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Bezier curves 
 


Information on Bezier curves by Darren Meyer. Check also the essay by Chris Bentley. [Google] [More] ⦿  
A Tribute to Pierre Bézier (19101999)
 This tribute to Bezier states: At least two mathematicians solved the problem before Bezier: Airplane designer James Ferguson, and engineer Paul de Casteljau who worked for Citroen. The latter's work is mathematically equivalent to Bezier, in fact the formula listed above is De Casteljau's. Unfortunately, their discoveries were closely guarded industrial secrets and were not published until after Bezier. [Google] [More] ⦿ 
Alex explains npoint Bezier curves graphically. [Google] [More] ⦿  
Londoner who created the experimental Bezierdriven Blended Alphabet in 2009. [Google] [More] ⦿  
Barsky and Bézier
 Pierre Bézier (born in Paris on 1 September 1910, died on 25 November 1999) was a friend of Brian Barsky, a famous graphics professor at Berkeley, and an exgraduate of my own university, McGill. Bézier gave Barsky a wonderful Bézier curve drawing, signed and dated. This is a thing of beauty. [Google] [More] ⦿ 
Japanese page explaining about the use of Bézier curves in type design. [Google] [More] ⦿  
L.S. Ng explains the mathematical aspects of truetype to type 1 Bezier conversions (quadratic to cubic). Conversions from quadratic Beziers (truetype Beziers) to cubic (type 1) Beziers. [Google] [More] ⦿  
A Bézier curve is a parametric curve frequently used in computer graphics and related fields such as type design. Bézier curves are used to model smooth curves that can be scaled indefinitely. The mathematical basis for Bézier curves is the Bernstein polynomial (1912), but its applicability to graphics was understood half a century later. Bézier curves were widely publicized in 1962 by the French engineer Pierre Bézier who used them to design automobile bodies at Renault. The study of these curves was first developed in 1959 by mathematician Paul de Casteljau using de Casteljau's algorithm, a numerically stable method to evaluate Bézier curves, at Citroën, another French automaker. TrueType fonts use Bézier splines composed of quadratic Bézier curves. Type 1 or PostScript fonts use cubic Bézier curves. Imaging systems like PostScript, Metafont, and SVG use Bézier splines composed of cubic Bézier curves for drawing curved shapes. OpenType fonts can use either kind, depending on the flavor of the font. [Google] [More] ⦿  
Essay by Steven Hollasch on possible conversions. [Google] [More] ⦿  
Model for curves that some like better than Bezier curves in terms of beauty and smoothness. See the discussion here. Image. [Google] [More] ⦿  
Frenchman Marie Alfred Cornu proposed his famous Cornu spirals in 1874, also known as Klothoids. They have the desirable property that curves can blend smoothly into lines, much more so than Bezier curves can blend into straight lines. Bezierbased outlines thus exhibit the characteristic kinks. To avoid them, one has to resort to design by hand. This has annoyed the designers of FF DIN Round (AlbertJan Pool, FontShop). Clearly, there is a need to broaden the palette of curves from which type designers can choose. The FF DIN Round designers "simulated" the klothoid by using more Beziers than normal, to trick the eye. However, the designers failed to note a remarkble property of Bezier curves. If one places the first three Bezier points on a line (the start point and two control points), and the last (end) point off the line, then the transition away from the line is "cubic", not "quadratic". Now, a simple math exercise shows that klothoids have precisely this cubic behavior that is so eyepleasing. Did Pool try this trick? Why was it not used? [Google] [More] ⦿  
Drawing circles in Postscript is not possible using just Beziers. For best approximations, see G. Adam Stanislav's page. [Google] [More] ⦿  
In the course of making Bezier curves, one wants to fit ellipses to sets of points. A fast and stable method is described in this paper: "Numerically stable direct least squares fitting of ellipses" (Radim Halir (Charles University) and Jan Flusser (Academy of Sciences of the Czech Republic)). [Google] [More] ⦿  
G. Scott Owen's page on Bezier curves, with an applet to boot. [Google] [More] ⦿  
Geometric Modelling
 Ken Joy (University of California at Davis) has lots of didactic online material on geometric modelling in general and on Bezier curves in particular. [Google] [More] ⦿ 
Geometry, Surfaces, Curves, Polyhedra
 Wonderful pages on various curves in 2d and 3d by University of Western Australia Professor Paul Bourke. Subpage on Bezier curves. [Google] [More] ⦿ 
Don Lancaster's great links on Bezier curves. Has a bibliography as well. [Google] [More] ⦿  
Harrisson
 
Jan Gerner
 
Jan Gerner
 
Several numerical procedures involving Bezier curves are explained by Jim Fitzsimmons. [Google] [More] ⦿  
Ken Joy
 
