Surface integral over an arbitrary triangle in 3D

Question

I have a triangular surface given in 3D with nodes $$[x_1,y_1,z_1]^T, [x_2,y_2,z_2]^T,[x_3,y_3,z_3]^T$$ I want to calculate the surface integral of a function $$\oint_s \vec{f}(x,y,z)\bullet d\vec{S}$$ How can I achieve this? Thanks in advance and have a nice day.

Answer 1

One way how to do it is to parametrize the points of a triangle as $$\vec P=\vec P_1+(\vec P_2-\vec P_1)u+(\vec P_3-\vec P_1)v.$$ I assume the orientation of triangle is consistent with numeration of vertices $123$. The normal $d\vec S$ is constant and can be found using vector product: $$ d\vec{S} = \partial_u\vec P\times\partial_v\vec P = (\vec P_2-\vec P_1)\times(\vec P_3-\vec P_1)\,du\,dv = \vec n\,du\,dv $$

So the integral can be found as: $$ \int_S\vec f(x,y,z)\cdot d\vec S = \int_0^1\int_0^{1-u} \vec n\cdot \vec f\Big(x(u,v),y(u,v),z(u,v)\Big)du\,dv, $$ where functions $x(u,v),y(u,v),z(u,v)$ are the coordinates of $\vec P(u,v)$