WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

BezierCurve
✖
BezierCurve

✖
BezierCurve[{pt1,pt2,…}]

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

Details and Options

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

A Bézier curve and its control points in 2D:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-nymihs
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-d5ur72
Out[2]=2

A Bézier curve and its control points in 3D:

In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-hlqhf
In[4]:=4
✖
https://wolfram.com/xid/0cf3js6tiak2lem-ow240
Out[4]=4

A composite Bézier curve and its control points:

In[5]:=5
✖
https://wolfram.com/xid/0cf3js6tiak2lem-dfjac0
In[6]:=6
✖
https://wolfram.com/xid/0cf3js6tiak2lem-b544qc
Out[6]=6

Scope  (17)Survey of the scope of standard use cases

Graphics  (11)

Specification  (4)

A single cubic Bézier curve:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-3ox1f9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-zhulbh
Out[2]=2

A composite Bézier curve:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-s6aykc
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-9m5poc
Out[2]=2

Bézier curves of different degrees:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-vvx2i7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-tww1t2
Out[2]=2

By default, a Bézier curve is open:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-olfw0d
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-g9j71y
Out[2]=2

Styling  (4)

Bézier curves with different thicknesses:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-rhselg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-54nygb
Out[2]=2

Thickness in scaled size:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-d1pl7g
Out[1]=1

Thickness in printer's points:

In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-pdg0bx
Out[2]=2

Dashed curves:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-zflhi8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-busjcw
Out[2]=2

Colored curves:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-qisk2m
Out[1]=1

Coordinates  (3)

Use Scaled coordinates:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-nb9g9m
Out[1]=1

Use ImageScaled coordinates in 2D:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-c0gi38
Out[1]=1

Use Offset coordinates in 2D:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-g2at7e
Out[1]=1

Regions  (6)

Embedding dimension:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-y220
Out[1]=1

Geometric dimension:

In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bx9tom
Out[2]=2

Point membership test:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-di0yv6
Out[1]=1

Arc length:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-lb51on
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bs1xd
Out[2]=2

Centroid:

In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-m2h1an
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cf3js6tiak2lem-sgjro6
Out[4]=4

Distance from a point:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-oc6hy
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bjikjq
Out[2]=2

The distance to the nearest point for the Bézier curve:

In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-da2zys
Out[3]=3

Signed distance from a point:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-cybvpc
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bm4ed
Out[2]=2

Signed distance to the Bézier curve:

In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-g7ezsg
Out[3]=3

A Bézier curve is bounded:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-cyjdcd
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-b5d6xy
Out[2]=2

Get its range:

In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-1pdcpd
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cf3js6tiak2lem-mu3hj3
Out[4]=4

Generalizations & Extensions  (3)Generalized and extended use cases

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

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-mfpx
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-f2dwk
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-kpb00m
Out[1]=1

Applications  (7)Sample problems that can be solved with this function

Graphics, Glyphs, etc.  (4)

Approximate a circle with 4 Bézier curves:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-ca8cal
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-hz9gd2
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-e55vjt
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-belmwk
In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-c5ehkq
In[4]:=4
✖
https://wolfram.com/xid/0cf3js6tiak2lem-ctda3p
Out[4]=4

Define the outline of a glyph:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-e2mw4j
Out[1]=1

Draw a tree plot:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-tr4gn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-j9jb0d
Out[2]=2

Use BezierCurve instead of lines to draw the edges:

In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-k6nwtu
In[4]:=4
✖
https://wolfram.com/xid/0cf3js6tiak2lem-b9ty3l
Out[4]=4

Interpolation  (1)

Choose 4 points to be interpolated:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-lvla
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-o8lerq
Out[2]=2

Compute distances between control points:

In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-nbqjn
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0cf3js6tiak2lem-jo0blv
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0cf3js6tiak2lem-fe6p8c

The formula for the interpolating Bézier curve:

In[6]:=6
✖
https://wolfram.com/xid/0cf3js6tiak2lem-f708xa

Solve the equations:

In[7]:=7
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bvrbfr
In[8]:=8
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bjgxsa
Out[8]=8

Show the interpolating curve:

In[9]:=9
✖
https://wolfram.com/xid/0cf3js6tiak2lem-mpw1e
Out[9]=9

Least Squares Fitting  (1)

