From Computer-Aided Design, vol. 22, November 1999: Pierre Etienne Bézier was born on September 1, 1910 in Paris. Son and grandson of engineers, he chose this profession too and enrolled to study mechanical engineering at the Ecole des Arts et Metiers and received his degree in 1930. In the same year he entered the Ecole Superieure d’Electricite and earnt a second degree in electrical engineering in 1931. In 1977, 46 years later, he received his DSc degree in mathematics from the University of Paris. In 1933, aged 23, Bézier entered Renault and worked for this company for 42 years. He started as Tool Setter, became Tool Designer in 1934 and Head of the Tool Design Office in 1945. In 1948, as Director of Production Engineering he was responsable for the design of the transfer lines producing most of the 4 CV mechanical parts. In 1957, he became Director of Machine Tool Division and was responsable for the automatic assembly of mechanical components, and for the design and production of an NC drilling and milling machine, most probably one of the first machines in Europe. Bézier become managing staff member for technical development in 1960 and held this position until 1975 when he retired. Bézier started his research in CADCAM in 1960 when he devoted a substantial amount of his time working on his UNISURF system. From 1960, his research interest focused on drawing machines, computer control, interactive free-form curve and surface design and 3D milling for manufactoring clay models and masters. His system was launched in 1968 and has been in full use since 1975 supporting about 1500 staff members today. Bézier’s academic career began in 1968 when he became Professor of Production Engineering at the Conservatoire National des Arts et Metiers. He held this position until 1979. He wrote four books, numerous papers and received several distinctions including the “Steven Anson Coons” of the Association for Computing Machinery and the “Doctor Honoris Causa” of the Technical University Berlin. He is an honorary member of the American Society of Mechanical Engineers and of the Societe Belge des Mecaniciens, ex-president of the Societe des Ingenieurs et Scientifiques de France, Societe des Ingenieurs Arts et Metiers, and he was one of the first Advisory Editors of “Computer-Aided Design”.

One of the oldest problems in mathematics is the interpolation problem.

Given a continuous function f defined on [a,b],find a function p from some class of “simple” functions such that

p(xî) = f(xî)

at some set of “nodes” xºo, x¡o, …, xño in [a,b].

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1

0.5

0.5

1

Special case: Given a continuous function f defined on [a,b], find the polynomial p of degree n or less such that

p(xî) = f(xî) at xºo, x¡o, …, xñoo in [a,b].

This is an “easy” problem, but the result is often not what we might expect:

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Carle RungeGöttingen, ca. 1911 … and it gets worse!

f(x) = (1 + 10ox €o)o — o⁄ 21 points

Can we do better if the nodes are chosen differently?Usually, but…

A theorem due to Georg Faber (1914):

Given any sequence of node sets

Uñ ={xºo, x¡o, …, xmñ } Ç [a,b], n = 1, 2, 3,…,

there exists a continuous function f on [a,b] such that the corresponding sequence of interpolating polynomials does not converge to f.

Berlin, 1885

Karl Weierstraß

Weierstraß Approximation Theorem

If f is continuous on [a ,b ], then for any e > 0 there exists a polynomial p such that

| f(x) - p(x) | < e for all x l [a ,b ].

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Polynomial approximation of oo|oxo|

