Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 480

B.8.1 Function integration for arc length computation
Given a function f ( x ), Gaussian quadrature can be used to produce an arbitrarily close approximation to

the integral of the function between two values, R a f ðxÞ

, if the function is sufficiently well behaved [ 16 ] .
Gaussian quadrature approximates the integral as a sum of weighted evaluations of the function at spe-
cific values (abscissas). The number of evaluations controls the error in the approximation. In its general
form, Gaussian quadrature incorporates a multiplicative function, W ( x ), which can condition some func-
tions for the approximation ( Eq. B.148 ). Gauss-Legendre integration is a special case in which W ( x )

¼

1.0, and it results in Equation B.149 . The code in Figure B.55 for n ¼ 10 duplicates that used in Chapter 3 ,

Section 2.1 for computing arc length. Figure B.56 gives the code for computing Gauss-Legendre weights
and abscissas for arbitrary n.

Z b

X n
i¼ 1 w i f ðx i Þ

ðWðxÞf ðxÞÞdx (B.148)

a

Z b

X n
i¼ 1 wf ðx i Þ

f ðxÞdx

(B.149)

a

/* -------------------------------------------------------------------
INTEGRATE FUNCTION
use gaussian quadrature to integrate square root of given function in
the given interval */
double integrate_func(polynomial_td *func,interval_td *interval)
{

double x[5]={.1488743389,.4333953941,.6794095682,.8650633666,.9739065285};
double w[5]={.2966242247,.2692667193,.2190863625,.1494513491,.0666713443};
double length, midu, dx, diff;
int i;
double evaluate_polynomial();
double u1,u2;

u1 = interval->u1;
u2 = interval->u2;

midu = (u1+u2)/2.0;
diff = (u2-u1)/2.0;
length = 0.0;
for (i=0; i<5; i++) {
dx = diff*x[i];
length += w[i]*(sqrt(evaluate_polynomial(func,midu+dx)) +
sqrt(evaluate_polynomial(func,midu-dx)));

}
length *= diff;
return (length);

}
FIGURE B.55