Coordinate Transformations

A lot of conventions as to how we do things are arbitrary. For example, driving on the left side of the road is not inherently better or worse than driving on the right side. But if users of the same road don’t all agree to drive on the same side, then the result is grief.

The same is true of computer graphics. For example, coordinate systems can be right-handed or left-handed; if you imagine placing your eye at the (0, 0, 0) point and looking in turn in the direction of the positive-X, positive-Y and positive-Z axes, if your gaze describes a clockwise rotation, then the coordinate system is right-handed, while anticlockwise means it is left-handed. One way to remember which is which is to hold your thumb, index finger and middle finger out at right angles to each other, calling the thumb direction positive-X, index finger pointing at positive-Y, and middle finger at positive-Z; if you do this with your right hand, then the coordinate system is (naturally) right-handed; while your left hand defines a left-handed coordinate system.

The recommended convention (used in most 3D software) is to define your model/scene in a right-handed coordinate system.

It is quite common in computer graphics to be working in a number of different coordinate systems. For example, a model of a car is defined in terms of its own model coordinate system. To place the car in a scene, perhaps moving along a road, involves transforming those model coordinates to the world coordinates of the scene. The placement of the “camera” (actually the representation of the eye position of the person viewing the scene) then involves transforming these world coordinates into eye coordinates, which are then finally transformed into normalized device coordinates and mapped to pixels on the user’s display.

And just to add to the fun, the car model itself may have multiple coordinate systems. For example, each wheel may be defined in its own child coordinate system, in which it rotates relative to its parent, namely the body of the car. Because the wheel is transformed relative to the car, it automatically gets the car transformation as well, so moving the car through the scene is sufficient to bring the wheels along, they don’t need to be separately repositioned.