searching for a method that help me to find the coordinates of a point on the ellipse with respect the cone coordinates.
The plane perpendicular to the plane of the ellipse and passing through vertex $V$, intersects the cone along two rays $VI$ and $VJ$, cutting the ellipse at $A$ and $B$. Segment $AB$ is then the major axis of the ellipse and its midpoint $O$ is the center of the ellipse.
To find minor axis $CD$, just intersect the cone with the line through $O$ perpendicular to plane $VIJ$. You can easily find all 3-D coordinates of these points, if cone and cutting plane are given.
You can then set up a local coordinate system on the plane of the ellipse, with origin $O$ and axis directions given by unit vectors $$ \hat x={A-O\over \overline{AO}},\quad \hat y={C-O\over \overline{CO}}. $$ With respect to this coordinate system the equation of the ellipse is $$ {x^2\over a^2}+{y^2\over b^2}=1, $$ where $a=\overline{AO}$ and $b=\overline{CO}$.
Any point $P$ on the ellipse, having local coordinates $(x_P,y_P)$ has then a position in space given by $$ P=x_P\,\hat x + y_P\,\hat y. $$