Minor documentation tweaks and test updates (#35)

Also, start testing on nightly again

Author: moble

.github/workflows/tests.yml

@@ -14,7 +14,7 @@ jobs:
       matrix:
         version:
           - '1'
-          # - 'nightly'
+          - 'nightly'
         os:
           - ubuntu-latest
         include:

docs/src/functions_of_time.md

@@ -190,7 +190,7 @@ confidence that other `ω⃗` functions would be correctly integrated also.
 ## Minimal rotation
 
 One common problem arises when a system must be rotated to align some axis with
-some direction, though the rotation of the system *about* that axis is
+some direction, while the rotation of the system *about* that axis is
 irrelevant.  To be specific, suppose we want to rotate our basis vectors
 ``\hat{x}, \hat{y}, \hat{z}`` so that ``\hat{z}`` points in a particular
 direction.  A naive approach may be to determine the direction in terms of
@@ -201,6 +201,13 @@ orientation will be extremely sensitive to the direction whenever it happens to
 be near the poles.  In such cases, the angular velocity of the system will be
 very high — potentially infinite, in principle.
 
+A *slightly* better approach would be to use ``(\alpha, \beta, \gamma) = (\phi,
+\theta, -\phi)``, which is the most direct rotation from the ``z`` axis to the
+point given by ``(\theta, \phi)``, and behaves better in the limit of small
+``\theta``.  However, this only works for rotations directly from the ``z``
+axis; the result depends on the choice of coordinates, and is not the best
+choice for tracking general motion of the target axis.
+
 Fortunately, it is possible to take *any* rotor ``R_\mathrm{axis}(t)`` that
 aligns the axis correctly, and compute another rotation that also aligns the
 axis, but has the smallest possible angular velocity.  This is called the
@@ -218,8 +225,8 @@ R(t) = R_\mathrm{axis}(t)\, \exp\left[\gamma(t) 𝐤 / 2 \right]
 ```
 This rotor also aligns the axis correctly, but otherwise has the smallest
 possible angular velocity.  Here, ``R_\mathrm{axis}`` may be constructed in any
-convenient way, including using spherical coordinates; the resulting ``R(t)``
-will be independent of such poor life choices.
+convenient way, including using spherical coordinates or even Euler angles; the
+resulting ``R(t)`` will be independent of such poor life choices.
 
 
 ## Derivatives and gradients

test/conversion.jl

@@ -62,7 +62,7 @@
 
         @testset "Euler phases" begin
             N = 5_000
-            ϵ = (T === Float16 ? 20eps(T) : 10eps(T))
+            ϵ = (T === Float16 ? 22eps(T) : 10eps(T))
 
             dumb_to_euler_phases(α, β, γ) = [exp(α*im), exp(β*im), exp(γ*im)]
 

test/math.jl

@@ -12,7 +12,7 @@
         Quaternion(c.re, [comp_i == component ? c.im : zero(real(c)) for comp_i in components]...)
     end
     @testset "Complex equivalence $T" for T in FloatTypes
-        ϵ = 10eps(T)
+        ϵ = (T === Float16 ? 20eps(T) : 10eps(T))
         scalars = [zero(T), one(T), -one(T)]
         for c in [a+b*im for a in scalars for b in scalars]
             # The following constructs a quaternion from the complex number `c` by equating `im`