OpenGL NURBS surface

Question

I'm learning OpenGL and I want to get a surface with a slight hump in the middle. I'm currently using this code and im not sure how to adjust the ctrl points to make it the way i want. Its currently like

and i would like to have it like this:

im not entirely sure what control points i should use and i'm confused on how it works.

#include <stdlib.h>
#include <GLUT/glut.h>

GLfloat ctrlpoints[4][4][3] = {
   {{-1.5, -1.5, 4.0}, {-0.5, -1.5, 2.0}, 
    {0.5, -1.5, -1.0}, {1.5, -1.5, 2.0}}, 
   {{-1.5, -0.5, 1.0}, {-0.5, -0.5, 3.0}, 
    {0.5, -0.5, 0.0}, {1.5, -0.5, -1.0}}, 
   {{-1.5, 0.5, 4.0}, {-0.5, 0.5, 0.0}, 
    {0.5, 0.5, 3.0}, {1.5, 0.5, 4.0}}, 
   {{-1.5, 1.5, -2.0}, {-0.5, 1.5, -2.0}, 
    {0.5, 1.5, 0.0}, {1.5, 1.5, -1.0}}
};

void display(void)
{
   int i, j;

   glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
   glColor3f(1.0, 1.0, 1.0);
   glPushMatrix ();
   glRotatef(85.0, 1.0, 1.0, 1.0);
   for (j = 0; j <= 8; j++) {
      glBegin(GL_LINE_STRIP);
      for (i = 0; i <= 30; i++)
         glEvalCoord2f((GLfloat)i/30.0, (GLfloat)j/8.0);
      glEnd();
      glBegin(GL_LINE_STRIP);
      for (i = 0; i <= 30; i++)
         glEvalCoord2f((GLfloat)j/8.0, (GLfloat)i/30.0);
      glEnd();
   }
   glPopMatrix ();
   glFlush();
}

void init(void)
{
   glClearColor (0.0, 0.0, 0.0, 0.0);
   glMap2f(GL_MAP2_VERTEX_3, 0, 1, 3, 4,
           0, 1, 12, 4, &ctrlpoints[0][0][0]);
   glEnable(GL_MAP2_VERTEX_3);
   glMapGrid2f(20, 0.0, 1.0, 20, 0.0, 1.0);
   glEnable(GL_DEPTH_TEST);
   glShadeModel(GL_FLAT);
}
void reshape(int w, int h)
{
   glViewport(0, 0, (GLsizei) w, (GLsizei) h);
   glMatrixMode(GL_PROJECTION);
   glLoadIdentity();
   if (w <= h)
      glOrtho(-5.0, 5.0, -5.0*(GLfloat)h/(GLfloat)w, 
               5.0*(GLfloat)h/(GLfloat)w, -5.0, 5.0);
   else
      glOrtho(-5.0*(GLfloat)w/(GLfloat)h, 
               5.0*(GLfloat)w/(GLfloat)h, -5.0, 5.0, -5.0, 5.0);
   glMatrixMode(GL_MODELVIEW);
   glLoadIdentity();
}

int main(int argc, char** argv)
{
   glutInit(&argc, argv);
   glutInitDisplayMode (GLUT_SINGLE | GLUT_RGB);
   glutInitWindowSize (500, 500);
   glutInitWindowPosition (100, 100);
   glutCreateWindow (argv[0]);
   init ();
   glutDisplayFunc(display);
   glutReshapeFunc(reshape);
   glutMainLoop();
   return 0;
}

Solution 1

EDIT: I thought you were experimenting, but I see the code came from the OpenGL tutorial. I glanced through it and understand your point now. It's difficult to learn the basics from there.

NURBS Background

The best way to get a grasp with NURBS is to play with it interactively. You'd then gain the intuition about the edge-defining points (on the edges), shape defining ones (every other), the tangent-relation between them and continuity. NURBS can be made of patches, stitched together at the edges, where continuity is highly controlled - namely you can ask for G3 for car's main body or C1 for a cheap game model. It's really difficult to get the concept from any description. It you want to get it this way, I'd highly recommend the trial of Rhino Nurbs Modeller. I used it years ago and now it seems abandoned, but still it is the software with one of the best NURBS support out there (Autodesk 3d Studio MAX and MAYA have worse). It might be a little time consuming though, for starters I'd recommend playing with something simpler; take the applet from the "Simple Bezier Curve Editor" page for a spin.

