View source for Bézier spline

A [[Spline|spline]] curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b1104701.png" />, where each polynomial spline segment is expressed in terms of [[Bernstein polynomials|Bernstein polynomials]] of a fixed degree. If the Bézier spline consists of only one segment, one speaks of a Bézier curve (cf. also [[Bézier curve|Bézier curve]]). Bézier splines and curves are mainly used in the field of computer aided geometric design (CAGD), which is concerned with the design, approximation and representation of curves and surfaces by a computer. The Bézier representation overcomes numerical and geometric drawbacks of other polynomial forms. Bézier curves and surfaces were independently developed by P. de Casteljau at Citroën (about 1959) and by P. Bézier at Rénault (about 1962) for the construction of car bodies.

Given an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b1104702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b1104703.png" />, the Bernstein polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b1104704.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b1104705.png" /> (cf. [[Bernstein polynomials|Bernstein polynomials]]) are defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b1104706.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b1104707.png" /></td> </tr></table>

In many applications, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b1104708.png" /> and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b1104709.png" />. Every polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047010.png" /> can be uniquely expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047012.png" />. Now, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047013.png" /> points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047015.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047017.png" />), the polynomial parametric curve

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is said to be a Bézier curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047020.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047021.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047022.png" /> are called Bézier points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047023.png" /> and they form the vertices of its so-called Bézier polygon. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047024.png" /> the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047025.png" /> lies in the convex hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047026.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047028.png" />, and the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047030.png" /> are tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047031.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047032.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047033.png" />. The following de Casteljau algorithm is an efficient and stable method for evaluating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047034.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047035.png" />: Setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047037.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047038.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047040.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047041.png" />.

Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047042.png" /> real values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047043.png" /> be given with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047045.png" />. Then a piecewise-polynomial continuous curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047046.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047048.png" />) is called a Bézier spline of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047049.png" /> if and only if each curve segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047051.png" />, is a Bézier curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047052.png" />, that is, it has a representation

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The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047054.png" />-continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047055.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047057.png" />. There are two concepts of continuity for the inner knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047058.png" />. First, one can use the usual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047059.png" />-continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047060.png" /> at the inner knots with respect to the given parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047061.png" /> for each coordinate function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047062.png" /> (cf. [[Spline|Spline]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047063.png" />-continuous (Bézier) splines are sufficient for many practical applications.

But, since a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047064.png" /> Bézier spline can have singularities, the weaker concept of geometric continuity was introduced. It is known that each rectifiable curve can be reparametrized so that the new parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047065.png" /> is arc length (see [[Natural parameter|Natural parameter]]). A curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047066.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047068.png" />-continuous at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047069.png" /> if and only if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047070.png" />-continuous at this point with respect to arc length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047071.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047072.png" />-continuity implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047073.png" />-continuity. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047074.png" /> (Bézier) spline is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047075.png" />-continuous at its inner knots. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047076.png" />-continuity can be characterized by tangent continuity. Furthermore, a Bézier spline <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047077.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047078.png" />-continuous if and only if it has a continuous Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]]) and a continuous curvature at each inner knot. Explicit formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047079.png" />-continuity involving Bézier points can be found in the references below. Special representations of cubic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047080.png" /> Bézier splines are cubic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047082.png" />-splines and cubic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110470/b11047084.png" />-splines (see [[#References|[a3]]] or [[#References|[a4]]]). For rational Bézier splines see the references below.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Boehm,   G. Farin,   J. Kahmann,   "A survey of curve and surface methods in CAGD"  ''Computer Aided Geometric Design'' , '''1'''  (1984)  pp. 1–60</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Encarnaçao,   W. Straßer,   R. Klein,   "Datenverarbeitung 1. Gerätetechnik, Programmierung und Anwendung graphischer Systeme" , R. Oldenbourg  (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Farin,   "Curves and surfaces for computer aided geometric design. A practical guide" , Acad. Press  (1993)  (Edition: Third)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Hoschek,   D. Lasser,   "Grundlagen der geometrischen Datenverarbeitung" , Teubner  (1992)  (Edition: Second)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.E Mortensen,   "Geometric modeling" , Wiley  (1985)</TD></TR></table>