I have a triangle that is extended in space and each edge is refracted differently, creating a 5 sided polytope. The initial and final triangles are the view radii segments of a frustum. We need to get the u and v along the triangle and the w depth in between the triangles of a point inside the polytope.
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You also can use Transfinite interpolation where you take the refracted triangle as a four sides polygon by having vertices at sides. https://en.wikipedia.org/wiki/Transfinite_interpolation – ali Jun 07 '18 at 06:35
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or this link:http://ebrary.free.fr/Mesh%20Generation/Handbook_of_Grid_%20Generation,1999/chap03.pdf – ali Jun 07 '18 at 08:11
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2I'm not clear on what shape you're trying to parameterize. Is it an oblique triangular prism? What's a refracted edge? Are the triangles parallel? Could you draw a picture of the shape? – gilgamec May 03 '19 at 07:45
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the term polytope is a broad term that covers a wide class of objects, and various definitions appear in the mathematical literature. Many of these definitions are not equivalent to each other, resulting in different overlapping sets of objects being called polytopes. – pmw1234 May 18 '23 at 12:49
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Let the two triangles be $ABC$ and $DEF$. They define planar frames to which you attach coordinates $(u,v)$, giving $P=A+u\vec{AB}+v\vec{AC}$ and $Q=D+u\vec{DE}+v\vec{DF}$. The triangles are filled with $u,v\ge0,u+v\le1$. Now for $w\in[0,1]$, interpolate to get $R=(1-w)P+wQ$. This is indeed trilinear interpolation.
Holding two coordinates constant, you get a line segment. Holding a single coordinate constant, you get a planar patch.
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I think I can do this using trilinear interpolation and treating two vertices on each side as merged and using my own volume calculations. I just need to interpolate some values given by the three points and a z. http://bmia.bmt.tue.nl/People/BRomeny/Courses/8C080/Interpolation.pdf
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