pst-bezier (21aug16)

git-svn-id: svn://tug.org/texlive/trunk@41899 c570f23f-e606-0410-a88d-b1316a301751

5 files changed, 553 insertions, 132 deletions

diff --git a/Master/texmf-dist/doc/generic/pst-bezier/Changes b/Master/texmf-dist/doc/generic/pst-bezier/Changes
index 66485d07ff9..149148692d8 100644
--- a/Master/texmf-dist/doc/generic/pst-bezier/Changes
+++ b/Master/texmf-dist/doc/generic/pst-bezier/Changes
@@ -1,10 +1,15 @@
 -- pst-bezier.tex ---
+0.02  2016-08-19 added macro \psRQBCmasse for a Bezier curve,
+                 definied by three weighted points
 0.01  2009-01-29 first CTAN version
 
 
 -- pst-bezier.sty ---
+0.02  2016-08-19 - load expl3 for floating point operations
+                 - define \pscalculation
 0.01  2009-01-29 first CTAN version
 
 
 -- pst-bezier.pro ---
+0.02  2016-08-19 added function tx@RQBCmasse for a Bezier curve
 0.01  2009-01-29 first CTAN version
diff --git a/Master/texmf-dist/doc/generic/pst-bezier/README b/Master/texmf-dist/doc/generic/pst-bezier/README.md
index c8f371e9078..23a4246c098 100644
--- a/Master/texmf-dist/doc/generic/pst-bezier/README
+++ b/Master/texmf-dist/doc/generic/pst-bezier/README.md
@@ -1,9 +1,9 @@
 Save the files pst-bezier.sty|tex in a directory, which is part of your 
 local TeX tree. pst-bezier.pro should be saved in ../texmf/dvips/pstricks/
 Then do not forget to run texhash to update this tree.
-pst-bezier needs pst-plot and pst-tricks, which should be part of your
-local TeX installation, otherwise get it from a CTAN server, f.ex.
-http://www.CTAN.org
+pst-bezier needs pst-plot and pstricks, which should be part of your
+local TeX installation, otherwise get it from a CTAN server
+http://mirror.CTAN.org
 
 
 Save the files
@@ -37,6 +37,3 @@ latex on the documentation file.
 
 The intermediate DVI file works only with viewers which can 
 interprete the embedded PostScript code.
-
-For another PDF output read the Introduction from
-the documentation.
diff --git a/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.bib b/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.bib
index d4c75592e2a..4493f87ffce 100644
--- a/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.bib
+++ b/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.bib
@@ -1,110 +1,148 @@
-@STRING{dtk	= {{D}ie {\TeX}nische {K}om{\"o}die} }
+@STRING{tugboat	= {TUGboat} }
+@STRING{beiprogramm	= {{\TeX}-Beiprogramm} }
+@STRING{bretter	= {Bretter, die die Welt bedeuten} }
+@STRING{dtk		= {{D}ie {\TeX}nische {K}om{\"o}die} }
+@STRING{editorial	= {Editorial} }
+@STRING{fremdebuehne	= {Von fremden B{\"u}hnen} }
+@STRING{fundus		= {Aus dem Fundus} }
+@STRING{hinterbuehne	= {Hinter der B{\"u}hne} }
+@STRING{leserbrief	= {Leserbrief(e)} }
+@STRING{magazin	= {Magazin} }
+@STRING{rezension	= {Rezensionen} }
+@STRING{schonimmer	= {Was Sie schon immer {\"u}ber {\TeX} wissen wollten \dots} }
+@STRING{theaterkasse	= {Von der Theaterkasse} }
+@STRING{theatertage	= {{\TeX}-Theatertage} }
 
-@Book{PostScript,
-  Author         = {Kollock, Nikolai G.},
-  Title          = {PostScript richtig eingesetzt: vom Konzept zum
-                   praktischen Einsatz},
-  Publisher      = {IWT},
-  Address        = {Vaterstetten},
-  year           = 1989,
+@Book{PSTricks2,
+  author	= {Herbert Vo\ss},
+  title		= {{\PST} {G}rafik f\"ur \TeX{} und \LaTeX},
+  edition	= {7},
+  publisher	= {DANTE -- Lehmanns},
+  year		= {2016},
+  address	= {Heidelberg/Berlin}
 }
 
-@Manual{pstricks,
-  Title          = {PSTricks - {\PS} macros for Generic TeX},
-  Author         = {Timothy Van Zandt},
-  Organization   = {},
-  Address        = {\url{http://www.tug.org/application/PSTricks}},
-  Note           = {},
-  year           = 1993,
+@Book{PSTricks-E,
+  author	= {Herbert Vo\ss},
+  title		= {{\PST} {G}raphics for \LaTeX},
+  edition	= {1},
+  publisher	= {UIT},
+  year		= {2011},
+  address	= {Cambridge}
 }
 
+@Book{companion04,
+  author	= {Frank Mittelbach and Michel Goosens et al},
+  title		= {The {\LaTeX} {C}ompanion},
+  edition	= {second},
+  publisher	= {Addison-Wesley Publishing Company},
+  year		= {2004},
+  address	= {Boston}
+}
 
-@Manual{pdftricks,
-  Title          = {PSTricks Support for pdf},
-  Author         = {Herbert Voss},
-  Organization   = {},
-  Address        = {\url{http://PSTricks.de/pdf/pdfoutput.phtml}},
-  Note           = {},
-  year           = 2002,
+@Book{unbound,
+  author	= {Alan Hoenig},
+  title		= {\TeX{} {U}nbound: \LaTeX{} \& \TeX{} {S}trategies, {F}onts, {G}raphics, and {M}ore},
+  publisher	= {Oxford University Press},
+  year		= {1998},
+  address	= {London}
 }
 