SeriesAè!!!!!!!!!!!1 z , 8z, 0, 20<E

1 z2

z28

z316

5 z4128

7 z5256

21 z61024

33 z72048

429 z832768

715 z965536

2431 z10262144

4199 z11524288

29393 z124194304

52003 z138388608

185725 z1433554432

334305 z1567108864

9694845 z162147483648

17678835 z174294967296

64822395 z1817179869184

119409675 z1934359738368

883631595 z20274877906944 O@zD21

p@x_D Normal@%D ê. z x2 1

1 12 H1 x2L

18 H1 x2L2

116 H1 x2L3

5128 H1 x2L4

7256 H1 x2L5

21 H1 x2L61024

33 H1 x2L72048

429 H1 x2L832768

715 H1 x2L965536

2431 H1 x2L10262144

4199 H1 x2L11524288

29393 H1 x2L124194304

52003 H1 x2L138388608

185725 H1 x2L14

33554432 334305 H1 x2L15

67108864 9694845 H1 x2L162147483648

17678835 H1 x2L174294967296

64822395 H1 x2L1817179869184

119409675 H1 x2L1934359738368

883631595 H1 x2L20274877906944

Plot@8Abs@xD, p@xD<, 8x, 1, 1<,PlotRange 80, 1<, PlotStyle 88<, 8Thickness@.008D, Hue@0D<<D;

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o|oxo| as the limit of polynomials(It’s not a Taylor series!)

Weierstraß’s proof was not a constructive proof. Over the next 30 years or so, numerous other famous mathematicians gave alternative proofs.

Runge, Picard, Volterra, Mittag-Leffler, Lebesgue, Landau, de La Vallée Poussin, and ...

Kharkov, Ukraine, 1911:

Sergi Bernstein (1880–1968)published a constructive proof, based on what have becomeknown as the Bernstein polynomials.

He proved that, for a given continuousfunction f on [0, 1], the followingsequence of polynomials converges uniformly to f on [0, 1]: ooñ Pñ(x) = ∑ f(ñoø)×(o˚˜o)x o(1o-ox) ˜ — o ± ‚ ooñ Pñ(x) = ∑ f(ñoø)obño≤ ̊(x) o ± ‚

0.2 0.4 0.6 0.8 1

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1

The cubic Bernstein polynomials ob£o≤ ̊ on [0,1].

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1

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1 0.5 0.5 1

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P∞

P¡ º

P¡ ∞

P™ º

Bernsteinapproximations of |oxo|

Meanwhile, in the rea l world…

In the shipbuilding and early aircraft industries, full-sizepatterns were laid out for the construction of curved parts. This practice became known as lofting because it was often done in large lofts above the factory floor. “Loftsmen” did this work by bending thin wooden planks called splines, which were held in place at discrete points called “ducks.”

Eventually loftsmen began to use mathematical techniques to lay out curves, first using conic sections.

The same design and manufacturing issues were present in theautomobile industry.

The term “lofting” is still used today, even though all thedesign work is done by computer.

“Lofting is the act of creating smooth,controlled curves which define the surfaces of a fuselage or any other similarstreamlined body.”

1945, Army Ballistic Research Center(The “Aberdeen Proving Ground”)

I. J. Schoenberg (1903–1990) initiated the mathematical theoryof splines.

“For the next 15 years, Schoenberg had splines all to himself. This changed around 1960, when computers became more widespread and splines first assumed their role as the premier tool for data fitting and computer-aided geometric design. Schoenberg’s more than 40 papers on splines after 1960 gave much impetus to the rapid development of the field.”

R. Askey and C. de Boor, In memoriam: I. J. Schoenberg (1903-1990), J. Approx. Theory 63 (1) (1990), 1-2.

Splines (nonparametric form)

Given n “knots” (xº,oyºo), (x¡o,oy¡o), …, (xño,oyño), a degree-k spline is a function S defined on [xºo,oxño] with the following properties:

1. S(xî) = yî, i = 1, 2, …, n.

2. S(x) is a polynomial of degree ≤ k on each subinterval [xî,oxîoòoo¡o]. 3. S has continuous derivatives up through order ko-o1 on [xºo,oxño].

S(x) = ∑cîB((x-xî)/ãx),

where B is a simple function like these:

k = 1 k = 3

If the xî’s areequally spaced,

2 1 1 2

1

3 2 1 1 2 3

1

2

3

4

and the coefficients cî satisfy a relatively simple system of linear equations. (When k = 1, cî = yî.)