Graphic designer and illustrator in Kiev, Ukraine. Her remarkable work includes some unnamed experimental Cyrillic typefaces (2014), and a Bezier curvature experimenta that is called Coordinate (2014). [Google] [More] ⦿  
Open Source Publishing (or: OSP)
 Free software project based in Belgium and run by four people (and I quote from their web page):

Paul Bourke
 
Peter Vollenweider
 
Pierre Bézier
 
Pierre Bézier
 
From ComputerAided Design, vol. 22, November 1999, an obituary: 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 freeform 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, expresident 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 "ComputerAided Design". Wikipedia link. [Google] [MyFonts] [More] ⦿  
Type and technology blog by Benicia, CAbased computer scientist Raph Levien, who is totally committed to free and open software. Software guru who was a lead developer for Gfonted and Spiro (a font editor), and helped out with Gimp, among many other things. Raph Levien is an expert on fonts and graphics technologies, and is currently an engineer with the Google Web Fonts project. The topic for his PhD in Computer Science from the University of California, Berkeley, is on better techniques for interactively designing curves, and he also used these tools to design Inconsolata, one of the fonts available on the font API (see CTAN). Raph is working on a revival of ATF Century Catalogue, and proposes it as a replacement for the skinny Computer Modern fonts used in TeX. Other fonts in the pipeline include Century Catalogue, Bruce Rogers' Centaur types, Museum Caps, LeBe Titling, LeBe Book, ATF Bodoni, ATF Franklin Gothic, and the monospaced programming font Inconsolata (2005; see also here and here for this relative of Franklin Gothic). In 2007, he finally published the Museum Fonts package based on historical metal Centaur fonts, all free. He writes:
 
Rechenzentrum Universität Zürich
 PostScript information and sample programs at RZU. Site by Peter Vollenweider with a ton of information. There is a crash course on Bezier curves, a type 1 version of Frutiger 47, and a random type 3 font, with line by line explanations. In German. [Google] [More] ⦿ 
Comparison between truetype and postscript at Zentrum Informatikdienste of the University of Zürich. Essay on Bezier curves as well. [Google] [More] ⦿  
Speed Punk
 A learning tool for better understanding Bézier curves, developed in 2011 at KABK by Jan Gerner, aka Yanone. See also his talk at ATypI 2013 in Amsterdam. [Google] [More] ⦿ 
Saltire Software's nice applet showing de Casteljau's construction for computing points on cubic splines (Bezier curves). [Google] [More] ⦿  
Professor at Brigham Young University (Utah), who has a beautiful set of course notes online for his graduate course on ComputerAided Geometric Design. It has virtually everything one needs to know about Bezier curves (chapter 2), and deals with PostScript, rational Bezier curves (with which one can make perfect circles), Bsplines, and advanced material. [Google] [More] ⦿  
Fun with Bezier curves in Flash. By Tomek Zemla. [Google] [More] ⦿  
Roger Whitlock explains about Bezier curves. [Google] [More] ⦿  
Urs Oswald
 
Urs Oswald: Bezier Curves
 URS Oswald explains Bezier curves in a simple and authoritative manner. Includes material on MetaPost. [Google] [More] ⦿ 
Yanone
 Jan Gerner (b. 1982, Dresden) from Weimar runs Yanone. He grew up in Addis Abbeba. He studied Media at the Bauhaus Universität Weimar in 2003. He still lives and works in Weimar. He designed the (free) informal sans family YanoneKaffeesatz (2005), which is analyzed by Gerrit van Aken. YanoneTagesschrift (2005) is a serif obtained by scanning the felt tip pen traces of a printed serif facea nice idea! The font is now at Schriftgestaltung. The pixel font Al Abdali 8 was created during his stay with Syntax in Amman/Jordan to match the growing need for joint Arabic/Latin typefaces. Dafont link. In 2006, Yanone started selling some fonts through MyFonts. These include Monospasz (2006, a manually produced monospace typewriter font family in 5 styles), Liebfraumilch (2009, connected hand) and Pochoir (2006, stencil). And in 2009, the popular free Kaffeesatz became a pay font, FF Kava, at FontFont. In 2010, Yanone published FF Amman for Latin and Arabica bit too angular for my taste, but it has its uses. In 2011, he obtained a Masters at KABK in the type and media program. His graduation typeface was Antithesis (2011)it consists of a slab serif, a connected script and a heavy sans. All three have a handprinted look and should be fine faces for signage. Page dedicated to Antithesis. The FontFont version, FF Antithesis, appeared in 2013. At ATypI 2013 in Amsterdam, he introduces Speed Punk, a learning tool to better understand the nature of Bézier curves and their curvature. See also here. Font Squirrel link. Dafont link. [Google] [MyFonts] [More] ⦿ 