-@Manual{miwi,
-  Title          = {References for \TeX{} and Friends},
-  Author         = {Michael Wiedmann and Peter Karp},
-  Organization   = {},
-  Address        = {\url{http://www.miwie.org/tex-refs/}},
-  Note           = {},
-  year           = 2003,
+@Book{tlgc2,
+  author	= {Michel Goosens and Frank Mittelbach and Sebastian Rahtz and Denis Roegel and Herbert Vo{\ss}},
+  title		= {The {\LaTeX} {G}raphics {C}ompanion},
+  publisher	= {{Addison-Wesley Publishing Company}},
+  edition	= 2,
+  year		= {2007},
+  address	= {Reading, Mass.}
 }
 
+@Article{girou:01:,
+  author	= {Denis Girou},
+  title		= {Pr\'esentation de {PST}ricks},
+  journal	= {Cahier {GUT}enberg},
+  year		= 1994,
+  volume	= {16},
+  month		= apr,
+  pages		= {21--70}
+}
 
-@Article{dtk02.2:jackson.voss:plot-funktionen,
-  author	= {Laura E. Jackson and Herbert Vo{\ss}},
-  title		= {Die {P}lot-{F}unktionen von {\texttt{pst-plot}}},
-  journal	= dtk,
-  year		= 2002,
-  volume	= {2/02},
-  altvolume	= 2,
-  altnumber	= 14,
-  month		= jun,
-  pages		= {27--34},
-  annote	= bretter,
-  keywords	= {},
-  abstract	= { Im letzten Heft wurden die mathematischen Funktionen von
-		  \PS~im Zusammenhang mit dem {\LaTeX}-Paket
-		  \texttt{pst-plot} zum Zeichnen von Funktionen beschrieben
-		  und durch Beispiele erl{\"a}utert. In diesem Teil werden
-		  die bislang nur erw{\"a}hnten Plot-Funktionen f{\"u}r
-		  externe Daten behandelt. }
+@Article{girou:02:,
+  author	= {{Timothy Van} Zandt and Denis Girou},
+  title		= {Inside {PST}ricks},
+  journal	= TUGboat,
+  year		= 1994,
+  volume	= {15},
+  month		= sep,
+  pages		= {239--246}
 }
 
-@Article{dtk02.1:voss:mathematischen,
-  author	= {Herbert Vo{\ss}},
-  title		= {Die mathematischen {F}unktionen von {P}ostscript},
-  journal	= dtk,
-  year		= 2002,
-  volume	= {1/02},
-  altvolume	= 1,
-  altnumber	= 14,
-  month		= mar,
-  pages		= {40-47},
-  annote	= bretter,
-  keywords	= {},
-  abstract	= { \PS, faktisch genauso alt wie {\TeX}, ist im
-		  Verh{\"a}ltnis dazu allgemein noch weniger bekannt, wenn es
-		  darum geht zu beurteilen, was es denn nun im eigentlichen
-		  Sinne ist. Au{\ss}erdem wird h{\"a}ufig vergessen, dass
-		  sich mit den \PS-Funktionen viele Dinge erledigen lassen,
-		  bei denen sonst auf externe Programme zur{\"u}ckgegriffen
-		  wird. Dies wird im Folgenden f{\"u}r die mathematischen
-		  Funktionen im Zusammenhang mit dem Paket \texttt{pst-plot}
-		  gezeigt. }
+@Book{PostScript,
+  Author         = {Kollock, Nikolai G.},
+  Title          = {PostScript richtig eingesetzt: vom {K}onzept zum
+                   praktischen {E}insatz},
+  Publisher      = {IWT},
+  Address        = {Vaterstetten},
+  year           = 1989,
 }
 
+@online{pstricks,
+  Title          = {PSTricks - {\PS} macros for generic {\TeX}},
+  Author         = {{Timothy Van} Zandt},
+  Organization   = {\TeX\ Users Group},
+  url        = {http://www.tug.org/application/PSTricks},
+  urldate={2016-08-21},
+  year           = 1993
+}
 
-@Book{companion,
-  author	= {Michel Goosens and Frank Mittelbach and Serbastian Rahtz and Denis Roegel and Herbert Vo\ss},
-  title		= {The {\LaTeX} {G}raphics {C}ompanion},
-  publisher	= {{Addison-Wesley Publishing Company}},
-  year		= {2007},
-  edition	= {2nd},
-  address	= {Reading, Mass.}
+@ctan{pst-plot,
+  Title          = {\texttt{pst-plot}: Plotting two dimensional functions and data},
+  Author         = {{Timothy Van} Zandt and Herbert Voß},
+  Organization   = {CTAN},
+  url        = {graphics/pstricks/generic/pst-plot.tex},
+  year           = 2016
 }
 
-@Book{PSTricks2,
-  author	= {Herbert Vo\ss},
-  title		= {\texttt{PSTricks} -- {G}rafik f\"ur \TeX{} und \LaTeX},
-  edition	= {5.},
-  publisher	= {DANTE/Lehmanns Media},
-  year		= {2008},
-  address	= {Heidelberg/Berlin}
+@ctan{multido,
+  Title          = {\texttt{multido.tex} - a loop macro, that supports fixed-point addition},
+  Author         = {{Timothy Van} Zandt},
+  Organization   = {CTAN},
+  url        = {/graphics/pstricks/generic/multido.tex},
+  year           = 1997
 }
 
