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Fix failed atan2 elimination by splitting SVD into two methods #583

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Fix failed atan2 elimination by splitting SVD into two methods #583

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91 changes: 43 additions & 48 deletions kurbo/src/affine.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -417,7 +417,7 @@ impl Affine {
// Frobenius norms, could be `0.5 (frob^2 + sqrt(frob^4 - 4 det^2))`. In terms of operations
// it's a wash: one fewer sqrt if the user actually wants the squared form, but it uses more
// muls. More importantly, that form has worse numeric conditioning.
self.svd().0.x
self.svd_singular_values().x
}

/// Compute the inverse transform.
Expand Down Expand Up @@ -476,8 +476,10 @@ impl Affine {
|| self.0[5].is_nan()
}

/// Compute the singular value decomposition of the linear transformation (ignoring the
/// translation).
/// Computes the singular values of the singular value decomposition of the linear
/// transformation (ignoring the translation).
///
/// Use [`Self::svd_post_scale_rotation`] to compute the rotation of the decomposition.
///
/// All non-degenerate linear transformations can be represented as
///
Expand All @@ -489,30 +491,11 @@ impl Affine {
/// decomposition" and is written `U Σ V^T`, where U and V^T are orthogonal (rotations) and Σ
/// is a diagonal matrix (a scaling).
///
/// Since currently this function is used to calculate ellipse radii and rotation from an
/// affine map on the unit circle, we don't calculate V^T, since a rotation of the unit (or
/// any) circle about its center always results in the same circle. This is the reason that an
/// ellipse mapped using an affine map is always an ellipse.
///
/// Will return NaNs if the matrix (or equivalently the linear map) is non-finite.
///
/// The first part of the returned tuple is the scaling, the second part is the angle of
/// rotation (in radians). The scaling along the x-axis is guaranteed to be greater than or
/// equal to the scaling along the y-axis.
//
// Note: though this does quite some computation, we are often interested only in specific
// components of the result. Hence this is marked `#[inline(always)]`, to give the compiler a
// good chance at eliminating dead code.
/// The returned vector contains the scaling (the diagonal of Σ). The scaling along the x-axis
/// is guaranteed to be greater than or equal to the scaling along the y-axis.
#[inline(always)]
pub(crate) fn svd(self) -> (Vec2, f64) {
pub(crate) fn svd_singular_values(self) -> Vec2 {
let [a, b, c, d, _, _] = self.0;
let a2 = a * a;
let b2 = b * b;
let c2 = c * c;
let d2 = d * d;
let ab = a * b;
let cd = c * d;
let angle = 0.5 * (2.0 * (ab + cd)).atan2(a2 - b2 + c2 - d2);

// Given matrix A = [ a c ]
// [ b d ]
Expand Down Expand Up @@ -549,13 +532,26 @@ impl Affine {
// and similarly σ2 = 1/2 |S1 - S2|
let s1 = ((a + d).powi(2) + (b - c).powi(2)).sqrt();
let s2 = ((a - d).powi(2) + (b + c).powi(2)).sqrt();
(
Vec2 {
x: 0.5 * (s1 + s2),
y: 0.5 * (s1 - s2).abs(),
},
angle,
)
Vec2 {
x: 0.5 * (s1 + s2),
y: 0.5 * (s1 - s2).abs(),
}
}

/// Computes the post-scale rotation in radians of the singular value decomposition of the
/// linear transformation (ignoring the translation).
///
/// This is the rotation represented by the `U`-term of the decomposition `U Σ V^T` (see
/// [`Self::svd_singular_values`] for more information about this decomposition).
///
/// The `V^T` rotation (i.e., the "pre-scale" rotation) is not computed: we currently only
/// require rotations to construct an ellipse from an affine map on the unit circle. A rotation
/// of the unit (or any) circle about its center always results in the same circle. This is the
/// reason that an ellipse mapped using an affine map is always an ellipse.
#[inline(always)]
pub(crate) fn svd_post_scale_rotation(self) -> f64 {
let [a, b, c, d, _, _] = self.0;
0.5 * (2.0 * (a * b + c * d)).atan2(a * a - b * b + c * c - d * d)
}

/// Returns the translation part of this affine map (`(self.0[4], self.0[5])`).
Expand Down Expand Up @@ -763,8 +759,10 @@ mod tests {
let a_no_translate = a.with_translation(Vec2::ZERO);

// translation should have no effect
let (scale, rotation) = a.svd();
let (scale_no_translate, rotation_no_translate) = a_no_translate.svd();
let scale = a.svd_singular_values();
let rotation = a.svd_post_scale_rotation();
let scale_no_translate = a_no_translate.svd_singular_values();
let rotation_no_translate = a_no_translate.svd_post_scale_rotation();
assert_near(scale.to_point(), scale_no_translate.to_point());
assert!((rotation - rotation_no_translate).abs() <= 1e-9);

