Simplify and optimize Ellipse::area (#559)

Instead of performing a singular value decomposition (via the
`Ellipse::radii` call), use the property that the area of a region
transformed by an affine is that area multiplied by the determinant of
the affine:

https://en.wikipedia.org/w/index.php?title=Determinant&oldid=1344268205#Geometric_meaning.

Author: tomcur

CHANGELOG.md

@@ -26,6 +26,7 @@ This release has an [MSRV][] of 1.85.
 
 - Improve performance of `RoundedRect::winding` and `RoundedRect::contains`. ([#534][] by [@tomcur][])
 - `Axis`, `Ellipse::from_affine`, `Insets`, `Line::reversed`, `Rect::min_x/max_x/min_y/max_y`, `RoundedRectRadii::abs/clamp/as_single_radius`, `RoundedRect::width/height/radii/rect/origin/center`, `Size::aspect_ratio`, `Stroke::new/with_join/with_miter_limit/with_start_cap/with_end_cap/with_caps` are all now `const`. ([#536][] by [@xStrom], [#539][] and [#561][] by [@tomcur][])
+- Improve performance of `Ellipse::area`. Note: for non-finite ellipses, this method may now return `f64::INFINITY` where it returned `f64::NAN` before. ([#559][] by [@tomcur][])
 
 ## Fixed
 
@@ -288,6 +289,7 @@ Note: A changelog was not kept for or before this release
 [#545]: https://github.com/linebender/kurbo/pull/545
 [#548]: https://github.com/linebender/kurbo/pull/548
 [#549]: https://github.com/linebender/kurbo/pull/549
+[#559]: https://github.com/linebender/kurbo/pull/559
 [#561]: https://github.com/linebender/kurbo/pull/561
 
 [Unreleased]: https://github.com/linebender/kurbo/compare/v0.13.0...HEAD

kurbo/src/ellipse.rs

@@ -235,10 +235,22 @@ impl Shape for Ellipse {
         .path_elements(tolerance)
     }
 
+    /// The area of the ellipse.
+    ///
+    /// This is always positive if the result is finite.
+    ///
+    /// If non-finite, this will be [`f64::INFINITY`] or [`f64::NAN`] depending on whether the
+    /// inner affine's [determinant](Affine::determinant) is [`f64::INFINITY`] (either positive or
+    /// negative) or [`f64::NAN`].
     #[inline]
     fn area(&self) -> f64 {
-        let Vec2 { x, y } = self.radii();
-        PI * x * y
+        // `Ellipse` is represented as a unit circle transformed by `Affine`. The transformed area
+        // of a region is `area * |det(affine)|`, see
+        // <https://en.wikipedia.org/w/index.php?title=Determinant&oldid=1344268205#Geometric_meaning>.
+        //
+        // A unit circle has area `PI`. Therefore, the area of this ellipse is PI multiplied by the
+        // affine's determinant.
+        PI * self.inner.determinant().abs()
     }
 
     /// Approximate the ellipse perimeter.