Geometric Primitives - 3D Math Primer for Graphics and Game Development - page 327

v1 = v [ 0 ] . z − v [ 2 ] . z ;

v2 = v [ 1 ] . z − v [ 2 ] . z ;

v3 = p . z − v [ 0 ] . z ;

v4 = p . z − v [ 2 ] . z ;

} e l s e i f ( f a b s ( n . y ) > = f a b s ( n . z ) ) {

/ / D i s c a r d y , p r o j e c t onto xz p l a n e

u1 = v [ 0 ] . z − v [ 2 ] . z ;

u2 = v [ 1 ] . z − v [ 2 ] . z ;

u3 = p . z − v [ 0 ] . z ;

u4 = p . z − v [ 2 ] . z ;

v [ 0 ] . x −

v1 = v [ 2 ] . x ;

v [ 1 ] . x − v2 = v [ 2 ] . x ;

v3 = p . x − v [ 0 ] . x ;

v4 = p . x − v [ 2 ] . x ;

} e l s e {

/ / D i s c a r d z , p r o j e c t onto xy p l a n e

u1 = v [ 0 ] . x − v [ 2 ] . x ;

u2 = v [ 1 ] . x − v [ 2 ] . x ;

u3 = p . x − v [ 0 ] . x ;

u4 = p . x − v [ 2 ] . x ;

v1 = v [ 0 ] . y − v [ 2 ] . y ;

v2 = v [ 1 ] . y − v [ 2 ] . y ;

v3 = p . y − v [ 0 ] . y ;

v4 = p . y − v [ 2 ] . y ;

}

/ / Compute denominator , check f o r i n v a l i d

f l o a t denom = v1 ∗ u2 − v2 ∗ u1 ;

i f ( denom == 0 . 0 f ) {

/ / Bogus t r i a n g l e − p r o b a b l y t r i a n g l e has z e r o a r e a

r e t u r n f a l s e ;

}

/ / Compute b a r y c e n t r i c c o o r d i n a t e s

f l o a t oneOverDenom = 1 . 0 f / denom ;

b [ 0 ] = ( v4 ∗ u2 − v2 ∗ u4 ) ∗ oneOverDenom ;

b [ 1 ] = ( v1 ∗ u3 − v3 ∗ u1 ) ∗ oneOverDenom ;

b [ 2 ] = 1 . 0 f − b [ 0 ] − b [ 1 ] ;

/ / OK
r e t u r n t r u e ;

}

Listing 9.6
Computing barycentric coordinates in 3D

Another technique for computing barycentric coordinates in 3D is based
on the method for computing the area of a 3D triangle using the cross
product, which is discussed in Section 9.6.2. Recall that given two edge
vectors e 1 and e 2 of a triangle, we can compute the area of the triangle as

/2. Once we have the area of the entire triangle and the areas of
the three subtriangles, we can compute the barycentric coordinates.

e 1 × e 2