-@Book{voss:math,
-  author	= {Herbert Vo\ss},
-  title		= {Mathematik mit \LaTeX},
-  publisher	= {{DANTE/Lehmanns Media}},
-  year		= {2009},
-  address	= {Heidelberg/Berlin}
+@inproceedings{GB16,
+  TITLE = {Mass points, {B}\'ezier curves and conics: a survey},
+  AUTHOR = {Lionel Garnier and Jean-Paul Bécar},
+  url = {http://ufrsciencestech.u-bourgogne.fr/$\sim$garnier/publications/adg2016/},
+  BOOKTITLE = {Eleventh International Workshop on Automated Deduction in Geometry},
+  ADDRESS = {Strasbourg, France},
+  SERIES = {Proceedings of ADG 2016},
+  PAGES = {97--116},
+  date = {2016-06},
+  urldate={2016-08-20},
+}
+
+@online{gb16a,
+  author={Lionel Garnier},
+  title={Courbes de Bézier et coniques},
+  url={http://ufrsciencestech.u-bourgogne.fr/~garnier/Migs/03_CourbesBezierPointsMassiquesEleve.pdf},
+  urldate={2016-08-20},
+}
+@online{gb16b,
+  author={Lionel Garnier and Jean-Paul Bécar and Lucie Drouton},
+  title={Surfaces canal et courbes de Bézier rationnelles quadratiques},
+  journal={Journées du Groupe de Travail en Modélisation Géométrique 2016},
+  address={Dijon},
+  url={http://ufrsciencestech.u-bourgogne.fr/~garnier/publications/hippocampe/64_GTMG2016_courbesBezierSurfacesCanal.pdf},
+  urldate={2016-08-20},
+}
+
+@PhdThesis{Bec97,
+author = {Jean-Paul Bécar},
+title = {Forme ({B}{R}) des coniques et de leurs faisceaux},
+school = {Universit\'e de Valenciennes et de Hainaut-Cambr\'esis, LIMAV},
+date = {1997-12-12},
+address= {Valenciennes, France},
 }
 
diff --git a/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.pdf b/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.pdf
index 79ca40cd86b..2bfbd642d15 100644
--- a/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.pdf
+++ b/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.pdf
diff --git a/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.tex b/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.tex
index 3be402bebd2..95fa8782722 100644
--- a/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.tex
+++ b/Master/texmf-dist/doc/generic/pst-bezier/pst-bezier-doc.tex
@@ -1,20 +1,34 @@
-%% $Id: pst-bezier-doc.tex 86 2009-01-29 10:34:00Z herbert $
-\documentclass[11pt,english,BCOR10mm,DIV12,bibliography=totoc,parskip=false,smallheadings
-    headexclude,footexclude,oneside]{pst-doc}
+%% $Id: pst-bezier-doc.tex 134 2009-09-27 12:28:50Z herbert $
+\documentclass[11pt,english,bibliography=totoc,parskip=false,smallheadings,
+    oneside]{pst-doc}
 \usepackage[utf8]{inputenc}
-\usepackage{pst-bezier}
 \usepackage{esvect}
 \let\vec\vv
+\usepackage{animate}
+\usepackage{pst-bezier}
+\usepackage{bbold}
+\addbibresource{pst-bezier-doc.bib}
 
 \let\pstBezierFV\fileversion
 \lstset{pos=l,wide=false,language=PSTricks,
     morekeywords={multidipole,parallel},basicstyle=\footnotesize\ttfamily}
+\definecolor{navy}{rgb}{0 0 0.5}
 %
+\def\bgImage{\pspicture[showgrid](0,1)(5,6)
+\psset{showpoints}
+\psbcurve[linecolor=blue,linewidth=0.01](1,1)%
+  (2,2)(3,1)(4,2)(4,4)(3,5)%
+  (2,4)(1,5)
+\psbcurve(1,1)(2,2)(3,1)(4,2)%
+  T{0.5}(4,4)(3,5)(2,4)(1,5)
+\endpspicture}
+\newtheorem{definition}{Definition}
+\def\dy{\displaystyle}
 \begin{document}
 
 \title{\texttt{pst-bezier}}
 \subtitle{A PSTricks package for drawing Bezier curves; v.\pstBezierFV}
-\author{Tobias Nähring \\Herbert Vo\ss}
+\author{Jean-Paul Bécar\\Lionel Garnier\\Tobias Nähring \\Manuel Luque\\Herbert Voß}
 \docauthor{}
 \date{\today}
 \maketitle
@@ -37,7 +51,7 @@ control points without simultaneously changing the interpolated
 points. Note that some control is possible via the
 \Lkeyword{curvature} option.
 
-The \Lcs{psbezier} macro gives full control over the
+The \Lcs{psbcurve} macro gives full control over the
 interpolation points and the control points of one Bezier polynominal
 of degree three (two interpolated points and two control
 points).
@@ -47,7 +61,7 @@ Thanks to: \\
     Jean-C\^ome Charpentier.
 \end{abstract}
 
-%% Author: Tobias N"ahring
+\clearpage
 
 \section{Introduction}
 
@@ -95,7 +109,7 @@ correctly installed the package:
 \usepackage{pstricks}
 \usepackage{pst-bezier}
 \begin{document}
- \begin{pspicture}(6,4)
+ \begin{pspicture}(0,-0.4)(6,2)
    \psbcurve(1,2)(5,2) % Draw just one straight line.
  \end{pspicture}
 \end{document}
@@ -107,8 +121,8 @@ In the most simple form you can specify any number of interpolation
 points as the argument of \Lcs{psbcurve}.
 
 \begin{LTXexample}
-\begin{pspicture}[showgrid=true](5,3)
-  \psbcurve(1,1)(2,2)(3,1)(4,2)
+\begin{pspicture}[showgrid](0,-0.4)(5,3)
+  \psbcurve[showpoints](1,1)(2,2)(3,1)(4,2)
 \end{pspicture}
 \end{LTXexample}
 
@@ -116,8 +130,8 @@ As usual, options can be specified within brackets.
 
 
 \begin{LTXexample}
-\begin{pspicture}[showgrid=true](5,3)
-  \psbcurve[showpoints=true](1,1)(2,2)(3,1)(4,2)
+\begin{pspicture}[showgrid](0,-0.4)(5,3)
+  \psbcurve[showpoints](1,1)(2,2)(3,1)(4,2)
 \end{pspicture}
 \end{LTXexample}
 
@@ -129,8 +143,8 @@ time labels are added to the points (this is just for the following
 description, it is not a feature of \Lcs{psbcurve}).
 
 \begin{LTXexample}
-\begin{pspicture}[showgrid=true](5,3)
-  \psbcurve[showpoints=true](1,1)(2,2)(3,1)(4,2)
+\begin{pspicture}[showgrid](0,-0.4)(5,3)
+  \psbcurve[showpoints](1,1)(2,2)(3,1)(4,2)
   \uput[-90](1,1){$\vec{p}_{0}=\vec{l}_{1}$}
   \uput[90](1.5,2){$\vec{r}_{1}$}
   \uput[90](2,2){$\vec{p}_{1}$}
@@ -181,8 +195,8 @@ background in {\color{blue}blue} color.
 
