template<typename T> quaternion<T> spherical(T const & rho, T const & theta, T const & phi1, T const & phi2);
template<typename T> quaternion<T> semipolar(T const & rho, T const & alpha, T const & theta1, T const & theta2);
template<typename T> quaternion<T> multipolar(T const & rho1, T const & theta1, T const & rho2, T const & theta2);
template<typename T> quaternion<T> cylindrospherical(T const & t, T const & radius, T const & longitude, T const & latitude);
template<typename T> quaternion<T> cylindrical(T const & r, T const & angle, T const & h1, T const & h2);

These build quaternions in a way similar to the way polar builds complex numbers, as there is no strict equivalent to polar coordinates for quaternions.

spherical is a simple transposition of polar, it takes as inputs a (positive) magnitude and a point on the hypersphere, given by three angles. The first of these, theta has a natural range of -pi to +pi, and the other two have natural ranges of -pi/2 to +pi/2 (as is the case with the usual spherical coordinates in R³). Due to the many symmetries and periodicities, nothing untoward happens if the magnitude is negative or the angles are outside their natural ranges. The expected degeneracies (a magnitude of zero ignores the angles settings...) do happen however.

cylindrical is likewise a simple transposition of the usual cylindrical coordinates in R³, which in turn is another derivative of planar polar coordinates. The first two inputs are the polar coordinates of the first C component of the quaternion. The third and fourth inputs are placed into the third and fourth R components of the quaternion, respectively.

multipolar is yet another simple generalization of polar coordinates. This time, both C components of the quaternion are given in polar coordinates.

cylindrospherical is specific to quaternions. It is often interesting to consider H as the cartesian product of R by R³ (the quaternionic multiplication as then a special form, as given here). This function therefore builds a quaternion from this representation, with the R³ component given in usual R³ spherical coordinates.

semipolar is another generator which is specific to quaternions. It takes as a first input the magnitude of the quaternion, as a second input an angle in the range 0 to +pi/2 such that magnitudes of the first two C components of the quaternion are the product of the first input and the sine and cosine of this angle, respectively, and finally as third and fourth inputs angles in the range -pi/2 to +pi/2 which represent the arguments of the first and second C components of the quaternion, respectively. As usual, nothing untoward happens if what should be magnitudes are negative numbers or angles are out of their natural ranges, as symmetries and periodicities kick in.

In this version of our implementation of quaternions, there is no analogue of the complex value operation arg as the situation is somewhat more complicated. Unit quaternions are linked both to rotations in R³ and in R⁴, and the correspondences are not too complicated, but there is currently a lack of standard (de facto or de jure) matrix library with which the conversions could work. This should be remedied in a further revision. In the mean time, an example of how this could be done is presented here for R³, and here for R⁴ (example test file).