To understand NURBS it's also good to consult Wikipedia Article about Bezier Curves. Once you get a grasp of relation between point position and the final curve shape you can easily generalize it to surfaces. I find this animation highly intuitive:

You can imagine the surface from your example as a set of four of those curves, with a cloth spanned on them. Using the applet I linked earlier, you can play with the position and get instant feedback on the resulting shape. Notice the t parameter - it is a coordinate along the curve and has a range of [0, 1]. The NURBS surface have two of those coordinates, by convention called u and v (that's important for the drawing function).

So, the ctrlpoints structure from the code holds all point coordinates. Simplifying for explanation, those are four cubic Bezier curves (the ones from the animation). For each curve you have four points within 3 dimensions. If you disregard the Y-axis, then all of them lie on a grid, with X and Z being: -1.5, -1.0, 1.0, 1.5. That explains a total of 32 values (4x4 for X plus 4x4 for Z).

The rest is the height, Y-values. In your case it is the second value of every point in ctrlpoints. To get your expected result, you can make all Y values equal on the edges (outer ones) and slightly raised in the middle (4 inner). You'll get:

Points used to render above image:

GLfloat ctrlpoints[4][4][3] = {
 {{-1.5, 1.0, -1.5},  {-0.5, 1.0,-1.5 }, {0.5, 1.0, -1.5 },   {1.5, 1.0,-1.5}}, 
 {{-1.5, 1.0, -0.5},  {-0.5, 2.0,-0.5 }, {0.5, 2.0, -0.5 },   {1.5, 1.0,-0.5}}, 
 {{-1.5, 1.0,  0.5},  {-0.5, 2.0, 0.5 }, {0.5, 2.0,  0.5 },   {1.5, 1.0, 0.5}}, 
 {{-1.5, 1.0,  1.5},  {-0.5, 1.0, 1.5 }, {0.5, 1.0,  1.5 },   {1.5, 1.0, 1.5}}
};
//        ^                   ^                 ^                    ^
//        |                   |                 |                    |
//        |                   |                 |                    |
//        \_________ Those are most relevant - Y-coord, height ______/

NURBS in OpenGL with GLUT - API walkthrough

I see OpenGL API is hiding quite relevant details. The NURBS surface is drawn using Evaluator and is defined with the Map function.

You're supposed to define control points in the init(void) function, like so:

glMap2f(GL_MAP2_VERTEX_3, 0, 1, 3, 4,
           0, 1, 12, 4, &ctrlpoints[0][0][0]);

Good explanation of the function can be found on the MSDN Site for glMap2f. We are passing control points, their type and the details such as array stride and order.

You can draw it using an Evaluator function. It takes two coordinates as arguments and gives back a point in 3d space. Those input coordinates are exactly the u and v I mentioned earlier, under the animation. In our example:

      glBegin(GL_LINE_STRIP); // we'll draw a line

      // take 31 samples of a cross-section of the surface
      for (i = 0; i <= 30; i++)
         // for each sample, evaluate a 3d point
         glEvalCoord2f((GLfloat)i/30.0, (GLfloat)j/8.0);

         // notice j is constant in the loop here, but
         // is being changed by the outer loop.
         //
         // j is iterated in 9 steps, so we'll end up
         // with 9 lines
      glEnd();

I deliberately omitted the outer loop, which is described here:

   // we want 9 lines
   for (j = 0; j <= 8; j++) {
      // OpenGL state machine will be used to draw lines

      glBegin(GL_LINE_STRIP);
      // inner loop for j-th line along X

      glBegin(GL_LINE_STRIP);
      // inner loop for j-th line along Z

      glEnd(); // done with the lines
   }

Working example

#include <stdlib.h>
#include <GL/glut.h>

GLfloat ctrlpoints[4][4][3] = {
 {{-1.5, 1.0, -1.5}, {-0.5, 1.0,-1.5 }, {0.5, 1.0, -1.5 }, {1.5, 1.0,-1.5}}, 
 {{-1.5, 1.0, -0.5}, {-0.5, 2.0,-0.5 }, {0.5, 2.0, -0.5 }, {1.5, 1.0,-0.5}}, 
 {{-1.5, 1.0,  0.5}, {-0.5, 2.0, 0.5 }, {0.5, 2.0,  0.5 }, {1.5, 1.0, 0.5}}, 
 {{-1.5, 1.0,  1.5}, {-0.5, 1.0, 1.5 }, {0.5, 1.0,  1.5 }, {1.5, 1.0, 1.5}}
};