 
 \begin{LTXexample}
-\pspicture[showgrid=true](5,3)
-\psset{showpoints=true}
+\pspicture[showgrid](0,-0.4)(5,3)
+\psset{showpoints}
 \psbcurve[linecolor=blue,linewidth=0.01](1,1)%
   (2,2)(3,1)(4,2)
 \psbcurve(1,1)l(2,1)(2,2)(3,1)r(4,1)(4,2)
@@ -192,8 +206,8 @@ background in {\color{blue}blue} color.
 \end{LTXexample}
 
 \begin{LTXexample}
-\pspicture[showgrid=true](5,3)
-\psset{showpoints=true}
+\pspicture[showgrid](0,-0.4)(5,3)
+\psset{showpoints}
 \psbcurve[linecolor=blue,linewidth=0.01](1,1)%
   (2,2)(3,1)(4,2)
 \psbcurve(1,1)(2,2)l(2,1)(3,1)(4,2)
@@ -208,7 +222,7 @@ interpolation point.
 
 
 \begin{LTXexample}
-\pspicture(5,3)
+\pspicture(0,-0.4)(5,3)
 \psbcurve(1,1)(2,2)l(2,1)(3,1)(4,2)
 \endpspicture
 \end{LTXexample}
@@ -220,10 +234,10 @@ interpolation point $\vec{p}_{k-1}$. In that way you get a smooth joint as
 demonstrated in the next example.
 
 \begin{LTXexample}
-\pspicture[showgrid=true](5,3)
+\pspicture[showgrid](0,-0.4)(5,3)
 \psbcurve[linecolor=blue,linewidth=0.01](1,1)%
   (2,2)(3,1)(4,2)
-\psset{showpoints=true}
+\psset{showpoints}
 \psbcurve(1,1)(2,2)L(2,1)(3,1)(4,2)
 \uput[-90](2,1){$\vec{l}_{2}$}
 \uput[0](2,2){$\vec{p}_{1}$}
@@ -236,8 +250,8 @@ automatically computed control points of the current Bezier spline.
 
 
 \begin{LTXexample}
-\pspicture[showgrid=true](5,3)
-\psset{showpoints=true}
+\pspicture[showgrid](0,-0.4)(5,3)
+\psset{showpoints}
 \psbcurve[linecolor=blue,linewidth=0.01](1,1)%
   (2,2)(3,1)(4,2)
 \psbcurve(1,1)(2,2)t{0.5}(3,1)(4,2)
@@ -252,8 +266,8 @@ point you can use the \Lnotation{tl}\Largb{t} or \Lnotation{tr}\Largb{t} modifie
 respectively, as demonstrated in the following two examples.
 
 \begin{LTXexample}
-\pspicture[showgrid=true](5,3)
-\psset{showpoints=true}
+\pspicture[showgrid](0,-0.4)(5,3)
+\psset{showpoints}
 \psbcurve[linecolor=blue,linewidth=0.01](1,1)%
   (2,2)(3,1)(4,2)
 \psbcurve(1,1)%
@@ -263,8 +277,8 @@ respectively, as demonstrated in the following two examples.
 
 
 \begin{LTXexample}
-\pspicture[showgrid=true](5,3)
-\psset{showpoints=true}
+\pspicture[showgrid](0,-0.4)(5,3)
+\psset{showpoints}
 \psbcurve[linecolor=blue,linewidth=0.01](1,1)%
   (2,2)(3,1)(4,2)
 \psbcurve(1,1)(2,2)tr{0.5}(3,1)(4,2)
@@ -278,8 +292,8 @@ of the modifier. In the next example a negative tension value leads to
 a rather surprising effect.
 
 \begin{LTXexample}
-\pspicture[showgrid=true](5,3)
-\psset{showpoints=true}
+\pspicture[showgrid](0,-0.4)(5,3)
+\psset{showpoints}
 \psbcurve[linecolor=blue,linewidth=0.01](1,1)%
   (2,2)(3,1)(4,2)
 \psbcurve(1,1)(2,2)ts{-0.5}(3,1)(4,2)
@@ -291,8 +305,8 @@ The default value of the tension can be set with the option
 
 
 \begin{LTXexample}
-\pspicture[showgrid=true](5,3)
-\psset{showpoints=true}
+\pspicture[showgrid](0,-0.4)(5,3)
+\psset{showpoints}
 \psbcurve[linecolor=blue,linewidth=0.01](1,1)%
   (2,2)(3,1)(4,2)
 \psbcurve[bcurveTension=0.5](1,1)%
@@ -307,8 +321,8 @@ middle of a \Lcs{psbcurve}. Just use the modifier \Lnotation{T}\Largb{t} for
 that purpose as shown in the following example.
 
 \begin{LTXexample}
-\pspicture[showgrid=true](5,6)
-\psset{showpoints=true}
+\pspicture[showgrid](0,0.6)(5,6)
+\psset{showpoints}
 \psbcurve[linecolor=blue,linewidth=0.01](1,1)%
   (2,2)(3,1)(4,2)(4,4)(3,5)%
   (2,4)(1,5)
@@ -324,7 +338,7 @@ The \texttt{linestyle} and \texttt{fillstyle} options (and several
 more) are respected by \Lcs{psbcurve} as the following example shows.
 
 \begin{LTXexample}
-\pspicture[showgrid=true](5,3)
+\pspicture[showgrid](0,-0.4)(5,3)
 \psbcurve[linestyle=dashed,
   linewidth=3pt,
   dash=0.5 0.2,
@@ -333,7 +347,7 @@ more) are respected by \Lcs{psbcurve} as the following example shows.
 \endpspicture
 \end{LTXexample}
 
-\section{Things that do not work (`known bugs')}
+\subsection{Things that do not work (`known bugs')}
 As already mentioned this project is something like an experiment. So,
 there are many things that do not work.
 
