Exact numerical computation of the rational general linear transformations

The rational, general-linear transformations can be computed exactly using rational, matrix arithmetic. A subset of these transformations can be expressed in QR form as the product of a rational, orthogonal matrix Q and a rational, triangular matrix R of homogeneous co-ordinates. We present here a derivation of a half-tangent formula that encodes all of the rational rotations. This presentation involves many fewer axioms than in previous, unpublished work and reduces the number of transrational numbers in the total trigonometric functions from three to two. The practical consequence of this is that rotational sensors, such as computer vision cameras, gyroscopes, lidar, radar, and sonar can all be calibrated in terms of rational half-tangents, hence all subsequent, general-linear, numerical computations can be carried out exactly. In this case the only error is sensor error, so computations can be carried out precisely to the physical limits of the sensor.