Drawing Sphere in OpenGL without using gluSphere()?

The code in the sample is quickly explained. You should look into the function void drawSphere(double r, int lats, int longs):

void drawSphere(double r, int lats, int longs) {
    int i, j;
    for(i = 0; i <= lats; i++) {
        double lat0 = M_PI * (-0.5 + (double) (i - 1) / lats);
        double z0  = sin(lat0);
        double zr0 =  cos(lat0);

        double lat1 = M_PI * (-0.5 + (double) i / lats);
        double z1 = sin(lat1);
        double zr1 = cos(lat1);

        glBegin(GL_QUAD_STRIP);
        for(j = 0; j <= longs; j++) {
            double lng = 2 * M_PI * (double) (j - 1) / longs;
            double x = cos(lng);
            double y = sin(lng);

            glNormal3f(x * zr0, y * zr0, z0);
            glVertex3f(r * x * zr0, r * y * zr0, r * z0);
            glNormal3f(x * zr1, y * zr1, z1);
            glVertex3f(r * x * zr1, r * y * zr1, r * z1);
        }
        glEnd();
    }
}

The parameters lat defines how many horizontal lines you want to have in your sphere and lon how many vertical lines. r is the radius of your sphere.

Now there is a double iteration over lat/lon and the vertex coordinates are calculated, using simple trigonometry.

The calculated vertices are now sent to your GPU using glVertex...() as a GL_QUAD_STRIP, which means you are sending each two vertices that form a quad with the previously two sent.

All you have to understand now is how the trigonometry functions work, but I guess you can figure it out easily.

One way you can do it is to start with a platonic solid with triangular sides - an octahedron, for example. Then, take each triangle and recursively break it up into smaller triangles, like so:

Once you have a sufficient amount of points, you normalize their vectors so that they are all a constant distance from the center of the solid. This causes the sides to bulge out into a shape that resembles a sphere, with increasing smoothness as you increase the number of points.

Normalization here means moving a point so that its angle in relation to another point is the same, but the distance between them is different. Here's a two dimensional example.

A and B are 6 units apart. But suppose we want to find a point on line AB that's 12 units away from A.

We can say that C is the normalized form of B with respect to A, with distance 12. We can obtain C with code like this:

#returns a point collinear to A and B, a given distance away from A. 
function normalize(a, b, length):
    #get the distance between a and b along the x and y axes
    dx = b.x - a.x
    dy = b.y - a.y
    #right now, sqrt(dx^2 + dy^2) = distance(a,b).
    #we want to modify them so that sqrt(dx^2 + dy^2) = the given length.
    dx = dx * length / distance(a,b)
    dy = dy * length / distance(a,b)
    point c =  new point
    c.x = a.x + dx
    c.y = a.y + dy
    return c

If we do this normalization process on a lot of points, all with respect to the same point A and with the same distance R, then the normalized points will all lie on the arc of a circle with center A and radius R.

Here, the black points begin on a line and "bulge out" into an arc.

This process can be extended into three dimensions, in which case you get a sphere rather than a circle. Just add a dz component to the normalize function.

If you look at the sphere at Epcot, you can sort of see this technique at work. it's a dodecahedron with bulged-out faces to make it look rounder.

I'll further explain a popular way of generating a sphere using latitude and longitude (another way, icospheres, was already explained in the most popular answer at the time of this writing.)

A sphere can be expressed by the following parametric equation:

F(u, v) = [ cos(u)*sin(v)*r, cos(v)*r, sin(u)*sin(v)*r ]

Where:

• r is the radius;
• u is the longitude, ranging from 0 to 2π; and
• v is the latitude, ranging from 0 to π.

Generating the sphere then involves evaluating the parametric function at fixed intervals.

For example, to generate 16 lines of longitude, there will be 17 grid lines along the u axis, with a step of π/8 (2π/16) (the 17th line wraps around).

The following pseudocode generates a triangle mesh by evaluating a parametric function at regular intervals (this works for any parametric surface function, not just spheres).

In the pseudocode below, UResolution is the number of grid points along the U axis (here, lines of longitude), and VResolution is the number of grid points along the V axis (here, lines of latitude)

var startU=0
var startV=0
var endU=PI*2
var endV=PI
var stepU=(endU-startU)/UResolution // step size between U-points on the grid
var stepV=(endV-startV)/VResolution // step size between V-points on the grid
for(var i=0;i<UResolution;i++){ // U-points
 for(var j=0;j<VResolution;j++){ // V-points
 var u=i*stepU+startU
 var v=j*stepV+startV
 var un=(i+1==UResolution) ? EndU : (i+1)*stepU+startU
 var vn=(j+1==VResolution) ? EndV : (j+1)*stepV+startV
 // Find the four points of the grid
 // square by evaluating the parametric
 // surface function
 var p0=F(u, v)
 var p1=F(u, vn)
 var p2=F(un, v)
 var p3=F(un, vn)
 // NOTE: For spheres, the normal is just the normalized
 // version of each vertex point; this generally won't be the case for
 // other parametric surfaces.
 // Output the first triangle of this grid square
 triangle(p0, p2, p1)
 // Output the other triangle of this grid square
 triangle(p3, p1, p2)
 }
}

If you wanted to be sly like a fox you could half-inch the code from GLU. Check out the MesaGL source code (http://cgit.freedesktop.org/mesa/mesa/).