@@ -342,10 +356,375 @@ there are many things that do not work.
 \item The control points are computed in a rather crude way (see
   above). The \Lkeyword{curvature} option is not recognised.
 \item If \Lkeyword{fillstyle} is set to \Lkeyword{solid} and
-  \Lkeyset{showpoints=true} then the fill color covers the interpolation and control points.
+  \Lkeyword{showpoints} then the fill color covers the interpolation and control points.
 \item arrow heads do not work.
 \end{itemize}
 
+\clearpage
+
+\section{Bezier curve with weighted points}
+
+\subsection{Mathemathical background}
+
+A mass point is a weighted point $\left(P;\omega\right)$ with $\omega \neq 0$ or a vector $\left(\overrightarrow{P};0\right)$ with a weight equal to $0$. A generic mass point is noted $\left(P;\omega\right)$.
+
+Using the quadratic Bernstein polynomials, a rational quadratic B\'ezier curve having three control 
+mass points $\left(P_{0};\omega_{0}\right)$, $\left(P_{1};\omega_{1}\right)$
+and $\left(P_{2};\omega_{2}\right)$, is defined as follow:
+
+\begin{definition}\label{fdef::DefRQBC_Fiorot}: Rational quadratic B\'ezier curve (BR curve)
+
+Let $\omega_{0}$, $\omega_{1}$ and $\omega_{2}$ be three real numbers. 
+Let $\left(P_{0};\omega_{0}\right)$, $\left(P_{1};\omega_{1}\right)$
+and $\left(P_{2};\omega_{2}\right)$ be three mass points, these points are not collinear.
+
+Define two sets $I = \left \{ i    |      \omega_i \neq 0 \right \}$ and 
+$J = \left \{ i      |      \omega_i = 0 \right \}$
+
+
+Define the function $\omega_{f}$ from $\left[0;1\right] $ to $\mathbb{R} $ as follows 
+
+\begin{equation}
+%\begin{array}{cccc}
+%\omega_{f}: & \left[0;1\right] & \longrightarrow & \mathbb{R} \\
+%& t & \longmapsto &\omega_{f}\left(t\right)=\dy\sum_{i\in I}\omega_{i}\times B_{i}\left(t\right)
+%\end{array}
+\omega_{f}\left(t\right)=\dy\sum_{i\in I}\omega_{i}\times B_{i}\left(t\right)
+\label{eq:DenominateurCbreBezier}
+\end{equation}
+
+A mass point $\left(M;\omega\right)$ or $\left(\overrightarrow{u};0\right)$
+belongs to the quadratic B\'ezier curve defined by the three control
+mass points $\left(P_{0};\omega_{0}\right)$, $\left(P_{1};\omega_{1}\right)$
+and $\left(P_{2};\omega_{2}\right)$, 
+if there is a real $t_{0}$ in $\left[0;1\right]$ such that:
+
+\begin{itemize}
+\item [$\bullet$] if $\omega_{f}\left(t_{0}\right)\neq0$ then we have
+
+\hspace*{-0.75cm}\begin{minipage}{1.0\textwidth}
+\begin{equation}
+\overrightarrow{OM}  = \dy \frac{1}{\omega_{f}\left(t_{0}\right)}\left(\dy \sum_{i\in I} \dy \omega_{i}  B_{i}\left(t_{0}\right)
+  \overrightarrow{OP_{i}} \right)
++\vspace{0.2cm}\dy \frac{1}{\omega_{f}\left(t_{0}\right)}\left( \sum_{i\in J}  B_{i}\left(t_{0}\right)  \overrightarrow{P_{i}}\right)
+\label{eq:DefRQBC_FiorotPoint}
+\end{equation}
+\end{minipage}
+
+\item [$\bullet$] if $\omega_{f}\left(t_{0}\right)=0$ then we have
+\begin{equation}
+\overrightarrow{u}=\sum_{i\in I}\omega_{i}B_{i}\left(t_{0}\right)\overrightarrow{OP_{i}}+\sum_{i\in J}B_{i}\left(t_{0}\right)\overrightarrow{P_{i}}\label{eq:DefRQBC_FiorotVecteur}
+\end{equation}
+
+\end{itemize}
+\hrulefill{}\end{definition}
+
+The  reduced discriminant of the denominator $\omega_{f}\left(t_{0}\right)$  is
+\begin{equation}
+\Delta'=\omega_{1}^{2}-\omega_{2} \omega_{0}\label{eq:DiscrimantReduitCBRQnonStandard}
+\end{equation}
+and we can state the following fundamental result:
+\begin{itemize}
+\item[$\star$] 
+if $\omega_{1}^{2}-\omega_{2} \omega_{0}=0$ then the 
+ denominator has one and only one root, the curve is a parabolic arc; 
+\item[$\star$]
+ if  $\omega_{1}^{2}-\omega_{2} \omega_{0}>0$ then the 
+ denominator has two distinct roots, the curve is  a hyperbolic arc;
+\item[$\star$]
+ if $\omega_{1}^{2}-\omega_{2} \omega_{0}<0$ then the 
+ denominator does not vanish,  the curve is  an elliptical arc. 
+\end{itemize}
+
+We can note w.l.o.g.\footnote{We can permute  the role of $P_0$ and $P_2$} that one of the weights can be equal to~$1$. If $\omega_0$ is not equal to $0$, we choose $\omega_0=1$, else, we choose $\omega_1=1$, and we can characterise the type of the conic from the mass points of the BR curve,  see Table~\ref{tab::TypeConicEtcbeBr}.
