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*)}, 1/2(*radius of the cone*)] }}, 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!