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Vectors, vertices, and quaternions

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Throughout Direct3D, vertices describe position and orientation. Each vertex in a primitive is described by a vector that gives its position, color, texture coordinates, and a normal vector that gives its orientation.

Quaternions add a fourth element to the [x, y, z] values that define a three-component-vector. Quaternions are an alternative to the matrix methods that are typically used for 3D rotations. A quaternion represents an axis in 3D space and a rotation around that axis. For example, a quaternion might represent a (1,1,2) axis and a rotation of 1 radian. Quaternions carry valuable information, but their true power comes from the two operations that you can perform on them: composition and interpolation.

Performing composition on quaternions is similar to combining them. The composition of two quaternions is notated like the following illustration.

illustration of quaternion notation

The composition of two quaternions applied to a geometry means "rotate the geometry around axis₂ by rotation₂, then rotate it around axis₁ by rotation₁." In this case, Q represents a rotation around a single axis that is the result of applying q₂, then q₁ to the geometry.

Using quaternion interpolation, an application can calculate a smooth and reasonable path from one axis and orientation to another. Therefore, interpolation between q₁ and q₂ provides a simple way to animate from one orientation to another.

When you use composition and interpolation together, they provide you with a simple way to manipulate a geometry in a manner that appears complex. For example, imagine that you have a geometry that you want to rotate to a given orientation. You know that you want to rotate it r₂ degrees around axis₂, then rotate it r₁ degrees around axis₁, but you don't know the final quaternion. By using composition, you could combine the two rotations on the geometry to get a single quaternion that is the result. Then, you could interpolate from the original to the composed quaternion to achieve a smooth transition from one to the other.

Coordinate systems and geometry