Expo Rational B-spline=
s
– ERBS (under constructions) <=
o:p>
<=
o:p>
T=
his page
is under construction and will be official published in May 2007. For the
moment there is only a very short introduction and the code of a C++ class =
for
an ERBS evaluator (at the end).
<=
o:p>
<=
o:p>
An “Expo-Rational B-spl=
ine ”
is related to a polynomial B-spline . It is defi=
ned by
a knot vector and a set of 5 intrinsic parameters.
=
=
=
=
=
ERBS is supposed
to be used in compound functions to define curves and surfaces.
<=
o:p>
<=
o:p>
<=
o:p>
In the example above is the red curve an ERBS -curve, while the green curves are local curves of
Bezier type. An ERBS -curve is a curve made by
blending local curves with domains defined by a knot vector.
<=
o:p>
<=
o:p>
<=
o:p>
<=
o:p>
<=
o:p>
This example surface is=
a
connected set of eight triangular Expo-Rational B-spli=
ne
surfaces. This surface is made by first approximating a sphere. Then the
surface is deformed by an editing process.
The use of the following
evaluator class will be described later.
//*******************************=
********************************************************* =
// GM_ERBSeval.h: interface for t=
he
ERBSeval class.
//*******************************=
********************************************************* =
#ifndef GM_ERBSEVAL_H_
#define GM_ERBSEVAL_H_
class ERBSeval
{
&=
nbsp; struct Double5
&=
nbsp; {
&=
nbsp; double p[5];
&=
nbsp; double & operator=
[](int i) {return p[i];}
&=
nbsp; };
=
int =
j;
double &nb=
sp; dt;
double &nb=
sp; _dt,
_sk, _sk1, _sk2;
int =
_m,
_der;
double &=
nbsp; _tk,
_dtk;
&=
nbsp; double * &n=
bsp; _b;
&=
nbsp; Double5* _a;
&=
nbsp; double &nb=
sp; alpha,
beta, gamma, lambda;
&=
nbsp; double phi(double t)
&=
nbsp; {
&=
nbsp; double d =3D pow(t*pow(1-t,gamma), alpha);=
&=
nbsp; if (d < 2.3e-308)
&=
nbsp; &nbs=
p; return 0.0;
&=
nbsp; else
&=
nbsp; &nbs=
p; return exp(-beta*pow(abs(t-lambda),
alpha*(1+gamma))/d);
&=
nbsp; }
&=
nbsp; double f2(double t)
&=
nbsp; {
&=
nbsp; if (t=3D=3Dlambda || t<2.3e-308 || t=3D=3D1.0=
) return
0.0;
&=
nbsp;
&=
nbsp; double h =3D (t-1/(1+gamma))*abs(t-lambda)/(t*(1-t));
&nbs=
p; if (t<lambda) =
h
-=3D 1.0;
&=
nbsp; else =
; h
+=3D 1.0;
&=
nbsp; h *=3D -_sk*alpha*beta*(1+gamma)/pow(t*pow(1-t,gamma),=
alpha);
&nbs=
p; if ((1+gamma)*alpha < 1)
&=
nbsp; {
&=
nbsp; &nbs=
p; double g =3D pow(abs(t-lambda),1-(1+gamma)*alph=
a)/h;
&=
nbsp; &nbs=
p; if (g < 2.3e-308) return 0.0;
&=
nbsp; &nbs=
p; else =
; &n=
bsp; return 1/g;
&=
nbsp; }
&=
nbsp; else if
((1+gamma)*alpha > 1)
&=
nbsp; &nbs=
p; return &nb=
sp; h
* pow(abs(t-lambda),(1+gamma)*alpha-1);
&=
nbsp; else
&=
nbsp; &nbs=
p; return &nb=
sp; h;
&=
nbsp; }
&=
nbsp; double integral(doub=
le
a, double b, d=
ouble
sum, double eps)
&=
nbsp; {
&=
nbsp; double M[16][16];
&=
nbsp; int i,j;
// double
sum =3D 0.5*(phi(a)+phi(b));
&=
nbsp; double t =
=3D b-a;
&=
nbsp; M[0][0] =3D sum*t;
&=
nbsp; for (i=3D1; i<16; i++)
&=
nbsp; {
&=
nbsp; &nbs=
p; double s =3D 0.0;
&=
nbsp; &nbs=
p; int =
;
k =3D 1 << i;
&=
nbsp; &nbs=
p; t
/=3D 2.0;
&=
nbsp; &nbs=
p; for (j=3D1; j<k; j+=3D2) s +=3D phi(a+t*j);
&=
nbsp; &nbs=
p; M[0][=
i] =3D
t*(sum +=3D s);
&=
nbsp; for (j=3D1; j<=3Di; j++)
&=
nbsp; {
&=
nbsp; &nbs=
p; b
=3D 1 << (j << 1);
&=
nbsp; &nbs=