+
+\begin{table}[!h]
+\begin{center}
+\begin{tabular}{|c||c|c|c|}\hline
+Conic & Three weighted points & Points and vectors \\ \hline \hline
+Parabola & $\left(P_{0};1\right)$, $\left(P_{1};\omega\right)$ 
+ $\left(P_{2};\omega^{2}\right)$ & $\left(P_{0};1\right)$,  $\left(\overrightarrow{P_{1}};0\right)^{\mathstrut^{\mathstrut}}_{\mathstrut_{\mathstrut}}$  $\left(\overrightarrow{P_{2}};0\right)$\\ \hline \hline
+ Ellipse & $\left(P_{0};1\right)$, $\left(P_{1};\omega_{1}\right)$, $\left(P_{2};\omega_{2}\right)$, $ \omega_{2}>\omega_{1}^{2} $ &  $\left(P_{0};1\right)$,   $\left(\overrightarrow{P_{1}};0\right)^{\mathstrut^{\mathstrut}}_{\mathstrut_{\mathstrut}}$  $\left(P_{2};1\right)$ \\ \hline \hline
+  Hyperbola & $\left(P_{0};1\right)$, $\left(P_{1};\omega_{1}\right)$   $\left(P_{2};\omega_{2}\right)$, $\omega_{2}<\omega_{1}^{2}$ & $\left(P_{0};1\right)$, $\left(\overrightarrow{P_{1}};0\right)^{\mathstrut^{\mathstrut}}_{\mathstrut_{\mathstrut}}$  $\left(P_{2};-1\right)$   \\ \cline{3-3}
+&   & $\left(\overrightarrow{P_{0}};0\right)$,   $\left(P_{1};1\right)$  and $\left(\overrightarrow{P_{2}};0\right)^{\mathstrut^{\mathstrut}}_{\mathstrut_{\mathstrut}}$  \\ \hline  \hline
+\end{tabular}
+\end{center}
+\caption{Types of conics defined by B\'ezier curves with control mass points.
+\hrulefill{}
+\label{tab::TypeConicEtcbeBr}}
+\end{table}
+
+From the  access rights used by Unix and Linux, we define a bijection $f$ between $\mathbb{F_2}^3-\left\lbrace\left(0,0,0\right)\right\rbrace$ and the set $\left\lbrace 1 ,2 , 3, 4, 5, 6, 7\right\rbrace$. From $\left(\omega_2,\omega_1,\omega_0\right)$, we define a triplet $\left(b_2,b_1,b_0\right)$ as follow: if $w_i\neq0$ then $b_i=1$ else $b_i=0$. Then 
+$$f\left(\omega_2,\omega_1,\omega_0\right)= b_2 \times 4+ b_1 \times 2+b_0$$
+
+If $f\left(\omega_2,\omega_1,\omega_0\right)=7$, the control points are weighted points: the curve is an elliptical arc, a parabolic arc or a hyperbolic arc. If $\left(\omega_2,\omega_1,\omega_0\right)=\left(1,-1,1\right)$, the parabolic arc is not bounded and for $t=\frac{1}{2}$, the mass point is a direction vector of the parabola axis. If $\left(\omega_2,\omega_1,\omega_0\right)=\left(1,-2,1\right)$, the hyperbolic arc is not bounded and there exists $t$ in $\left]0,1\right[$ such as the mass point is a direction vector of one of the asymptotes of the hyperbola. \\
+If $f\left(\omega_2,\omega_1,\omega_0\right)=1$, the first control point is  a weighted point, the others are vectors: the curve is a parabolic arc. The B\'ezier curve is defined by
+ \begin{equation}
+\begin{cases}
+ \dy \frac{1}{\omega_0\, B_0\left(t_{0}\right)}\left( \omega_{0}\,  B_{0}\left(t_{0}\right) 
+ \overrightarrow{OP_{0}} + B_{1}\left(t_{0}\right)  \overrightarrow{P_{1}}+ B_{2}\left(t_{0}\right)  
+ \overrightarrow{P_{2}}\right) & \text{ if }t_0\in\left[0,1\right[  \\[1ex]
+\overrightarrow{P_2} & \text{ if }t_0=1\\
+ \end{cases}
+\label{eq:parabola}
+\end{equation}
+If $f\left(\omega_2,\omega_1,\omega_0\right)=4$, the B\'ezier curve can be defined in the same way.\\
+If $f\left(\omega_2,\omega_1,\omega_0\right)=2$, the intermediate control point is  a weighted point, the others are vectors: the curve is a branch of a hyperbola. The B\'ezier curve is defined by
+ \begin{equation}
+\begin{cases}
+ \dy \frac{1}{\omega_1\, B_1\left(t_{0}\right)}\left( \omega_{1}\,  B_{1}\left(t_{0}\right) \overrightarrow{OP_{1}}+ B_{0}\left(t_{0}\right)  \overrightarrow{P_{0}}+ B_{2}\left(t_{0}\right)  \overrightarrow{P_{2}}\right) & \text{ if }t_0\in\left]0,1\right[ 
+\\[1ex]
+\overrightarrow{P_0} & \text{ if }t_0=0\\[1ex]
+\overrightarrow{P_2} & \text{ if }t_0=1
+ \end{cases}
+\label{eq:branchHyperbola}
+\end{equation}
+and the centre of the hyperbola is $P_1$. The vector $\overrightarrow{P_0}$ is a direction vector of an asymptote of the hyperbola whereas the vector $\overrightarrow{P_2}$ is a direction vector of the other asymptote.\\
+If $f\left(\omega_2,\omega_1,\omega_0\right)=5$, the intermediate control point is    a vector,   the others are weighted points: the curve is an elliptical arc. The B\'ezier curve is defined by
