In

engineering analysis it is often necessary to

numerically evaluate surface integrals over a particular geometric shape. The

triangle is one of the most widely used shapes used to evaluate surface integrals arising in techniques such as the

Finite Element Method (

FEM) and the

Method of Moments (

MoM) in

electromagnetics.

**Numerical Quadrature Over Triangles** is typically done by sampling the
function being integrated at discrete points on the triangle, multiplying each samples by specially chosen quadrature weights, and summing these up, i.e.

I = (1/A) ∫_{s} F(**r**_{i}) ds ~
Σ_{i} w_{i} F(**r**_{i})

where F(**r**_{i}) is the function sampled at quadrature point
**r**_{i} on the triangle, and w_{i} is the quadrature weight for that point. The quadrature operation results in a quantity that is normalized by the triangle's area A, hence the (1/A) coefficient.

I will present a set of quadrature points and weights useful for performing typical quadratures. These quadrature points are generalized into what are called **barycentric coordinates**, also commonly referred to as area coordinates. Given a triangle defined by the three vertexes **v**_{1}, **v**_{2} and **v**_{3}, any point **r** inside that triangle can be written in terms of the barycentric coordinates β_{1}, β_{2} and β_{3} as

**r** = β_{1} **v**_{1} + β_{2} **v**_{2} + β_{3} **v**_{3}

Given a set of barycentric coordinates (quadrature points), each location **r**_{i} on the triangle may be found and the integrated function sampled. The weights to be presented are what are called Gauss-Legendre quadrature weights, the derivation of which I will not show here. Read numerical integration for a more in-depth discussion on this.

Below are tables of sampling points and weights. The points are chosen
to lie in the interior of the triangle. There exist quadrature formulas
that have points on the triangle edge, however they are not suitable for the analysis that I do so I don't have them on hand.

One-point quadrature (center of triangle):
--------------------------------------------------
i beta_1 beta_2 beta_3 w_i
--------------------------------------------------
1 0.33333333 0.33333333 0.33333333 1.00000000
Four-point quadrature:
--------------------------------------------------
i beta_1 beta_2 beta_3 w_i
--------------------------------------------------
1 0.33333333 0.33333333 0.33333333 0.28125000
2 0.73333333 0.13333333 0.13333333 0.26041666
3 0.13333333 0.73333333 0.13333333 0.26041666
4 0.13333333 0.13333333 0.73333333 0.26041666
Seven-point quadrature:
--------------------------------------------------
i beta_1 beta_2 beta_3 w_i
--------------------------------------------------
1 0.33333333 0.33333333 0.33333333 0.22500000
2 0.05961587 0.47014206 0.47014206 0.13239415
3 0.47014206 0.05961587 0.47014206 0.13239415
4 0.47014206 0.47014206 0.05961587 0.13239415
5 0.79742699 0.10128651 0.10128651 0.12593918
6 0.10128651 0.79742699 0.10128651 0.12593918
7 0.10128651 0.10128651 0.79742699 0.12593918

May the Force Be With You. Go evaluate some integrals over triangles!