Finding the face of an icosahedron in which a vector falls in

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In summary, the conversation discusses finding the face of an icosahedron that a given vector falls in, where the vector represents a point on the sphere. Various methods are suggested, including checking angles between the vector and vertex vectors, computing a representative vector for each face and selecting the one with the smallest dot product, and projecting the vector onto each face and selecting the face with the largest projection. It is noted that the last method may not always work as the projection onto the correct face could be zero. The conversation ends with a suggestion to search for "voronoi" for further help.
  • #1
machete
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I am thinking about, given an icosahedron (polyhedron of 20 triangular faces) that "represents" a sphere (some may call it a Fuller Projection) and a vector describing a point in the sphere, finding the face of the icosahedron in which the vector falls in.

The vector is unitary, the vectors describing the polyhedron are unitary, so I thought that I could do this by checking the angles between my vector and every vector of the vertices, then checking which three vertex vectors were the nearest, and voila', those are the vertices of my triangular face.

Wrong. In some conditions (near one of the vertices) two of the nearest vertices are not necessarily those of the triangle one would look for.

If any of you knows a way to do this, I would be very grateful!
 
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  • #2
Welcome to PF! Never having done such a calculation, I am not sure of my
ground here, but since no one else has yet replied, here is a suggestion
that might work: Let a, b, and c be unit vectors (relative to an origin at
the centre of the sphere) representing the vertices of one of the
triangular faces of the icosahedron. Then the vector V=a+b+c should be
perpendicular to the triangular face and point to the middle of the face.
Now compute one such representative vector V for each of the faces. I should
think that the correct face is the one whose representative vector has the
smallest dot product with your test vector pointing to some point on the
sphere. Points on edges or vertices of the icosahedron would of course
belong to two or three faces respectively.

If this does not work, then perhaps a Google search for "voronoi" might
help you further. I hope this is of some help in getting started.
 
  • #3
pkleinod said:
Now compute one such representative vector V for each of the faces. I should
think that the correct face is the one whose representative vector has the
smallest dot product with your test vector pointing to some point on the
sphere.

Oops. I meant the LARGEST dot product.
 
  • #4
You could project the vector onto the plane of each face, and see which plane has the largest projection.

Edit: I think this is the same as what pleknoid is saying.
 
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  • #5
maze said:
You could project the vector onto the plane of each face, and see which plane has the largest projection.
Hmmm. Does this really work? Consider a point on the sphere whose vector is perpendicular to one of the faces (i.e. the reference vector for the face, as defined in my post above). The projection of this vector onto its face would be zero, whereas the projection onto the other faces (excluding an opposite face) would not be zero. i.e. the largest projection would select the wrong face.
 

Related to Finding the face of an icosahedron in which a vector falls in

1. What is an icosahedron?

An icosahedron is a three-dimensional geometric shape that has 20 triangular faces, 30 edges, and 12 vertices.

2. How can a vector fall within an icosahedron?

A vector can fall within an icosahedron if its direction aligns with one of the triangular faces of the shape.

3. What is the significance of finding the face of an icosahedron in which a vector falls in?

Finding the face of an icosahedron in which a vector falls in can help determine the direction and magnitude of the vector, as well as its relationship to the shape's geometry.

4. How is this process useful in scientific research?

This process is useful in various fields of science, such as physics and engineering, where understanding the direction and magnitude of vectors is important in analyzing and predicting phenomena.

5. Are there any real-life applications for this concept?

Yes, this concept is used in various applications, such as computer graphics and animation, navigation systems, and satellite tracking, to name a few.

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