Jeff Molofee - NeHe's OpenGL Tutorials

GLfloat res[4]; // Hold Calculated Results

res[0]=M[ 0]*v[0]+M[ 4]*v[1]+M[ 8]*v[2]+M[12]*v[3];

res[1]=M[ 1]*v[0]+M[ 5]*v[1]+M[ 9]*v[2]+M[13]*v[3];

res[2]=M[ 2]*v[0]+M[ 6]*v[1]+M[10]*v[2]+M[14]*v[3];

res[3]=M[ 3]*v[0]+M[ 7]*v[1]+M[11]*v[2]+M[15]*v[3];

v[0]=res[0]; // Results Are Stored Back In v[]

v[1]=res[1];

v[2]=res[2];

v[3]=res[3]; // Homogenous Coordinate

}

The function to load the object is simple, just calling readObject, and then setting up the connectivity and the plane equations for each face.

int InitGLObjects() // Initialize Objects

{

if (!readObject("Data/Object2.txt", obj)) // Read Object2 Into obj

{

return FALSE; // If Failed Return False

}

setConnectivity(obj); // Set Face To Face Connectivity

for (int i=0; i < obj.nFaces; i++) // Loop Through All Object Faces

calculatePlane(obj, obj.pFaces[i]); // Compute Plane Equations For All Faces

return TRUE; // Return True

}

Finally, KillGLObjects is a convenience function so that if you add more objects, you can add them in a central place.

void KillGLObjects() {

killObject( obj );

}

All of the other functions don't require any further explanantion. I have left out the standard NeHe tutorial code, as well as all of the variable definitions and the keyboard processing function. The commenting alone explains these sufficiently.

Some things to note about the tutorial:

• The sphere doesn't stop shadows being projected on the wall. In reality, the sphere should also be casting a shadow, so seeing the one on the wall won't matter, it's hidden. It's just there to see what happens on curved surfaces :)

• If you are noticing extremely slow frame rates, try switching to fullscreen mode, or setting your desktop colour depth to 32bpp.

• Arseny L. writes: If you are having problems with a TNT2 in Windowed mode, make sure your desktop color depth is not set to 16bit. In 16bit color mode, the stencil buffer is emulated, resulting in sluggish performance. There are no problems in 32bit mode (I have a TNT2 Ultra and I checked it). I've got to admit this was a lengthy task to write out this tutorial. It gives you full appreciation for the work that Jeff puts in! I hope you enjoy it, and give a huge thanks to Banu who wrote the original code! IF there is anything that needs further explaining in here, you are welcome to contact me (Brett), at brettporter@yahoo.com.

* DOWNLOAD Visual C++Code For This Lesson.

* DOWNLOAD LinuxCode For This Lesson. (Conversion by Jay Groven)

This tutorial is intended to introduce you to Bezier Surfaces in the hopes that someone more artistic than myself will do something really cool with them and show all of us. This is not intended as a complete Bezier patch library, but more as proof of concept code to get you familiar with how these curved surfaces actually work. Also, as this is a very informal piece, I may have occasional lapses in correct terminology in favor of comprehensability; I hope this sits well with everyone. Finally, to those of you already familiar with Beziers who are just reading this to see if I screw up, shame on you ;-), but if you find anything wrong by all means let me or NeHe know, after all no one's perfect, eh? Oh, and one more thing, none of this code is optimised beyond my normal programming technique, this is by design. I want everyone to be able to see exactly what is going on. Well, I guess that's enough of an intro. On with the show!

Ok, it will be very hard to understand Beziers without at least a basic understanding of the math behind it, however, if you just don't feel like reading this section or already know the math, you can skip it. First I will start out by describing the Bezier curve itself then move on to how to create a Bezier Patch.

Odds are, if you've ever used a graphics program you are already familiar with Bezier curves, perhaps not by that name though. They are the primary method of drawing curved lines and are commonly represented as a series of points each with 2 points representing the tangent at that point from the left and right. Here's what one looks like:

This is the most basic Bezier curve possible (longer ones are made by attaching many of these together (many times without the user realizing it)). This curve is actually defined by only 4 points, those would be the 2 ending control points and the 2 middle control points. To the computer, all the points are the same, but to aid in design we often connect the first and the last two, respectively, because those lines will always be tangent to the endpoint. The curve is a parametric curve and is drawn by finding any number of points evenly spaced along the curve and connecting them with straight lines. In this way you can control the resolution of the patch (and the amount of computation). The most common way to use this is to tesselate it less at a farther distance and more at a closer distance so that, to the viewer, it always appears to be a perfectly curved surface with the lowest possible speed hit.

Bezier curves are based on a basis function from which more complicated versions are derived. Here's the function:

t + (1 – t) = 1

Sounds simple enough huh? Well it really is, this is the Bezier most basic Bezier curve, a 1st degree curve. As you may have guessed from the terminology, the Bezier curves are polynomials, and as we remember from algebra, a 1st degree polynomial is just a straight line; not very interesting. Well, since the basis function is true for all numbers t, we can square, cube, whatever, each side and it will still be true right? Well, lets try cubing it.

(t + (1-t))^3 = 1^3

t^3 + 3*t^2*(1-t) + 3*t*(1-t)^2 + (1-t)^3 = 1

This is the equation we use to calculate the most common Bezier, the 3rd degree Bezier curve. This is most common for two reasons, a) it's the lowest degree polynomial that need not necesarily lie in a plane (there are 4 control points) and b) the tangent lines on the sides are not dependant on one another (with a 2nd degree there would be only 3 control points). So do you see the Bezier curve yet? Hehe, me neither, that's because I still need to add one thing.

Ok, since the entire left side is equal to 1, it's safe to assume that if you add all the components they should still equal one. Does this sound like it could be used to descide how much of each control point to use in calculating a point on the curve? (hint: just say yes ;-) ) Well you're right! When we want to calculate the value of a point some percent along the curve we simply multiply each part by a control point (as a vector) and find the sum. Generally, we'll work with 0 <= t <= 1, but it's not technically necesary. Confused yet? Here's the function:

P1*t^3 + P2*3*t^2*(1-t) + P3*3*t*(1-t)^2 + P4*(1-t)^3 = P new

Because polynomials are always continuous, this makes for a good way to morp between the 4 points. The only points it actually reaches though are P1 and P4, when t = 1 and 0 respectively.

Now, that's all well and good, but how can I use these in 3D you ask? Well it's actually quite simple, in order to form a Bezier patch, you need 16 control points (4*4), and 2 variables t and v. What you do from there is calculate a point at v along 4 of the parallel curves then use those 4 points to make a new curve and calculate t along that curve. By calculating enough of these points, we can draw triangle strips to connect them, thus drawing the Bezier patch.