Projection/Cone

Whiteout

Hi,

I am trying to project a point x in 3D Space onto a point x' on the surface of a cone. x' should be the point on the cone's surface closest to the original point x such that the normalized vector x-x' is the cone's surface normal in the point x'.

The Cone is parameterize in two distinct ways...

(1) - the cone's apex a e R

Josh Code

Maybe this high level overview of a solution would help.

Simplify the problem. For example, you could rotate the whole space around such that the axis of the cone is the y axis. You could also transform so that the point x is on a plane like z=0. Don't forget this transformation because it will have to be reversed after you solve this simpler problem.

Look at that plane z=0. The cone's axis is on it along with the point x. The cone intersects the plane along 2 lines. Also, the closest point between x and the cone is on this plane too. In fact, the closest point between x and the cone is the same as the closest point between x and those lines.

Solve for the closest point on the plane. You can use basic linear algebra for this. The coordinates would be of the form (X,Y,0).

Now, perform the reverse of the earlier transformations on (X,Y,0) to get your point x prime.

I'm working on a ray tracer that uses a similar technique to make it easier to implement the scaling, translation, rotation... of shapes while each shape's ray tracing functions are implemented assuming it is centred on the origin, unscaled, and not rotated.


: Hi,
:
: I am trying to project a point x in 3D Space onto a point x' on the
: surface of a cone. x' should be the point on the cone's surface
: closest to the original point x such that the normalized vector x-x'
: is the cone's surface normal in the point x'.
:
: The Cone is parameterize in two distinct ways...
:
: (1) - the cone's apex a e R?
: - the cone's normal n e R?
: - the cone's opening semi-angle alpha
:
: (2) - a point p e R? on the cone's axis but not the apex
: - a normal n e R? pointing from p towards the axis
: - the ortohgonal distance s from from p to the cone's surface
: - the opening semi-angle alpha
:
: I have thought quite a lot about this problem but have not come up
: with a satisfying result. Does anybody know a solution and is
: willing to help out?
:
: Thanks a lot...
: