Tetrahedron voronoi cells volume
Given a tetrahedron described with it's vertices, for every vertex I need to find which fraction of tetrahedron's volume is closer that vertex than any other.
I managed to solve same problem for a triangle considering only two cases:
For acute triangle you can find radius of circumcircle that is a hypotenuse for six triangles you can split triangle into, then just compute their areas using known hypotenuse and leg
For obtuse triangle you can find areas belonging to acute angles which are right triangle with one known leg and an angle and subtract them from full area for area closes to last angle
But there seems to be overwhelmingly many cases for tetrahedron. Is there any solution not involving actually intersecting tetrahedron with all orthogonal bisecting planes.
geometry
asked Nov 25 at 1:46
SteelRaven
26625
add a comment |
Given a tetrahedron described with it's vertices, for every vertex I need to find which fraction of tetrahedron's volume is closer that vertex than any other.
I managed to solve same problem for a triangle considering only two cases:
For acute triangle you can find radius of circumcircle that is a hypotenuse for six triangles you can split triangle into, then just compute their areas using known hypotenuse and leg
For obtuse triangle you can find areas belonging to acute angles which are right triangle with one known leg and an angle and subtract them from full area for area closes to last angle
But there seems to be overwhelmingly many cases for tetrahedron. Is there any solution not involving actually intersecting tetrahedron with all orthogonal bisecting planes.
geometry
asked Nov 25 at 1:46
SteelRaven
26625
1
Sorry, for 1) what does it mean that you can find areas belonging to acute angles which are "right triangle"? I just google Voronoi cells, it seems soo interesting. and how do you divide triangle into 6 smaller triangle in 2)?
– mathnoob
Nov 25 at 2:23
1
@mathnoob thanks for images, so in first case you can divide it like in third picture and get areas from radius and half of side, and in second case you can find $S_b$ for example as you know $beta$ and $c/2$.
– SteelRaven
Nov 25 at 2:43
add a comment |
Given a tetrahedron described with it's vertices, for every vertex I need to find which fraction of tetrahedron's volume is closer that vertex than any other.
I managed to solve same problem for a triangle considering only two cases:
For acute triangle you can find radius of circumcircle that is a hypotenuse for six triangles you can split triangle into, then just compute their areas using known hypotenuse and leg
For obtuse triangle you can find areas belonging to acute angles which are right triangle with one known leg and an angle and subtract them from full area for area closes to last angle
But there seems to be overwhelmingly many cases for tetrahedron. Is there any solution not involving actually intersecting tetrahedron with all orthogonal bisecting planes.
geometry
asked Nov 25 at 1:46
SteelRaven
26625
Given a tetrahedron described with it's vertices, for every vertex I need to find which fraction of tetrahedron's volume is closer that vertex than any other.
I managed to solve same problem for a triangle considering only two cases:
For acute triangle you can find radius of circumcircle that is a hypotenuse for six triangles you can split triangle into, then just compute their areas using known hypotenuse and leg
For obtuse triangle you can find areas belonging to acute angles which are right triangle with one known leg and an angle and subtract them from full area for area closes to last angle
But there seems to be overwhelmingly many cases for tetrahedron. Is there any solution not involving actually intersecting tetrahedron with all orthogonal bisecting planes.
geometry
geometry
asked Nov 25 at 1:46
SteelRaven
26625
asked Nov 25 at 1:46
SteelRaven
26625
edited Nov 25 at 2:40
asked Nov 25 at 1:46
SteelRaven
26625
asked Nov 25 at 1:46
SteelRaven
26625
asked Nov 25 at 1:46
SteelRaven
26625
26625
1
Sorry, for 1) what does it mean that you can find areas belonging to acute angles which are "right triangle"? I just google Voronoi cells, it seems soo interesting. and how do you divide triangle into 6 smaller triangle in 2)?
– mathnoob
Nov 25 at 2:23
1
@mathnoob thanks for images, so in first case you can divide it like in third picture and get areas from radius and half of side, and in second case you can find $S_b$ for example as you know $beta$ and $c/2$.
– SteelRaven
Nov 25 at 2:43
add a comment |
1
Sorry, for 1) what does it mean that you can find areas belonging to acute angles which are "right triangle"? I just google Voronoi cells, it seems soo interesting. and how do you divide triangle into 6 smaller triangle in 2)?
– mathnoob
Nov 25 at 2:23
1
@mathnoob thanks for images, so in first case you can divide it like in third picture and get areas from radius and half of side, and in second case you can find $S_b$ for example as you know $beta$ and $c/2$.
– SteelRaven
Nov 25 at 2:43
1
1
Sorry, for 1) what does it mean that you can find areas belonging to acute angles which are "right triangle"? I just google Voronoi cells, it seems soo interesting. and how do you divide triangle into 6 smaller triangle in 2)?
– mathnoob
Nov 25 at 2:23
Sorry, for 1) what does it mean that you can find areas belonging to acute angles which are "right triangle"? I just google Voronoi cells, it seems soo interesting. and how do you divide triangle into 6 smaller triangle in 2)?
– mathnoob
Nov 25 at 2:23
1
1
@mathnoob thanks for images, so in first case you can divide it like in third picture and get areas from radius and half of side, and in second case you can find $S_b$ for example as you know $beta$ and $c/2$.
– SteelRaven
Nov 25 at 2:43
@mathnoob thanks for images, so in first case you can divide it like in third picture and get areas from radius and half of side, and in second case you can find $S_b$ for example as you know $beta$ and $c/2$.
– SteelRaven
Nov 25 at 2:43
add a comment |
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1
Sorry, for 1) what does it mean that you can find areas belonging to acute angles which are "right triangle"? I just google Voronoi cells, it seems soo interesting. and how do you divide triangle into 6 smaller triangle in 2)?
– mathnoob
Nov 25 at 2:23
1
@mathnoob thanks for images, so in first case you can divide it like in third picture and get areas from radius and half of side, and in second case you can find $S_b$ for example as you know $beta$ and $c/2$.
– SteelRaven
Nov 25 at 2:43