4.1 Vector treatment

We are given three 3-vectors , , , defining the three points vertices of the triangle. Using elementary vector subtraction, we find two edges of the triangle, and , using Equations 1 and 2.

$$\mathbf{E}_{1}=\mathbf{P}_{3}-\mathbf{P}_{2}$$ (1)

$$\mathbf{E}_{2}=\mathbf{P}_{2}-\mathbf{P}_{1}$$ (2)

The definition of the surface normal vector is a unit-length vector which is perpendicular to any vector in the plane of the surface. Thus an unnormalised surface normal vector can be obtained by taking the cross-product of any two vectors in the plane of the surface. As and are both vectors in the plane of the surface, Equation 3 gives one such unnormalised surface normal vector.

$$\mathbf{n}=\mathbf{E}_{1}\times \mathbf{E}_{2}$$ (3)

The unnormalised surface normal vector of Equation 3 is now normalised in Equation 4, where is used to denote the magnitude of the vector .

$$\hat{\mathbf{n}}=\frac{1}{\left\vert \mathbf{n}\right\vert }\mathbf{n}$$ (4)

Figure 9: Geometry of the given triangle and its surface normal.

If the points , and are defined in a clockwise order and the surface normal is required to be in the ``upward'' direction, as depicted in Figure 9, then the vectors , and form a right-handed set, as is usually desired.