Tuesday, December 21, 2010

A Graphics Programmer's Toolbox, part 1 (Vectors)

Everyone needs to know the basic tools of their art. I thought I'd like to share with you my equivalents of a screwdriver, a hammer and maybe a saw, some of the everyday tools of a graphics programmer.


A vector is just a fancy word for a point in space. So in 2d, a vector is just a co-ordinate pair (x, y). In 3d it's (x, y, z) and in 4d it's (x, y, z, w). A mental image of four dimensions is not important here - don't even try.

Some vectors are given a special name. The vector in the middle of the space, the zero-vector, is called the origin. In 3d, that's (0, 0, 0). The vector to the right, (1, 0, 0), is called i. The up-vector, (0, 1, 0) is j. The remaining vector, (0, 0, 1), is named k and it's the backwards-vector.

Sum, Difference and Multiplication

These points, or vectors, can be added and subtracted. Adding a vector to another is just summing the coordinates of the vectors; if a = (X, Y) and b = (x, y), then a + b = (X+x, Y+y). Subtracting is done in a similar way, a - b = (X-x, Y-y). Multi-dimensional calculation is not really that much more complicated. It's basically just the same, but doing the operations to all co-ordinates, instead of just one.

Sum and difference are easy, but with multiple components (co-ordinates) there are many kind of vectors multiplications with different meanings. Complicated? I didn't say anything; not yet.

Multiplying by a scalar just multiplies the distance of the point with respect to the origin. This scalar is just a fancy name for a real number; you might have noticed mathematicians love fancy words. So in other words, if a = (x, y, z) and r is a real number, then a*r = (x*r, y*r, z*r). If r is negative, you can see that the point will also be mirrored with respect to the origin.

The componentwise product works the same way as sum and addition, but it's a bit rarer in the mathematical context. It's very useful in computer graphics though, mostly in lighting calculations. This product is denoted with , and if a = (R, G, B, A) and b = (r, g, b, a), then ab = (R*r, G*g, B*b, A*a). Mathematicians call this product the Hadamard product (Jacques Hadamard, 1865 to 1963). Componentwise product is fine too.

There are also at least two more very useful products.

Dot and Cross Product

The dot product is the screwdriver of the toolbox. This product is written with the symbol •, and it's the sum of the components' products; if a = (X, Y, Z) and b = (x, y, z), then ab = X*x + Y*y + Z*z. Notice that the result is a scalar, or a real number. The usefulness of this product comes mainly from the fact that if a and b are both unit-length (X2 + Y2 + Z2 = 12), then, denoting the angle between b and a with α, it holds that ab = cos α. Generally, if a and b are not necessarily of unit length, then ab / |a||b| = cos α, where |v| denotes the length of a vector.

In 3d there's the cross product. Its main purpose is outputting a vector that's orthogonal to two given vectors, i.e. getting their normal vector. It can also be used to calculate the sine of the angle between two vectors. A funny fact is that a cross product only exists in 3 and 7 dimensions. The formula for cross product is pretty symmetric; if a = (X, Y, Z) and b = (x, y, z), then a × b = (Yz - Zy, Zx - Xz, Xy - Yx). Notice that this product is a bit special in that a × b = - b × a, so you can't just switch the order mindlessly. It is also true, like stated before, that a × b is orthogonal to both a and b. In mathematical notation, (a × b) • a = 0 and (a × b) • b = 0. For the direction of the cross product vector there's the right-hand rule, well described in this wiki article.

Continuing soon...
It's a known fact that once you open a Wikipedia page, you just can't stop reading. I'll set here a trap for you and I bet you can't oppose it:
Euclidean vector
Unit vector
The Gram-Schmidt algorithm

I'm planning to continue with matrices, bases, coordinate transforms, quaternions and such. If you'd like to suggest a topic or ask a question, please comment! :)