CODEWITHSUNDEEP
computational-geometrylinear-algebra
emesx
emesx

Reputation: 12755

How to determine a semi-sphere's point x-y-z coordinates?

I'm having serious problems solving a problem illustrated on the pic below. Let's say we have 3 points in 3D space (blue dots), and the some center of the triangle based on them (red dot - point P). We also have a normal to this triangle, so that we know which semi-space we talking about.

I need to determine, what is the position on a point (red ??? point) that depends on two angles, both in range of 0-180 degrees. Doesnt matter how the alfa=0 and betha=0 angle is "anchored", it is only important to be able to scan the whole semi-sphere (of radius r).

https://i.sstatic.net/a1h1B.png

If anybody could help me, I'd be really thankful.

Kind regards, Rav

Upvotes: 3

Views: 1033

Answers (2)

antonakos
antonakos

Reputation: 8361

From the drawing it looks as if the position of the point on the sphere is given by a form of spherical coordinates. Let r be the radius of the sphere; let alpha be given relative to the x-axis; and let beta be the angle relative to the x-y-plane. The Cartesian coordinates of the point on the sphere are:

x = r * cos(beta) * cos(alpha)
y = r * cos(beta) * sin(alpha)
z = r * sin(beta)

Edit

But for a general coordinate frame with axes (L, M, N) centered at (X, Y, Z) the coordinates are (as in dmuir's answer):

(x, y, z) = 
   (X, Y, Z) 
   + r * cos(beta) * cos(alpha) * L 
   + r * cos(beta) * sin(alpha) * M 
   + r * sin(beta) * N

The axes L and N must be orthogonal and M = cross(N, L). alpha is given relative to L, and beta is given relative to the L-M plane. If you don't know how L is related to points of the triangle, then the question can't be answered.

Upvotes: 1

dmuir
dmuir

Reputation: 644

You need to find two unit length orthogonal vectors L, M say, in the plane of the triangle as well as the the unit normal N. The points on the sphere are

r*cos(beta)*cos(alpha) * L + r*cos(beta)*sin(alpha)*M + r*sin(beta)*N

Upvotes: 0

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