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How to determine the coordinates of the hemisphere xyz?

I am having serious problems resolving the problem illustrated in the figure below. Say we have 3 points in three-dimensional space (blue points) and some center of the triangle based on them (red point of P). We also have a normal to this triangle, so we know which half-space we are talking about.

I need to determine what is the position at the point (red dot), which depends on two angles, as in the range of 0-180 degrees. It does not matter how it is “attached” to alpha = 0 and betha = 0, it is only important to be able to scan the entire hemisphere (radius r).

http://i.stack.imgur.com/a1h1B.png

If anyone could help me, I would be very grateful.

Regards, Rav

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2 answers

It can be seen from the figure that the position of the point on the sphere is determined by the shape of the spherical coordinates . Let be the rradius of the sphere; let it be alphaset relative to the x axis; and let betabe the angle relative to the xy plane. The Cartesian coordinates of a point on a sphere are:

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

Edit

But for a common coordinate system with axes (L, M, N)centered on the (X, Y, Z)coordinates (as in dmuir answer):

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

Axes Land Nmust be orthogonal and M = cross(N, L). alphais set relatively L, and betais set relative to the plane L- M. If you do not know how it Lis connected with the points of the triangle, then the question cannot be answered.

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L, M, , , N.

r * cos () * cos () * ​​L + r * cos () * sin () * ​​M + r * sin () * N

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