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This article provides a useful perspective of understanding, representing, and manipulating 3D orientation and rotation for geometric computing. Coordinate-free geometric programming and affine geometry, which makes a distinction between points and vectors and defines operations for combining them, inspires our approach. Based upon affine geometry, Goldman and DeRose pioneered a method of writing graphics programs that are independent of the choice of reference coordinate frames. The study on geometric algebra pursues a similar goal with various geometric primitives rather than just vectors and points.