Fix glm::is_normalized epsilon

The existing comparison bound of $\epsilon^2$ is improperly scaled for
testing an epsilon of the squared vector magnitude. Let $\epsilon$ be
our specified epsilon and $\delta$ be the permissible delta of the
squared magnitude. Thus, for a nearly-normalized vector, we have

$$\begin{align}
\sqrt{1 + \delta} &=  1 + \epsilon        \\
          \delta  &= (1 + \epsilon)^2 - 1 \\
          \delta  &=  \epsilon^2 + 2\epsilon
\text{ .}\end{align}$$

Since we only care about small epsilon, we can assume
that $\epsilon^2$ is small and just use $\delta = 2\epsilon$. And in
fact, [this is the bound used by GLM][GLM#isNormalized] (MIT license)
... except they're using `length` and not `length2` for some reason.

[GLM#isNormalized]: b06b775c1c/glm/gtx/vector_query.inl (L102)

If we stick an epsilon of `1.0e-6` into the current implementation,
this gives us a computed delta of `1.0e-12`: smaller than the `f32`
machine epsilon, and thus no different than direct float comparison
without epsilon. This also gives an effetive epsilon of `5.0e-13`;
*much* less than the intended `1.0e-6` of intended permitted slack!
By doing a bit more algebra, we can find the effective epsilon is
$\sqrt{\texttt{epsilon}^2 + 1} - 1$. This patch makes the effective
epsilon $\sqrt{2\times\texttt{epsilon} + 1} - 1$ which still isn't
*perfect*, but it's effectively linear in the domain we care about,
only really making a practical difference above an epsilon of 10%.

TL;DR: the existing `is_normalized` considers a vector normalized if
the squared magnitude is within `epsilon*epsilon` of `1`. This is wrong
and it should be testing if it's within `2*epsilon`. This PR fixes it.

For absence of doubt, a comparison epsilon of $\texttt{epsilon}^2$ is
correct when comparing squared magnitude against zero, such as when
testing if a displacement vector is nearly zero.
This commit is contained in:
Christopher Durham 2024-01-12 22:54:09 -05:00
parent 0b89950fca
commit f578181351

View File

@ -41,7 +41,10 @@ pub fn is_comp_null<T: Number, const D: usize>(v: &TVec<T, D>, epsilon: T) -> TV
/// Returns `true` if `v` has a magnitude of 1 (up to an epsilon).
pub fn is_normalized<T: RealNumber, const D: usize>(v: &TVec<T, D>, epsilon: T) -> bool {
abs_diff_eq!(v.norm_squared(), T::one(), epsilon = epsilon * epsilon)
// sqrt(1 + epsilon_{norm²} = 1 + epsilon_{norm}
// ==> epsilon_{norm²} = epsilon_{norm}² + 2*epsilon_{norm}
// For small epsilon, epsilon² is basically zero, so use 2*epsilon.
abs_diff_eq!(v.norm_squared(), T::one(), epsilon = epsilon + epsilon)
}
/// Returns `true` if `v` is zero (up to an epsilon).