A cubic spline

“Nobody used splines because nobody knew how to compute with them, even in the late 1960s.” — J. Ferguson

“During the early sixties at the Renault corporation, where I was an engineer at the time, I went to see my supervisor to tell him that I had found a new mathematical method for drawing curves that would replace all previous rough calculations and other lathed shapes and models. He saw my project, looked at me, and said, ‘Monsieur Bézier, if your thing worked, the Americans would already be using it.”

In the late 1950s, the computer began to replace drafting/lofting splines. Several people are considered pioneers, including:

de Casteljau at Citroën, Birkhoff , Garabedian, de Boor at General Motors,

J. Ferguson at Boeing, M. Sabin at British Aircraft,

andPierre Bézier at Renault.

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Cubic Bézier Curve Segment

Given four “control points” P¡o, P™o, P£o, P¢o,

B(t) = (1-t) ‹ P¡ + 3ot(1-t) € P™ + 3ot € (1-t)P£ + t ‹ P¢

Bézier would not be nearly so well known today had there not been an importantdevelopment in the 1980s…

} bind def/Mdot { moveto 0 0 rlineto stroke} bind def/Mtetra { moveto lineto lineto lineto fill} bind def/Metetra { moveto lineto lineto lineto closepath gsave fill grestore 0 setgray stroke} bind def/Mistroke { flattenpath 0 0 0 { 4 2 roll pop pop } { 4 -1 roll 2 index sub dup mul

4 -1 roll 2 index sub dup mul add sqrt 4 -1 roll add 3 1 roll } { stop } { stop } pathforall pop pop currentpoint stroke moveto currentdash 3 -1 roll add setdash} bind def/Mfstroke { stroke currentdash pop 0 setdash} bind def/Mrotsboxa { gsave dup /Mrot exch def Mrotcheck Mtmatrix

dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def grestore Msboxa 3 -1 roll /Mtmatrix exch def /Mrot 0 def} bind def/Msboxa { newpath 5 -1 roll Mvboxa pop Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Msboxa1 5 -3 roll Msboxa1 Mboxrot

Dr. John Warnock , inventor of PostScript and founder of Adobe Systems.

BézierBézier

From Computer-Aided Design, vol. 22, November 1999: Pierre Etienne Bézier was born on September 1, 1910 in Paris. Son and grandson of engineers, he chose this profession too and enrolled to study mechanical engineering at the Ecole des Arts et Metiers and received his degree in 1930. In the same year he entered the Ecole Superieure d’Electricite and earnt a second degree in electrical engineering in 1931. In 1977, 46 years later, he received his DSc degree in mathematics from the University of Paris. In 1933, aged 23, Bézier entered Renault and worked for this company for 42 years. He started as Tool Setter, became Tool Designer in 1934 and Head of the Tool Design Office in 1945. In 1948, as Director of Production Engineering he was responsable for the design of the transfer lines producing most of the 4 CV mechanical parts. In 1957, he became Director of Machine Tool Division and was responsable for the automatic assembly of mechanical components, and for the design and production of an NC drilling and milling machine, most probably one of the first machines in Europe. Bézier become managing staff member for technical development in 1960 and held this position until 1975 when he retired. Bézier started his research in CADCAM in 1960 when he devoted a substantial amount of his time working on his UNISURF system. From 1960, his research interest focused on drawing machines, computer control, interactive free-form curve and surface design and 3D milling for manufactoring clay models and masters. His system was launched in 1968 and has been in full use since 1975 supporting about 1500 staff members today. Bézier’s academic career began in 1968 when he became Professor of Production Engineering at the Conservatoire National des Arts et Metiers. He held this position until 1979. He wrote four books, numerous papers and received several distinctions including the “Steven Anson Coons” of the Association for Computing Machinery and the “Doctor Honoris Causa” of the Technical University Berlin. He is an honorary member of the American Society of Mechanical Engineers and of the Societe Belge des Mecaniciens, ex-president of the Societe des Ingenieurs et Scientifiques de France, Societe des Ingenieurs Arts et Metiers, and he was one of the first Advisory Editors of “Computer-Aided Design”.