<p>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. <a href="http://bmia.bmt.tue.nl/People/BRomeny/Courses/8C080/Interpolation.pdf" rel="nofollow noreferrer">http://bmia.bmt.tue.nl/People/BRomeny/Courses/8C080/Interpolation.pdf</a></p>

CC BY-SA 3.0
2017-10-09T20:15:46.180Z
2017-10-09T20:15:46.180Z
user5884
<p>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. </p>

CC BY-SA 4.0
2018-06-07T06:35:22.193Z
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
CC BY-SA 4.0
2018-06-07T08:11:44.587Z
or this link:http://ebrary.free.fr/Mesh%20Generation/Handbook_of_Grid_%20Generation,1999/chap03.pdf
ali
CC BY-SA 4.0
2019-05-03T07:45:29.243Z
I'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
CC BY-SA 4.0
2023-05-18T12:49:00.340Z
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
<mathematics>
Barycentric coordinates for 5-sided polytope of two triangles and three trapezoids
user5884
# Barycentric coordinates for 5-sided polytope of two triangles and three trapezoids

(Asked by: username_0 on 2017-10-09)

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.

**Tags:** mathematics

> **Comments:**
> **username_1**: 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  
> **username_1**: or this link:http://ebrary.free.fr/Mesh%20Generation/Handbook_of_Grid_%20Generation,1999/chap03.pdf  
> **username_2**: I'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?  
> **username_3**: 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.  

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## 1 answers

### Answer by username_0 on 2017-10-09. Score: 0

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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