Generate a list of points to be approximated:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-d724oz
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-cpd7xj
Out[2]=2

Fit to a cubic Bézier curve, using Bernstein polynomials:

In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-c4ycf1
Out[3]=3

Show the data with the curve:

In[4]:=4
✖
https://wolfram.com/xid/0cf3js6tiak2lem-fgaefe
Out[4]=4

Construct control points from the coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bzazf0
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0cf3js6tiak2lem-ta4hn
Out[6]=6

Show the data with the curve:

In[7]:=7
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bcozo7
Out[7]=7

Geometric Invariances  (1)

Linear transition from one Bézier curve to another:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-deyq9b
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-hnitoa
In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-cb2cyk
Out[3]=3

Properties & Relations  (11)Properties of the function, and connections to other functions

A Bézier curve always interpolates the endpoints:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-cz6aeg
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bcy7va
Out[2]=2

A Bézier curve with degree 1 is equivalent to Line:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-fblwnd
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-cj72vq
Out[2]=2

A Bézier curve is affine invariant:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-wdkjj
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-3n32i
In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-b0i8ws
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-xqbu0p
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-hjoxqi
In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-gkb8s4
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-coznj8
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-dv8tcs
In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-ojl06
In[4]:=4
✖
https://wolfram.com/xid/0cf3js6tiak2lem-hosjau
Out[4]=4

The cubic Bernstein polynomials:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-cltgwx
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-1ghus
In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bwxk19
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cf3js6tiak2lem-ivlhbf
In[5]:=5
✖
https://wolfram.com/xid/0cf3js6tiak2lem-5lyc5
Out[5]=5

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

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-rxu5a
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-n3dkz
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-zx0il
In[4]:=4
✖
https://wolfram.com/xid/0cf3js6tiak2lem-kwryn
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-dq9607
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-kdbgp
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-bpkw27
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-92bb0
Out[2]=2

A single BezierCurve is a special case of BSplineCurve:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-dju1g9
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-fn3syw
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-f3rjwe
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3js6tiak2lem-c89t0n
Out[2]=2

The boundaries of the surface form Bézier curves:

In[3]:=3
✖
https://wolfram.com/xid/0cf3js6tiak2lem-be0h7q
Out[3]=3

Interactive Examples  (1)Examples with interactive outputs

A simple Bézier curve editor:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-f03g97
Out[1]=1

Neat Examples  (2)Surprising or curious use cases

A random collection of cubic Bézier curves:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-twyu6
Out[1]=1
In[58]:=58
✖
https://wolfram.com/xid/0cf3js6tiak2lem-4mx20
Out[58]=58

A composite Bézier flower:

In[1]:=1
✖
https://wolfram.com/xid/0cf3js6tiak2lem-hr5eo
Out[1]=1
Wolfram Research (2008), BezierCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/BezierCurve.html.
✖
Wolfram Research (2008), BezierCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/BezierCurve.html.

Text

Wolfram Research (2008), BezierCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/BezierCurve.html.

✖
Wolfram Research (2008), BezierCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/BezierCurve.html.

CMS

Wolfram Language. 2008. "BezierCurve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BezierCurve.html.

✖
Wolfram Language. 2008. "BezierCurve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BezierCurve.html.

APA

Wolfram Language. (2008). BezierCurve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BezierCurve.html

✖
Wolfram Language. (2008). BezierCurve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BezierCurve.html

BibTeX

@misc{reference.wolfram_2025_beziercurve, author="Wolfram Research", title="{BezierCurve}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/BezierCurve.html}", note=[Accessed: 02-June-2025 ]}

✖
@misc{reference.wolfram_2025_beziercurve, author="Wolfram Research", title="{BezierCurve}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/BezierCurve.html}", note=[Accessed: 02-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_beziercurve, organization={Wolfram Research}, title={BezierCurve}, year={2008}, url={https://reference.wolfram.com/language/ref/BezierCurve.html}, note=[Accessed: 02-June-2025 ]}

✖
@online{reference.wolfram_2025_beziercurve, organization={Wolfram Research}, title={BezierCurve}, year={2008}, url={https://reference.wolfram.com/language/ref/BezierCurve.html}, note=[Accessed: 02-June-2025 ]}