brush-strokes

This library provides a toolkit for dealing with Bézier curves and Bézier splines, with a particular focus on computing the outline of a brush stroke in the spirit of Knuth's METAFONT.

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Bézier curves

This library mainly deals with quadratic and cubic Bézier curves, with Math.Bezier.Quadratic and Math.Bezier.Cubic providing basic operations such as differentiation, curvature, subdivision, and closest-point queries.

Bézier splines

Math.Bezier.Spline contains functionality for working with Bézier splines (made up of straight lines and quadratic/cubic Bézier curve). For example:

• subdividing a spline,
• deleting points from a spline,
• distinguishing between open and closed splines.

Arc-length parametrisation

Math.Bezier.ArcLength implements several numerical integration strategies for computing the arc length of a Bézier curve, and inverts the arc-length map via Newton–Raphson for arc-length re-parametrisation:

Integrator	Description
GaussLegendre	Gauss–Legendre quadrature
ClenshawCurtis	Clenshaw–Curtis quadrature
TanhSinh	Tanh–sinh (double-exponential) quadrature
Gravesen	Gravesen's adaptive polygon/chord method

These methods are also extended to Bézier splines.

Brush stroking

Math.Bezier.Stroke implements brush stroking, in which:

• the base path is a Bézier spline,
• the brush shape is a closed Bézier spline whose parameters can vary as the brush moves along the path; the parameters are themselves Bézier functions of the base path Bézier parameter.

Brush shapes are defined in Calligraphy.Brushes; predefined shapes include circular, elliptical, and teardrop brushes.

The stroking algorithm:

• Computes outline points by numerically solving the envelope equation using (forward-mode) automatic differentiation. (See Math.Bezier.Stroke.EnvelopeEquation for details.)
• Detects cusps via real root isolation algorithms implemented using interval arithmetic.
• Splits up the base path at cuspidal points.
• Fits a cubic Bézier spline to the computed outline, using Hoschek's least-squares curve fitting algorithm (see Math.Bezier.Cubic.Fit).
• Joins up adjacent stroke segments at corners.

Cusps

The most complex (and computationally intensive) step in the above algorithm is the computation of cusps, using real root isolation with interval arithmetic.

The cusp equation is formulated in Math.Bezier.Stroke.EnvelopeEquation using implicit differentiation of the envelope equation. (There is also a secondary cusp equation corresponding to corners of the brush, if the brush has corners.)

The cusps benchmark suite compares the performance of the root isolation algorithms for cusp finding on a selection of brush strokes that arose in testing.

Root finding

Math.Roots implements a few basic 1D root-finding algorithms:

• A quadratic equation solver,
• A Newton–Raphson implementation with Armijo line search,
• Laguerre's method,
• A modified Halley M2 method by Cordero, Ramos, Torregrosa (2020).

A very basic n-dimensional Newton's method is also implemented, calling out to eigen for inversion of the Jacobian.

Real root isolation

Math.Root.Isolation provides general-purpose real root isolation algorithms for general multi-dimensional non-linear systems.

Available algorithms include bisection, the interval Newton method, and narrowing methods (e.g. Kubica (2017) and Goldsztejn & Goualard (2010)).

For interval Newton, two methods are available for inverting the interval Jacobian:

• Gauss–Seidel step (with option to use the complete Gauss–Seidel method, Montanher, Domes, Schichl, Neumaier (2017)),
• (2D only) a highly-performant linear-programming approach, kindly contributed by Walter Figueiredo Mascarenhas; see cbits/lp_2d.cpp.

It is possible to customise the pipeline to choose in which order and when to apply the different algorithms when performing branch-and-bound style search.

Interval arithmetic

Math.Interval provides rigorously rounded intervals and n-dimensional interval boxes.

Three back-ends for computing correctly-rounded interval arithmetic operations are selectable at build time via Cabal flags:

Flag	Description
(default)	Use rounded-hw
use-fma	Use Kahan TwoSum and FMA TwoProd
use-simd	Use 128-bit SIMD registers

Automatic differentiation

Math.Algebra.Dual implements k-th order dual numbers for forward-mode automatic differentiation in n variables, used throughout the library for derivatives, curvature, and critical-point detection.

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Other utilities

Arc-length test suite

The test-arc-length test executable can output relative error data to a CSV file, useful for visualising which Béziers each integrator struggles with:

cabal run test-arc-length -- csv --output errors.csv INTEGRATOR [--grid-size STEP]

Arc-length benchmark suite

The bench-arc-length executable compares the speed and accuracy of various arc-length integrators across a selection of quadratic and cubic Béziers.

On top of command-line output, it also writes the results to bench_results.csv. These results can be visualised using the included helper Python script plot_bench.py (requires numpy and matplotlib):

python src/arc-length/bench/plot_bench.py bench_results.csv