+ \begin{equation}
+ \dy \frac{1}{\omega_0\, B_0\left(t_{0}\right)+\omega_2\, B_2\left(t_{0}\right)}\left( \omega_{0}\,  B_{0}\left(t_{0}\right) \overrightarrow{OP_{0}} + B_{1}\left(t_{0}\right)  \overrightarrow{P_{1}}+ \omega_2\, B_{2}\left(t_{0}\right)  \overrightarrow{OP_{2}}\right),\;\; t_0\in\left[0,1\right] 
+\label{eq:ellipse}
+\end{equation}
+and the tangent vector to the curve at $P_0$ or $P_2$ is parallel to $\overrightarrow{P_1}$.
+
+\subsection{Syntax}
+
+\begin{BDef}
+\Lcs{psRQBCmasse}\OptArgs\Largr{$x_0,y_0$}\Largr{$x_1,y_1$}\Largr{$x_2,y_2$}\Largb{$w_0,w_1,w_2$}
+\end{BDef}
+
+For the coordinates of the points all possible kinds of coordinates are possible, like polar, PostScript, nodes, \ldots
+
+\subsection{Three weighted orthogonal points}
+\begin{LTXexample}[pos=t]
+\begin{pspicture}[showgrid](-6,-6.4)(3,3)
+\psclip{\psframe(-6,-6)(3,3)}
+  \psRQBCmasse[linecolor=blue](2,0)(2,2)(0,2){1,-1,1}
+  \psRQBCmasse[linecolor=navy,autoTrace](2,0)(2,2)(0,2){1,1,1}
+  \rput(P0){$P_0$}\uput[r](P1){$P_1$}\uput[r](P2){$P_2$}
+\endpsclip%
+\end{pspicture}
+\end{LTXexample}
+
+
+
+\subsection{Half-ellipse}
+\begin{LTXexample}[pos=t]
+\begin{pspicture}[showgrid](-3,-2.4)(3,2)
+\psframe(-3,-2)(3,2)
+\psRQBCmasse[linecolor=red,autoTrace](2,0)(0,1)(-2,0){1,0,1}
+\uput[r](P0P1){$\overrightarrow{P_1}$} \uput[r](P2){$P_2$}
+\rput(P1P2){$\overrightarrow{P_{1}}$} \uput[r](P0){$P_0$}
+\psRQBCmasse[linecolor=orange,autoTrace=false](2,0)(0,-1)(-2,0){1,0,1}
+\end{pspicture}
+\end{LTXexample}
+
+
+\clearpage
+
+\subsection{Half-parabola}
+\subsubsection{Point $P_2$ and two vectors}
+
+\begin{LTXexample}[pos=t]
+\begin{pspicture}[showgrid](-3,-3.4)(3,3)
+\psclip{\psframe(-3,-3)(3,3)}
+  \psRQBCmasse[linecolor=red,autoTrace](2,0)(0,1)(-1,0){0,0,1}
+  \uput[r](P1P2){$\overrightarrow{P_1}$} \uput[r](P2){$P_2$}  
+  \uput[r](P0P2){$\overrightarrow{P_0}$}
+  \psRQBCmasse[linecolor=orange,autoTrace=false](2,0)(0,-1)(-1,0){0,0,1}
+  \uput[r](P1P2){$\overrightarrow{P_1}$} \uput[r](P2){$P_2$}
+  \uput[r](P0P2){$\overrightarrow{P_0}$}
+\endpsclip
+\end{pspicture}
+\end{LTXexample}
+
+\subsubsection{Point $P_0$ and two vectors}
+
+\begin{LTXexample}[pos=t]
+\begin{pspicture}[showgrid](-3,-3.4)(3,3)
+\psclip{\psframe(-3,-3)(3,3)}
+  \psRQBCmasse[linecolor=red,autoTrace](2,0)(0,1)(-1,0){1,0,0}
+  \uput[r](P0P1){$\overrightarrow{P_1}$} \uput[r](P0){$P_0$}
+  \uput[r](P0P2){$\overrightarrow{P_2}$}
+  \psRQBCmasse[linecolor=orange,autoTrace=false](2,0)(0,-1)(-1,0){1,0,0}
+\endpsclip%
+\end{pspicture}
+\end{LTXexample}
+
+\clearpage
+
+\subsection{Branch of a hyperbola}
+\begin{LTXexample}[pos=t]
+\begin{pspicture}[showgrid](-3,-3.4)(3,3)
+\psclip{\psframe(-3,-3)(3,3)}
+  \psRQBCmasse[linecolor=red,autoTrace](1,1)(0,0)(-1,1){0,1,0}
+  \uput[r](P0){$\overrightarrow{P_0}$} \uput[r](0,-0.5){$P_1$}
+  \uput[r](P2){$\overrightarrow{P_2}$}
+  \psRQBCmasse[linecolor=orange,autoTrace=false](1,1)(0,0)(-1,1){0,-1,0}
+\endpsclip%
+\end{pspicture}
+\end{LTXexample}
+
+\subsection{Parabola}
+\begin{LTXexample}[pos=t]
+\psset{unit=0.5}
+\begin{pspicture}(-14,-3.4)(15,10)
+\psclip{\psframe(-14,-3)(15,10)}
+  \psRQBCmasse[linecolor=red,autoTrace](0,6)(-13,0)(-1,-1){1,1,1}
+  \psRQBCmasse[linecolor=orange](0,6)(-13,0)(-1,-1){1,-1,1}
+  \uput[u](P0){$P_0$}\uput[l](P1){$P_1$}\uput[d](P2){$P_2$}
+\endpsclip
+\end{pspicture}
+\end{LTXexample}
+
+
+\clearpage
+
+\subsection{Ellipse}
+\begin{LTXexample}[pos=t]
+\psset{unit=0.5}
+\begin{pspicture}(-14,-3.4)(15,10)
+\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=5,gridlabels=0pt]
+%\psplotImp[linewidth=0.5pt,linecolor=blue,algebraic](-6,-3)(15,10)%
+  %{ -0.044*x^2-0.161*y^2 + 0.075*x*y + 0.074*x + 0.797*y + 1}
+\psRQBCmasse[nPoints=20,autoTrace,showpoints](0,6)(-13,0)(-1,-1){1,0.5,1}
+\psRQBCmasse[nPoints=40,linecolor=red,showpoints](0,6)(-13,0)(-1,-1){1,-0.5,1}
+\psaxes[labelFontSize=\scriptscriptstyle]{->}(0,0)(-14,-3)(15,10)
+\end{pspicture}
+\end{LTXexample}
+
+
+\subsection{Complete circle}
+\begin{LTXexample}[pos=t]
+\psset{unit=1}
+\begin{pspicture}(-4,-4.4)(4,4)
