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,
has a natural range of
and the other two have natural ranges of
+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.
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
+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 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
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