Abstract
In this tutorial we explain about complex numbers and quaternions which are an extension of complex numbers, and compare their properties of operations. More importantly, for rotations in a two dimensional space we review how a vector is transformed by a rotation angle and induce the corresponding rotation matrix. Consequently, we show that a unit complex number can represent a rotation in a two dimensional space, and the rotated vector can be calculated by complex number multiplication. For rotations in a three dimensional space, on the other hand, we review how a vector is transformed by a rotation axis and an angle, and induce the Rodrigues’ formula. Consequently, we show that a unit quaternion can represent a rotation in a three dimensional space, and the rotated vector can be calculated by quaternion conjugation. Meanwhile, we also summarize how to convert a unit complex number and a unit quaternion to other rotation representations, and vice versa.
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