Following on from Euler’s Theorem,Euler–Rodrigues parameters  the axis and angle parameters that represent a rotation can be represented using what are known as Euler–Rodrigues parameters and will be described in this chapter along with the resulting rotation matrixRotation matrix in terms of Euler–Rodrigues parameters. Four real numbers are required for Euler–Rodrigues parameters. Whereas a rotation matrix requires nine real numbers to represent a rotation with six redundant numbers, Euler–Rodrigues parameters have just one redundant parameter. Euler–Rodrigues parameters provide a tidy way for constructing the rotation matrix, i.e. a compact way of representing the expressions for the components of the rotation matrix. They also provide the first use of half angles for rotation parameters. More importantly though is the mechanism of combining two sets of parameters to form a third set of Euler–Rodrigues parameters, i.e. combining rotations. Euler did not succeed in doing this, with Rodrigues being the first to provide an expression for combining rotations. The details of combining Euler–Rodrigues parameters are given in the next chapter. Interestingly, Euler was initially credited with the use of half angles, but recent examination of his works [HC89] indicate that this is incorrect and that Rodrigues was the first to use half angles and to define this four parameter set.