
754 Chapter 16 Numerical Methods
on that triangle be denoted v
T
(x, y) = α
T
x + β
T
y + γ
T
; then the approximating
integral to I is
˜
I =
T
α
2
T
+ β
2
T
Area(T )
Since α
T
and β
T
are linear functions of the interior values v
i
,
˜
I is quadratic in
v
i
. Minimizing a quadratic function can be done by solving a linear system (set
derivatives equal to zero) or by the conjugate gradient method (equivalent to solving
the linear system, but uses root-finding techniques).
16.8 Fast Function Evaluation
A handful of functions that are typically expensive to compute occur frequently in
computer graphics and games applications: computing the length of a vector (re-
quires a square ...