I need help developing parametric equations for a cone which is about an arbitrary vector and it's vertex offset from the origin.
Recall that a cone in its cross section is just a circle and a circle has a standard parametrization involving an angle (call it t) and a radius r.
If the radius is proportional to the height of the cone and the the height and angle are independent parameters (you will have since the cone is two-dimensional like taking a piece of paper and folding it), then can you find the parametrization?
(Hint: If you don't know the parametrization of a circle, think about how a circle is defined with trig functions).
Using a = angle about the circle
t = opening angle of the cone
h = height of cone
x = h*tant(t)*cos(a)
y = h*tant(t)*sin(a)
z = h
However that represents a cone which rotates about the Z axis with its vertex and the origin (or can be rearranged for any of the other axis). I need to parametrize a cone that rotates about an arbitrary vector V. To explain more I need to parametrize a cone which has had 6 rotations applied to it. I could multiply the above equations by the rotation matrix, but with there being 6 rotations it gets messy. Since the rotations are "known" values I want to apply the rotations to the x - axis ... V = R(a1,a2,a3,a4,a5,a5)*[1;0;0]..... and have the parametric equations for a cone that rotates about V.
Any ideas for parametrizing a cone that rotates about the arbitrary vector V?
Think about how you would rotate a point using a matrix and then design a rotation matrix that rotates <x,y,z> into <x',y',z'> and take that parametrization (you can keep everything terms of your existing parameterization and that will be in your vector).
There are formulas for this like in here:
Rotation matrix - Wikipedia, the free encyclopedia