# Calculation of the coordinates on the surface of a tilted cone

I have a mathematical problem (which I am trying to solve with Mathematica).

I want to tilt a cone around its base point as in my example, where I have used Mathematica's Cone-function and spherical coordinates for the tip of the cone:

Manipulate[Graphics3D[{
Point[{0,0,0}](*visualisation of the base point*),
{Opacity[0.5],Gray,
Cone[{{0,0,0}(*base point*),
{Sin[theta1]*Cos[phi1],Sin[theta1]*Sin[phi1],Cos[theta1]}
(*tip point*)},
}}, Axes->True, PlotRange->{{-1,1},{-1,1},{-1,1}},
AxesLabel->{"x","y","z"}],
{theta1,0,Pi},{phi1,0,2*Pi}]


But now I have to find another solution for this problem which allows me to get information about the coordinates on the surface of the cone, e.g. I want to know that for

theta=0,    x=0, y=0 => z=1,
theta=Pi,   x=0, y=0 => z=-1,
theta=Pi/2, x=0, y=0 => z={-1/2,1/2} etc.


To solve this problem, I have tried to use coordinate transformations for the equation

z^2 = (x^2+y^2)/Tan[halfApertureAngle]^2.


And I have tried to solve the general equation of a cone with analytical geometry by using the equation

((x-s).a)^2=Norm[x-a]^2*Norm[a]^2*Cos[halfApertureAngle]^2,


with x: position vector, a: vector from the tip pointing to the base of the cone, s: vector from the origin of the coordinate system to the cone tip.

Unfortunately, none of my attempts have worked. Does somebody of you know how I could solve this problem? I would be grateful for every hint and help!

-
You know the parametric equations of the cone, no? (see equations 1-3) – J. M. Nov 14 '11 at 12:47
Oh, I am so silly. I did the correct coordinate transformation but when I plotted the cone, I made a stupid mistake. Because you have z^2 in the equation of a cone, you get two possible solutions. One of the cones is tilted around its base point (as I wanted) and the other cone, whose tip touches the tip of the first cone, not. Unfortunately, I plotted the wrong cone and thought that my transformation was wrong. Well, I apologize for bothering you and the other forum users with this problem! – partial81 Nov 15 '11 at 6:46