void display(void)
{
   int i, j;

   glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
   glColor3f(1.0, 1.0, 1.0);
   glPushMatrix();
   glRotatef(25.0, 1.0, 1.0, 1.0);
   for (j = 0; j <= 8; j++) {
      glBegin(GL_LINE_STRIP);
      for (i = 0; i <= 30; i++)
         glEvalCoord2f((GLfloat)i/30.0, (GLfloat)j/8.0);
      glEnd();
      glBegin(GL_LINE_STRIP);
      for (i = 0; i <= 30; i++)
         glEvalCoord2f((GLfloat)j/8.0, (GLfloat)i/30.0);
      glEnd();
   }
   glPopMatrix();
   glFlush();
}

void init(void)
{
   glClearColor(0.0, 0.0, 0.0, 0.0);
   glMap2f(GL_MAP2_VERTEX_3, 0, 1, 3, 4,
           0, 1, 12, 4, &ctrlpoints[0][0][0]);
   glEnable(GL_MAP2_VERTEX_3);
   glMapGrid2f(20, 0.0, 1.0, 20, 0.0, 1.0);
   glEnable(GL_DEPTH_TEST);
   glShadeModel(GL_FLAT);
}

void reshape(int w, int h)
{
   glViewport(0, 0, (GLsizei) w, (GLsizei) h);
   glMatrixMode(GL_PROJECTION);
   glLoadIdentity();
   if (w <= h)
      glOrtho(-5.0, 5.0, -5.0*(GLfloat)h/(GLfloat)w, 
               5.0*(GLfloat)h/(GLfloat)w, -5.0, 5.0);
   else
      glOrtho(-5.0*(GLfloat)w/(GLfloat)h, 
               5.0*(GLfloat)w/(GLfloat)h, -5.0, 5.0, -5.0, 5.0);
   glMatrixMode(GL_MODELVIEW);
   glLoadIdentity();
}

int main(int argc, char** argv)
{
   glutInit(&argc, argv);
   glutInitDisplayMode(GLUT_SINGLE | GLUT_RGB);
   glutInitWindowSize(500, 500);
   glutInitWindowPosition(100, 100);
   glutCreateWindow(argv[0]);
   init();
   glutDisplayFunc(display);
   glutReshapeFunc(reshape);
   glutMainLoop();
   return 0;
}

Solution 2

Even though it's hard to match in completeness and presentation the answer of Rekin, there's something to be clarified:

The 'R' in NURBS stands for rational. This requires using homogeneous coordinates where each control point [x,y,z] is assigned a weight 1/w, which will be used to divide all the other elements, making NURBS control points really vectors of four elements. It's through this fourth element one can represent exactly circles, torii and spheres with NURBS, where as with regular bezier or spline curves one can only approximate circles.

The lucky thing is that with openGL one will quickly be at least familiar with the use of the element w, and this will eventually lead to understanding. (or illusion of that...).

If one uses a real NURBS modeler, one is really required to import also the fourth component, unless the modeler is programmed to export a non-rational surface approximation represented by 3 element vectors alone.

Comments

[Rekin]

You seem heading at the right direction. Remind me of one thing: is Y-axis UP in OpenGL?

[Knitex]

in this case im pretty sure it is

[Rekin]

Ok, the answer is pretty much comprehensive now. If you want any more details or have any doubts I'll be glad to clear that out.

[Rekin]

Ok, now I'm aware where the original code came from, I'll add some more explanation

[Knitex]

This is awesome. Thanks a lot!

[Preet Kukreti]

This is a beautifully presented answer. Much respect.

[Rekin]

+1 - Indeed, in any real world scenario the weights cannot be omitted; CAD packages tend to produce a lot of elliptic curves (pipes, rounds, cross-secions, joints) and have strict requirements about the deviation. The elliptic curves can be exactly represented (not approximated) only using aforementioned extension - control point weights. It's not the end though, one cannot represent exactly all types of curves, a projection of a curve on a NURBS surface can produce a shape, which in hand cannot be represented exactly using weighted Bezier splines.

[Rekin]

ctd.: So the CAD packages have to approximate the projected shapes. They add extra knots to compensate a significant angular or positional deviation. This all means, using NURBS it's easy to produce a non water-thight mesh (the one with surface edges ideally aligned) which in hand complicates the fabrication (one has to merge the vertexes in final mesh before feeding to the CAM machine). That also can be problematic in rendering, where Global Illumination photons slip through the holes.