Calculate the volume of on object given by a set of points.

Object: Tetrahedron Cube Octohedron Dodecahedron Icosahedron Random 10 Volume: calculated actual

The volume is calculated by using the Divergence theorem.

${\displaystyle {\iiint }_V(\mathrm{\nabla }\cdot \mathbf{F})dV={\iint }_S(\mathbf{F}\cdot \mathbf{n})dS}$

where the left hand integral is over the internal volume and the right hand integral is over the surface, $\mathbf{n}$ is the outward unit length surface normal, $\mathrm{\nabla }$ is the divergence, $\mathrm{div}\mathbf{F}=\mathrm{\nabla }\cdot \mathbf{F}=(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z})\cdot \mathbf{F}$

If $F$ is the field giving the position vector $\mathbf{F}(x,y,z)=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$ then $\mathrm{div}\mathbf{F}=3$, so the left hand integral is just ${\iiint }_{V}3dV$, three times the volume.

Hence to find the volume of a surface we just need to calculate ${\iint }_S(\mathbf{F}\cdot \mathbf{n})dS$. If the surface is polyhedron with set pologonal faces $P_1,\dots P_n$ the integral can be reduced to a sum over all the faces. The dot product $\mathbf{F}\cdot \mathbf{n}$ is constant for all points on the face, so the integral over any face $P$ is just $area(P)\mathbf{v}\cdot \mathbf{n}$ where $\mathbf{v}$ is any vertex on the surface. Hence the volume is given by

${\displaystyle \frac{1}{3}\sum _{i=1}^{n}area(P_i){\mathbf{v}}_i\cdot {\mathbf{n}}_i}$

where ${\mathbf{v}}_i$ is a vectex of the i-th face and ${\mathbf{n}}_1,\dots ,{\mathbf{n}}_n$ are the outward pointing normals.

If ${\mathbf{a}}_i$, ${\mathbf{b}}_i$, ${\mathbf{c}}_i$ are the three vertices of a triangular face the (non unit length) normal can be calculate as ${\mathbf{n}}_i=({\mathbf{b}}_i-{\mathbf{a}}_i)\times ({\mathbf{c}}_i-{\mathbf{a}}_i)$. The unit length normal is ${\hat{\mathbf{n}}}_i={\mathbf{n}}_i/|{\mathbf{n}}_i|$ and area of the triangle is $\frac{1}{2}|{\mathbf{n}}_i|$. The dot product becomes ${\mathbf{a}}_i\cdot {\hat{\mathbf{n}}}_i$. The whole volume becomes

${\displaystyle \frac{1}{6}\sum _{i=1}^n\pm {\mathbf{a}}_i\cdot {\mathbf{n}}_i}$,

where the signs are chosen so the normals are outwards.

Readers comments

vectex of the ith face

David Seed Fri Jun 17 2016

please give a simple workd example. with a diagram i presume you mean any vertex in the i-th face and that v.n represents the vertical dstance from the face to the origin. but perhaps the outward facng sense requires the origin to be outside the solid i dont

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