Aleph::BezierCurve Class Reference

Quadratic and cubic Bézier curves with exact rational arithmetic. More...

#include <geom_algorithms.H>

Classes
struct	SplitResult
	De Casteljau subdivision: split a cubic Bézier at parameter t into two cubic Béziers (left and right). More...

Static Public Member Functions
static Point	quadratic (const Point &p0, const Point &p1, const Point &p2, const Geom_Number &t)
	Evaluate a quadratic Bézier at parameter t.
static Point	cubic (const Point &p0, const Point &p1, const Point &p2, const Point &p3, const Geom_Number &t)
	Evaluate a cubic Bézier at parameter t.
static Array< Point >	sample_quadratic (const Point &p0, const Point &p1, const Point &p2, const size_t n)
	Sample a quadratic Bézier into n+1 points (t = 0, 1/n, ..., 1).
static Array< Point >	sample_cubic (const Point &p0, const Point &p1, const Point &p2, const Point &p3, const size_t n)
	Sample a cubic Bézier into n+1 points.
static SplitResult	split_cubic (const Point &p0, const Point &p1, const Point &p2, const Point &p3, const Geom_Number &t)
static Rectangle	control_bbox (const Point &p0, const Point &p1, const Point &p2, const Point &p3)
	Compute the bounding box of a cubic Bézier's control polygon.
static Polygon	approximate_quadratic (const Point &p0, const Point &p1, const Point &p2, const size_t n)
	Approximate a quadratic Bézier as a polyline (polygon without closing).
static Polygon	approximate_cubic (const Point &p0, const Point &p1, const Point &p2, const Point &p3, const size_t n)
	Approximate a cubic Bézier as a polyline.

Detailed Description

Quadratic and cubic Bézier curves with exact rational arithmetic.

A Bézier curve is a parametric curve defined by control points.

• Quadratic (3 control points): B(t) = (1-t)²P₀ + 2t(1-t)P₁ + t²P₂
• Cubic (4 control points): B(t) = (1-t)³P₀ + 3t(1-t)²P₁ + 3t²(1-t)P₂ + t³P₃

Since the parameter t ∈ [0,1] is rational, evaluation is exact.

Example

Point p = BezierCurve::cubic(Point(0,0), Point(1,2), Point(2,2), Point(3,0),
                            Geom_Number(1) / Geom_Number(2));
auto poly = BezierCurve::approximate_cubic(Point(0,0), Point(1,2), Point(2,2), Point(3,0), 32);

Definition at line 11133 of file geom_algorithms.H.

Member Function Documentation

approximate_cubic()

static Polygon Aleph::BezierCurve::approximate_cubic ( const Point &  p0, const Point &  p1, const Point &  p2, const Point &  p3, const size_t  n  )

inlinestatic

Approximate a cubic Bézier as a polyline.

Definition at line 11275 of file geom_algorithms.H.

References Aleph::Polygon::add_vertex(), Aleph::divide_and_conquer_partition_dp(), and sample_cubic().

approximate_quadratic()

static Polygon Aleph::BezierCurve::approximate_quadratic ( const Point &  p0, const Point &  p1, const Point &  p2, const size_t  n  )

inlinestatic

Approximate a quadratic Bézier as a polyline (polygon without closing).

Definition at line 11263 of file geom_algorithms.H.

References Aleph::Polygon::add_vertex(), Aleph::divide_and_conquer_partition_dp(), and sample_quadratic().

control_bbox()

static Rectangle Aleph::BezierCurve::control_bbox ( const Point &  p0, const Point &  p1, const Point &  p2, const Point &  p3  )

inlinestatic

Compute the bounding box of a cubic Bézier's control polygon.

Definition at line 11243 of file geom_algorithms.H.

References Aleph::divide_and_conquer_partition_dp(), Aleph::Point::get_x(), and Aleph::Point::get_y().

Referenced by TEST_F().

cubic()

static Point Aleph::BezierCurve::cubic ( const Point &  p0, const Point &  p1, const Point &  p2, const Point &  p3, const Geom_Number &  t  )

inlinestatic

Evaluate a cubic Bézier at parameter t.

Definition at line 11154 of file geom_algorithms.H.

References Aleph::divide_and_conquer_partition_dp(), Aleph::Point::get_x(), and Aleph::Point::get_y().

Referenced by sample_cubic(), TEST_F(), TEST_F(), and TEST_F().

quadratic()

static Point Aleph::BezierCurve::quadratic ( const Point &  p0, const Point &  p1, const Point &  p2, const Geom_Number &  t  )

inlinestatic

Evaluate a quadratic Bézier at parameter t.

Definition at line 11138 of file geom_algorithms.H.

References Aleph::divide_and_conquer_partition_dp(), Aleph::Point::get_x(), and Aleph::Point::get_y().

Referenced by sample_quadratic(), TEST_F(), and TEST_F().

sample_cubic()

static Array< Point > Aleph::BezierCurve::sample_cubic ( const Point &  p0, const Point &  p1, const Point &  p2, const Point &  p3, const size_t  n  )

inlinestatic

Sample a cubic Bézier into n+1 points.

Definition at line 11188 of file geom_algorithms.H.

References ah_domain_error_if, cubic(), Aleph::divide_and_conquer_partition_dp(), and Aleph::Array< T >::reserve().

Referenced by approximate_cubic(), and TEST_F().

sample_quadratic()

static Array< Point > Aleph::BezierCurve::sample_quadratic ( const Point &  p0, const Point &  p1, const Point &  p2, const size_t  n  )

inlinestatic

Sample a quadratic Bézier into n+1 points (t = 0, 1/n, ..., 1).

Definition at line 11176 of file geom_algorithms.H.

References ah_domain_error_if, Aleph::divide_and_conquer_partition_dp(), quadratic(), and Aleph::Array< T >::reserve().

Referenced by approximate_quadratic(), and TEST_F().

split_cubic()

static SplitResult Aleph::BezierCurve::split_cubic ( const Point &  p0, const Point &  p1, const Point &  p2, const Point &  p3, const Geom_Number &  t  )

inlinestatic

Definition at line 11209 of file geom_algorithms.H.

References Aleph::divide_and_conquer_partition_dp(), Aleph::Point::get_x(), Aleph::Point::get_y(), and Aleph::BezierCurve::SplitResult::left.

Referenced by TEST_F().

---

The documentation for this class was generated from the following file:

• geom_algorithms.H