p; M[j][i-j]
=3D (b*M[j-1][i-j+1]-M[j-1][i-j])/(b-1.0);
&=
nbsp; }
&=
nbsp; &nbs=
p; if (abs(M[i][0]-M[i-1][0]) < eps) return M[i][0];
&=
nbsp; }
&=
nbsp; return M[15][0];
&=
nbsp; }
&=
nbsp;
&=
nbsp; void interpolate(int=
i, double p0, =
double
p1, double f0, double
f1)
&=
nbsp; {
&=
nbsp; _a[i][0]
=3D p0;
&=
nbsp; _a[i][1]
=3D f0;
&=
nbsp; _a[i][2]
=3D 3*(p1-p0) - 2*f0 - f1;
&=
nbsp; _a[i][3]
=3D -2*(p1-p0) + f0 + f1;
&=
nbsp; _a[i][4]
=3D (p0+p1)/2 + (f0-f1)/12;
&=
nbsp; }
&=
nbsp; void _init(int m)
&=
nbsp; {
&=
nbsp; int i,j;
&=
nbsp; double p0, p1;
&=
nbsp; double f0, f1;
&=
nbsp; _b =3D new
double [m+1];
&=
nbsp; _a =3D new
Double5[m];
&=
nbsp; _m =3D m;
&=
nbsp; _dt =3D 1.0/m;
&=
nbsp; _sk =3D 1.0;
&=
nbsp; _b[0]
=3D p0 =3D f0 =3D 0.0;
&=
nbsp; for (i=3D1; i<m; i++)
&=
nbsp; {
&=
nbsp; double t =3D _dt*i;
&=
nbsp; p1
=3D phi(t);
&nbs=
p; &=
nbsp; f1 =3D f2(t)*p1;
&nbs=
p; &=
nbsp; _b[i]
=3D integral(t-_dt, t, 0.5*(p0+p1), 1e-17);
&nbs=
p; &=
nbsp; inter=
polate(i-1,
p0, p1, _dt*f0, _dt*f1);
&=
nbsp; p0 =3D p1;
&=
nbsp; f0 =3D f1;
&=
nbsp; }
&=
nbsp; _b[m]
=3D integral(_dt*(m-1), 1, 0.5*p0, 1e-17);
&=
nbsp; interpolate(m-1,
p0, p1, _dt*f0, 0.0);
&=
nbsp; for (i=3D1; i<=3Dm; i<<=3D1)
&=
nbsp; for (j=3Dm; j>=3Di; j-=3D1)
&=
nbsp; &nbs=
p; _b[j]
+=3D _b[j-i];
&=
nbsp; _sk =3D 1/_b[m];
&=
nbsp; }
public :
&=
nbsp; ERBSeval(int n, double =
a=3D1, double b=3D1, double=
g=3D1, double l=3D0.5)
&=
nbsp; {
&=
nbsp; _der
=3D 1;
&=
nbsp; set(a,b,g,l);
&=
nbsp; _init(n);
&=
nbsp; set_t(0,1);
&=
nbsp; }
&=
nbsp; virtual
&=
nbsp; ~ERBSeval()
&=
nbsp; {
&=
nbsp; delete _b;
&=
nbsp; delete _a;
&=
nbsp; }
&=
nbsp; double operator()(double t)
&=
nbsp; {
&=
nbsp; t =3D (t-_tk)/(_dtk); =
// translere/skalere parameteren 1 <=
/span>
&=
nbsp; j =3D min(int (t*_m),_m-1); &=
nbsp; // finner lokale intervallet
&=
nbsp; dt =3D (t - j*_dt)/_dt; &=
nbsp; // translere/skalere parameteren 2 <=
/span>
&=
nbsp; if (dt > 0.5)
&=
nbsp; &nbs=
p; return _sk*(_b[j+1] -
_dt*(_a[j][4]-dt*(_a[j][0]+dt*(_a[j][1]/2+dt*(_a[j][2]/3+dt*_a[j][3]/4)))))=
;
&=
nbsp; else
&=
nbsp; &nbs=
p; return _sk*(_b[j] +
_dt*dt*(_a[j][0]+dt*(_a[j][1]/2+dt*(_a[j][2]/3+dt*_a[j][3]/4))));
&=
nbsp; }
&=
nbsp; double der1(double t) { =
return
_sk*phi(t)/_dtk; }
&=
nbsp; double der2(double t) { =
double
p =3D phi(t); if (p < 1e-300) return 0;
&=
nbsp; &nbs=
p; &=
nbsp; else return
_sk*p*(1-2*t)/(4*t*t*(1-t)*(1-t)*_dtk*_dtk);}
&nbs=
p; double der1() =
{
return _sk1*(_a[j][0] + dt*(_a[j][1]+dt*(=
_a[j][2]+dt*_a[j][3])));
}
&=
nbsp; double der2()&n=
bsp; {
return _sk2*(_a[j][1] +
dt*(2*_a[j][2]+dt*3*_a[j][3])); }
&=
nbsp; double Sk()&nbs=
p; {
return _sk; }
&=
nbsp; double der(int i)
&=
nbsp; {
&=
nbsp; switch (i)
&=
nbsp; {
&=
nbsp; case 1: return
der1();
&=
nbsp; case 2: return
der2();
&=
nbsp; }
&=
nbsp; }
&=
nbsp; void set(double a=3D1, double b=3D1, double=
g=3D1, double l=3D0.5) <=
/p>
&=
nbsp; { alp=
ha=3Da;
beta=3Db; gamma=3Dg; lambda=3Dl; }
=
void set_t(double tk, double dtk)
&=
nbsp; {_tk=3Dtk;
_dtk=3Ddtk; _sk1 =3D _sk/_dtk; _sk2=3D_sk/(_dt*_dtk*_dtk);}
};
template <class T>
T round(T t)
{
&nbs=
p; if (abs(t)<T(1e-6)) return T(0);
&nbs=
p; else &=
nbsp; &nbs=
p; return t;
}
#endif // GM_ERBSEVAL_H_
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