Circle and sphere by NURBS functions of OpenGL

OpenGL
1

It is known that circles and spheres can be represented exactly by NURBS curve and surface respectively. I confirmed that this works correctly using the algorithm of NURBS. But, I could not obtain the correct results using the NURBS functions of OpenGL. I cannot understand why. Attached below is the image comparing the correct result and wrong result created by the OpenGL NURBS function.

OGLImage
OGLImage518×670 33.3 KB

The following is the OpenGL code.

void _OpenGLDrawCircleByNURBS(void)
{
	GLfloat pXyz[9][4]={ 1.0, 0.0, 0.0,  1.0,
					  	 1.0, 1.0, 0.0,  0.707107,
					 	 0.0, 1.0, 0.0,  1.0,
					    -1.0, 1.0, 0.0,  0.707107, 
					    -1.0, 0.0, 0.0,  1.0,
					    -1.0,-1.0, 0.0,  0.707107,
					     0.0,-1.0, 0.0,  1.0, 
					     1.0,-1.0, 0.0,  0.707107,
					     1.0, 0.0, 0.0,  1.0}; 
	GLfloat uKnotVect[]={0.0, 0.0, 0.0, 0.25, 0.25, 0.5, 0.5, 0.75, 0.75, 1.0, 1.0, 1.0};
	GLUnurbsObj *theNurb;

	theNurb=gluNewNurbsRenderer();
	gluNurbsProperty(theNurb,GLU_SAMPLING_TOLERANCE,25.0);
	gluNurbsProperty(theNurb,GLU_DISPLAY_MODE,GLU_OUTLINE_PATCH);
	gluBeginCurve(theNurb);
	gluNurbsCurve(theNurb,
				  12,uKnotVect,
				  4,
				  &pXyz[0][0],
			 	  3,
			   	  GL_MAP1_VERTEX_4);
	gluEndCurve(theNurb);
}
2

Multiply X/Y/Z by W. E.g.

for (int i=0; i<9; i++)
    for (int j=0; j<3; j++)
        pXyz[i][j] *= pXyz[i][3];

The book is treating w as a weight, i.e. <x,y,z,w> is interpreted as w*<x,y,z,1> = <w*x,w*y,w*z,w>.

3

A million thanks.
It works beautifully also in rendering sphere, cylinder, torus, etc. using gluNurbsSurface( ).