WARNING: This section was written at two drastically disparate times, and not proofread for consistency nor completely checked for accuracy. Please help me to improve this, send comments and suggestions to me!

This is the second in a series of a tutorial on ray tracing (and anything else that looks like fun). In the first installment I covered the basics of ray tracing, what it is and all that, and showed you how to derive an intersection formula for a ray and a sphere.

Because this is not a language specific text (although, if I can gather examples from people out there I'll include them), you'll have to figure out the particular implementation on your own. If you really get in a jam I'll provide help, and probably even C and C++ code to do all of this.

'Nuff with the intro, on with the real stuff: At the end of the last episode, you probably had some code that gave you a circle in the middle of the screen.... D r a w n v e r y s l o w l y ... And you said to yourself, "Why do I need these pages and pages of arithmetic to draw a stupid circle in the middle of the screen. I could have done that a thousand ways and been faster. And you would have been right. So, in order to give you a quick and easy sense of accomplishment, this time I want to talk about color.

No, this isn't gonna be one of those sensitive talks about racial issues, this is about how you make that ball look like it exists in the real world.

When the ray from your eye hits the object, we color the pixel some color based on a bunch of things. In the last episode you colored it based solely on the fact taht it hit something. Obviously this gave you a very boring answer.

In the real physical world color comes from the light that isn't absorbed and therefore bounces off when it hits an object. At this stage we'll break the way that light reflects off of various objects down into four components.

- Ambient light. In the real world, light comes from a source (the sun, for instance) and shines on things. If this were all that happened, however, you wouldn't be able to see anything in the shadows. Light bounces off of anything that isn't dark matte black, and therefore those shadows aren't completely dark. Rather than simulate the bounces (done in radiosity simulations), most renderers have a minimum lighting level that everything uses. Ambient light is it.
- Lambertian reflection. If you notice, the moon seems brightest at the center, even if it isn't full (yeah, I know that this isn't strictly true, but it's something everyone can relate to). The same is true for chalk, Nerf(tm) balls, and other dull matte surfaces. When a surface is not terribly reflective and is evenly lit, the color on it is related to perpendicular it is to the light: proportional to the cosine of the angle between the viewer's eye and a line perpendicular to the surface.
- Specular reflection. Shiny surfaces reflect the actual light sources with some blurring. For instance, the classic computer image of the plastic ball has a bright spot on it, the sharper the difference between that spot and the surrounding color, the harder the ball looks. There are several systems to describe this, the most popular is known as "Phong" illumination.

For this first section, we'll deal with ambient and diffuse reflection.

First, if you're using a VGA or other display with a limited number of colors, make a routine that at the beginning of your program sets your palette to give you as many gray scales as your hardware can handle and at the end of your program restores as many of them as you'll need to have a working computer.

First, add an ambient light to your routine; something that lets you enter in how bright it is in general in your scene. We'll deal with the color components later, but you might want to divide up all of your routines into three components (red, green and blue), treat them individually, but make everything put the same number into them for starters.

Now add in something that asks for the brightness of your object, and multiply that by your ambient light and use that color as the color of your point.

point brightness = ambient brightness * color brightness

Now you've got overly complex control over how bright that very boring very slowly drawn circle is.

I promise, things are going to get more exciting soon.

When you calculated the intersection with the sphere, the calculation routine returned a "time" element to you, how long it took your ray to hit the sphere. If we multiply the "delta" values of your ray and add in the origin of your ray, we have the point at which that ray hits the sphere (obviously).

Sooooo, if we define a light relative to the sphere, take the angle between that light and the sphere, and add to our previous brightness this brightness, we should have a shaded diffuse ball:

point brightness = (ambient brightness + light brightness * cos(angle between normal and eye)) * color brightness

Those of you, like me, who barely made it through linear algebra are now asking "this is great, but howintheheck are we supposed to deternine the angle between the surface normal and the vector to the eye.

I'm glad you asked. It turns out that the cosine of the angle between two "normalized" vectors is their "dot product".

"normalized" means that the lengths of the vectors is 1. What this means to those of you who don't remember Pythagorus is that the square root of the sum of the squares of the components of this vector is one. In other words:

square root(x * x + y * y + z * z) = 1.0.

So to get there, you find the length of the vector (see that square root thingie up above...), and divide each of the components by one.

"dot product" means the sum of each of the respective components of one vector times each of the components in the other vector. In other words:

dot product(V1, V2) = V1.x * V2.x + V1.y * V2.y + V1.z * V2.z

So, if we create a light somewhere in space, find the "surface normal" vector to the point of intersection on the sphere (draw a line from the center of the sphere to the point where it hit), create a vector to the light from the point of intersection, normalize those two, get the dot product, multiply that times a light value and add it to the previously calculated brightness, we've got a light value for each point.

Next time: We add Phong shading for specular highlights and make this sucker look plastic.

As usual, I want more than "more, more!", I want feedback. Have you implemented anything? Did you have trouble understanding anything? Have you found any technical errors?

This is part of the Graphics Without Greek collection in the home pages of Dan Lyke , reachable at danlyke@flutterby.com