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The paper derives a parametric definition of the set of third-order orthonormal real matrices.The derivation is done in several partial steps. First a generalized unit matrix is introduced as the simplest case of an orthonormal matrix along with some of its properties and, subsequently, the properties of orthonormal matrices are proved that will be needed.The derivation itself of a parametric definition of third-order orthonormal matrices is based on the numbers of zero entries that are theoretically possible. Therefore, it is first proved that a third-order square matrix with the number of non-zero entries different from nine, eight, five, or three cannot be orthonormal.The number of different ways in which the set of third-order orthonormal matrices can be pa­ra­me­te­ri­zed is greater than one. The concepts of a rotation matrix and a flop-enabling rotation matrix are introduced to motivate the parameterization chosen.Given the product of two rotation matrices and one flop-enabling rotation matrix, it is first proved that it is a third-order orthonormal matrix. In the last part of the paper, it is then proved that such a product already includes, as special cases, all the third-order orthonormal matrices. It is thus a parametric definition of all third-order orthonormal matrices.

Parameterization of orthonormal third-order matrices for linear calibration