...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
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).