Is it Possible to Parametrize A Skewed Cone?

[Karnage1993]

I would like to parametrize a skewed cone from a given vertex with an elliptical base, however I cannot seem to find the general formula for it. The parametrization given in http://mathworld.wolfram.com/EllipticCone.html produces a cone but not with the right vertex, ie, it is only a cone with an elliptical base, and not a skew cone with an elliptical base. The cone I want to parametrize is the cone on the right hand side in http://en.wikipedia.org/wiki/File:Cone_3d.png , except with an ellipse base and not a circle. Any suggestions?

 

[LCKurtz]

I would like to parametrize a skewed cone from a given vertex with an elliptical base, however I cannot seem to find the general formula for it. The parametrization given in http://mathworld.wolfram.com/EllipticCone.html produces a cone but not with the right vertex, ie, it is only a cone with an elliptical base, and not a skew cone with an elliptical base. The cone I want to parametrize is the cone on the right hand side in http://en.wikipedia.org/wiki/File:Cone_3d.png , except with an ellipse base and not a circle. Any suggestions?

Sure. Do you know how to parameterize the ellipse in the xy plane with a single parameter, like $\theta$? Then let (p,q,r) be the vertex. Parameterize the straight line from (p,q,r) to a point on the bottom ellipse as a function of t. The result will be a parameterization of the cone using those two parameters. Is that enough of a hint?

 

[Karnage1993]

Yes, that works! Although, I had to include the z-component of the ellipse parametrization as 0 in order to make a straight line. Thanks for the help.

 

[LCKurtz]

Yes, that works! Although, I had to include the z-component of the ellipse parametrization as 0 in order to make a straight line. Thanks for the help.

Yes, an ellipse in the xy plane would have z=0. You're welcome.