BUILT-IN MATHEMATICA SYMBOL

BezierCurve

is a graphics primitive that represents a Bézier curve with control points

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MORE INFORMATION

BezierCurve can be used in both Graphics and Graphics3D (two- and three-dimensional graphics).

The positions of control points can be specified either in ordinary coordinates as or , or in scaled coordinates as Scaled or Scaled.

In two dimensions, Offset and ImageScaled can be used to specify coordinates.

BezierCurve by default represents a composite cubic Bézier curve.

SplineDegree->d specifies that the underlying polynomial basis should have maximal degree d.

With SplineDegree->d, BezierCurve with d+1 control points yields a simple degree-d Bézier curve. With fewer control points, a lower-degree curve is generated. With more control points, a composite Bézier curve is generated. »

Curve thickness can be specified using Thickness or AbsoluteThickness, as well as Thick and Thin. »

Curve dashing can be specified using Dashing or AbsoluteDashing, as well as Dashed, Dotted, etc. »

Curve shading or coloring can be specified using CMYKColor, GrayLevel, Hue, Opacity, or RGBColor. »

Individual coordinates and lists of coordinates in BezierCurve can be Dynamic objects.

EXAMPLES

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Basic Examples (1)

A Bézier curve and its control points in 2D:

A Bézier curve and its control points in 3D:

A composite Bézier curve and its control points:

A Bézier curve and its control points in 2D:

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A Bézier curve and its control points in 3D:

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A composite Bézier curve and its control points:

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Scope (11)

A single cubic Bézier curve:

A composite Bézier curve:

Bézier curves of different degrees:

By default, a Bézier curve is open:

A closed Bézier curve automatically adds the first control point at the end:

Bézier curves with different thicknesses:

Thickness in scaled size:

Thickness in printer's points:

Dashed curves:

Colored curves:

Use Scaled coordinates:

Use ImageScaled coordinates in 2D:

Use Offset coordinates in 2D:

Generalizations & Extensions (3)

A single Bézier curve with degree d requires d+1 control points:

With fewer control points, a lower-degree curve is generated:

With more control points, a composite Bézier curve is generated:

Applications (7)

Approximate a circle with 4 Bézier curves:

A quadratic Bézier curve can be converted into a cubic Bézier curve:

Define the outline of a glyph:

Draw a tree plot:

Use BezierCurve instead of lines to draw the edges:

Choose 4 points to be interpolated:

Compute distances between control points:

Compute normalized parameters with respect to the distances (chord length parametrization):

Since a Bézier curve interpolates endpoints, you only need to compute two intermediate points:

The formula for the interpolating Bézier curve:

Solve the equations:

Show the interpolating curve:

Generate a list of points to be approximated:

Fit to a cubic Bézier curve, using Bernstein polynomials:

Show the data with the curve:

Construct control points from the coefficients:

Show the data with the curve:

Linear transition from one Bézier curve to another:

Properties & Relations (11)

A Bézier curve always interpolates the endpoints:

A Bézier curve with degree 1 is equivalent to Line:

A Bézier curve is affine invariant:

A single Bézier curve lies in the convex hull of the control points:

In 3D, a Bézier curve with planar control points lies in the plane:

The cubic Bernstein polynomials:

A Bézier curve can be constructed from the sum of the Bernstein polynomials:

A Bézier curve generated from the average of two sets of control points:

The new curve is indeed the average of two Bézier curves:

A composite Bézier curve may not be smooth at the point where two segments meet:

By making the adjacent points collinear, we can get a smooth composite Bézier curve:

A single BezierCurve is a special case of BSplineCurve:

In 3D, a single Bézier surface patch can be generated using BSplineSurface:

The boundaries of the surface form Bézier curves:

Interactive Examples (1)

A simple Bézier curve editor:

Neat Examples (2)

A random collection of cubic Bézier curves:

A composite Bézier flower: