GPU Pro 6

32 I Geometry Manipulation

function D(u, v) is then evaluated using the B-spline basis functions B

2

i

:

D(u, v)=

2



i=0

2



j=0

B

2

i

(T (u))B

2

j

(T (v))d

i,j

,

where the subpatch domain parameters ˆu, ˆv are given by the linear transforma-

tion T ,

ˆu = T (u)=u −u +

1

2

and ˆv = T (v)=v −v +

1

2

.

In order to obtain the displaced surface normal N

f

(u, v), the partial deriva-

tives of f(u, v) are required:

∂

∂u

f(u, v)=

∂

∂u

s(u, v)+

∂

∂u

N

s

(u, v)D(u, v)+N

s

(u, v)

∂

∂u

D(u, v).

In this case,

∂

∂u

N

s

(u, v) would involve the computation of the Weingarten equa-

tion, which is costly. Therefore, we approximate the partial derivatives of f(u, v)

(assuming small displacements) by

∂

∂u

f(u, v