Expand All @@ -777,7 +775,8 @@ mod tests {
// singular affine
let a = Affine::new([0., 0., 0., 0., 5., 6.]);
assert_eq!(a.determinant(), 0.);
let (scale, rotation) = a.svd();
let scale = a.svd_singular_values();
let rotation = a.svd_post_scale_rotation();
assert_eq!(scale, Vec2::new(0., 0.));
assert_eq!(rotation, 0.);
}
Expand All @@ -787,40 +786,36 @@ mod tests {
// Test a few known singular values.
let mat = |a, b, c, d| Affine::new([a, b, c, d, 0., 0.]);

let s = mat(1., 0., 0., 1.).svd().0;
let s = mat(1., 0., 0., 1.).svd_singular_values();
assert_near(s.to_point(), Point::new(1., 1.));

let s = mat(1., 0., 0., -1.).svd().0;
let s = mat(1., 0., 0., -1.).svd_singular_values();
assert_near(s.to_point(), Point::new(1., 1.));

let s = mat(1., 1., 1., 1.).svd().0;
assert_near(s.to_point(), Point::new(2., 0.));

let s = mat(1., 1., 1., 1.).svd().0;
let s = mat(1., 1., 1., 1.).svd_singular_values();
Comment on lines -796 to +795

@tomcur tomcur May 12, 2026

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This test was duplicated. Removed one, updated the other.

assert_near(s.to_point(), Point::new(2., 0.));

let s = mat(0., 0., 1., 0.).svd().0;
let s = mat(0., 0., 1., 0.).svd_singular_values();
assert_near(s.to_point(), Point::new(1., 0.));

// The singular values are the scaling of the affine map. So let's test that.
let s = Affine::scale_non_uniform(4., 8.)
.then_rotate_about(42_f64.to_radians(), (-2., 50.))
.svd()
.0;
.svd_singular_values();
assert_near(s.to_point(), Point::new(8., 4.));

// Correctly handles negative scaling (singular values are necessarily non-negative).
let s = Affine::scale_non_uniform(-20., 3.).svd().0;
let s = Affine::scale_non_uniform(-20., 3.).svd_singular_values();
assert_near(s.to_point(), Point::new(20., 3.));
let s = Affine::scale_non_uniform(-20., -3.).svd().0;
let s = Affine::scale_non_uniform(-20., -3.).svd_singular_values();
assert_near(s.to_point(), Point::new(20., 3.));
let s = Affine::scale_non_uniform(20., -3.).svd().0;
let s = Affine::scale_non_uniform(20., -3.).svd_singular_values();
assert_near(s.to_point(), Point::new(20., 3.));

// One more property: given a full-rank transform, the product of its singular values
// should be equal to its absolute determinant.
let m = mat(10., 9., -2.5, 3.3333);
let s = m.svd().0;
let s = m.svd_singular_values();
let prod = s.x * s.y;
let det = m.determinant().abs();
assert!(
Expand Down
20 changes: 12 additions & 8 deletions kurbo/src/ellipse.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -78,7 +78,7 @@ impl Ellipse {
#[inline]
#[must_use]
pub fn with_radii(self, new_radii: Vec2) -> Ellipse {
let rotation = self.inner.svd().1;
let rotation = self.inner.svd_post_scale_rotation();
let translation = self.inner.translation();
Ellipse::private_new(translation, new_radii.x, new_radii.y, rotation)
}
Expand All @@ -91,7 +91,7 @@ impl Ellipse {
#[inline]
#[must_use]
pub fn with_rotation(self, rotation: f64) -> Ellipse {
let scale = self.inner.svd().0;
let scale = self.inner.svd_singular_values();
let translation = self.inner.translation();
Ellipse::private_new(translation, scale.x, scale.y, rotation)
}
Expand Down Expand Up @@ -125,23 +125,23 @@ impl Ellipse {
/// consider using [`Ellipse::major_radius`] or [`Ellipse::minor_radius`] instead.
#[inline]
pub fn radii(&self) -> Vec2 {
self.inner.svd().0
self.inner.svd_singular_values()
}

/// Returns the major radius of this ellipse.
///
/// This metric is also known as the semi-major axis.
#[inline]
pub fn major_radius(&self) -> f64 {
self.inner.svd().0.x
self.inner.svd_singular_values().x
}

/// Returns the minor radius of this ellipse.
///
/// This metric is also known as the semi-minor axis.
#[inline]
pub fn minor_radius(&self) -> f64 {
self.inner.svd().0.y
self.inner.svd_singular_values().y
}

/// The ellipse's rotation, in radians.
Expand All @@ -150,15 +150,18 @@ impl Ellipse {
/// an ellipse with the two radii on the x and y axes.
#[inline]
pub fn rotation(&self) -> f64 {
self.inner.svd().1
self.inner.svd_post_scale_rotation()
}

/// Returns the radii and the rotation of this ellipse.
///
/// Equivalent to `(self.radii(), self.rotation())` but more efficient.
#[inline]
pub fn radii_and_rotation(&self) -> (Vec2, f64) {
self.inner.svd()
(
self.inner.svd_singular_values(),
self.inner.svd_post_scale_rotation(),
)
}

/// Is this ellipse [finite]?
Expand Down Expand Up @@ -224,7 +227,8 @@ impl Shape for Ellipse {
type PathElementsIter<'iter> = iter::Chain<iter::Once<PathEl>, ArcAppendIter>;

fn path_elements(&self, tolerance: f64) -> Self::PathElementsIter<'_> {
let (radii, x_rotation) = self.inner.svd();
let radii = self.inner.svd_singular_values();
let x_rotation = self.inner.svd_post_scale_rotation();
Arc {
center: self.center(),
radii,
Expand Down
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