Find the projection of the vector on the icosahedron or on its expanded surface

Question

Consider a sphere centered at the origin and an inscribed icosahedron oriented so that the two most distant vertices lie on the Z coordinate axis, and one of the edges emerging from any of these vertices lies in the XZ plane.

Consider a given vector xoccurring in the center of the sphere. The direction given by this vector pierces the surface of the icosahedron at a certain point.

Is there an elegant way to find a punctured face (given all the faces listed) and then find the piercing point on that face? This task is necessary for projecting the piercing point onto the flat unfolded (undisclosed) surface of the icosahedron.

Answer 1

, , - .

20 , 12 . 3 , , . 3- . 3x3, :

    [ x1 x2 x3 ]
T = [ y1 y2 y3 ]
    [ z1 z2 z3 ]

( SO google 3x3 ). 20 . :

B = V * T ()

3 B (?), .

, B - , . B . I, :

I = B * T

2D- . 2x3 T.

. 3 , . , . , . , , , . 1 , .

, , .

Answer 2

, , , . 12 - 10 , ( , ).

, , - - , - . , . ( , ), , - , .

. , (, , ). angular , , , , - , .

, ... . , , .