+\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=5,gridlabels=0pt]
+\psRQBCmasse[autoTrace](0,3)(3,3)(3,0){1,1,2}
+\psRQBCmasse[linecolor=red](0,3)(3,3)(3,0){1,-1,2}
+\psaxes[labelFontSize=\scriptscriptstyle]{->}(0,0)(-4,-4)(4,4)
+\end{pspicture}
+\end{LTXexample}
+
+
+\begin{LTXexample}[pos=t]
+\psset{unit=1.5}
+\begin{pspicture}(-4,-4.4)(4,4)
+\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=5,gridlabels=0pt]
+\psRQBCmasse[autoTrace](0,3)(3,0)(0,-3){1,0,1}
+\uput[u](-0.25,3){$P_0$}
+\uput[u](-0.25,-3.5){$P_2$}
+\uput[u](3,3){$\overrightarrow{P_1}$}
+\uput[u](3,-3.5){$\overrightarrow{P_1}$}
+\uput[u](2.5,0){$\overrightarrow{P_1}$}
+\psRQBCmasse[linecolor=red](0,3)(-3,0)(0,-3){1,0,1}
+\psaxes[labelFontSize=\scriptscriptstyle,linewidth=0.01]{->}(0,0)(-4,-4)(4,4)
+\end{pspicture}
+\end{LTXexample}
+We get a circle because we have
+
+\begin{equation}
+\left\lbrace
+\begin{array}{rcl}
+\omega_0\times\omega_2\times P_0 P_2^2 &= &4\times\overrightarrow{P_1}^2 \\[0.2cm]
+\overrightarrow{P_0 P_2} &\perp & \overrightarrow{P_1}
+\end{array}
+\right.
+\end{equation}
+
+\clearpage
+
+
+\subsection{Animations}
+
+\subsubsection{$w_0=1$, $w_2=1$ and a variable $w_1$}
+
+With the beginning of $w_1=0$
+the curves are swapped. In the case of Bezier curves $w_1 = 0$ gives only
+the $[P_0 P_2]$ segment. Using the mass points, the point $P_1$ no longer exists but we get the vector $\overrightarrow{P_1}$.
+
+
+\bigskip
+\begin{center}
+\begin{animateinline}[controls,loop,palindrome,
+                      begin={\begin{pspicture}(-4,-4)(10,4)},
+                     end={\end{pspicture}}]{3}% 3 images/s
+\multiframe{40}{rA=2.0+-0.1,rB=-2.0+0.1}{%
+  \psgrid[subgriddiv=0,gridcolor=lightgray,griddots=5,gridlabels=0pt]
+  \psclip{\psframe(-4,-4)(10,4)}
+    \psRQBCmasse[autoTrace,linewidth=1.5pt](0,-1)(1,0)(0,1){1,\rA,1}
+    \uput[u](P2){$P_2$}\uput[l](P1){$P_1$}\uput[d](P0){$P_0$}
+    \psRQBCmasse[linecolor=red,linewidth=1.5pt](0,-1)(1,0)(0,1){1,\rB,1}
+    \psaxes[labelFontSize=\scriptscriptstyle,linewidth=0.01]{->}(0,0)(-4,-4)(10,4)
+    \rput(8,3){$w_1=\rA$}%
+  \endpsclip
+}
+\end{animateinline}
+\end{center}
+
+\begin{lstlisting}
+\begin{animateinline}[controls,loop,palindrome,
+                      begin={\begin{pspicture}(-4,-4)(10,4)},
+                      end={\end{pspicture}}]{3}% 3 images/s
+\multiframe{40}{rA=2.0+-0.1,rB=-2.0+0.1}{%
+  \psgrid[subgriddiv=0,gridcolor=lightgray,griddots=5,gridlabels=0pt]
+  \psclip{\psframe(-4,-4)(10,4)}
+    \psRQBCmasse[autoTrace,linewidth=1.5pt](0,-1)(1,0)(0,1){1,\rA,1}
+    \uput[u](P2){$P_2$}\uput[l](P1){$P_1$}\uput[d](P0){$P_0$}
+    \psRQBCmasse[linecolor=red,linewidth=1.5pt](0,-1)(1,0)(0,1){1,\rB,1}
+    \psaxes[labelFontSize=\scriptscriptstyle,linewidth=0.01]{->}(0,0)(-4,-4)(10,4)
+    \rput(8,3){$w_1=\rA$}%
+  \endpsclip
+}
+\end{animateinline}
+\end{lstlisting}
+
+
+
+\clearpage
+
+\subsubsection{$w_0=1$, $\left |w_1\right|=1$ and a variable $w_2$}
+
+%L'utilisation de $\left |w_1\right|$ permet d'obtenir les deux arcs et donc toute la conique.
+The use of $\left |w_1\right|$ provides both arcs and the whole cone.
+
+\bigskip
+\begin{center}
+\begin{animateinline}[controls,loop,palindrome,
+                      begin={\begin{pspicture}(-8,-4)(4,4)},
+                      end={\end{pspicture}}]{3}% 3 images/s
+\multiframe{80}{rA=4.0+-0.1}{%
+  \psgrid[subgriddiv=0,gridcolor=lightgray,griddots=5,gridlabels=0pt]
+  \psclip{\psframe(-8,-4)(4,4)}
+    \psRQBCmasse[autoTrace,linewidth=1.5pt](0,-1)(1,0)(0,1){1,1,\rA}
+    \uput[u](P2){$P_2$}\uput[l](P1){$P_1$}\uput[d](P0){$P_0$}
+    \psRQBCmasse[linecolor=red,linewidth=1.5pt](0,-1)(1,0)(0,1){1,-1,\rA}
+    \psaxes[labelFontSize=\scriptscriptstyle,linewidth=0.01]{->}(0,0)(-8,-4)(4,4)
+    \rput[rb](3.5,3){$w_2=\rA$}%
+  \endpsclip
+}
+\end{animateinline}
+\end{center}
+
+\begin{lstlisting}
+\begin{animateinline}[controls,loop,palindrome,
+                      begin={\begin{pspicture}(-8,-4)(4,4)},
+                      end={\end{pspicture}}]{3}% 3 images/s
+\multiframe{80}{rA=4.0+-0.1}{%
+  \psgrid[subgriddiv=0,gridcolor=lightgray,griddots=5,gridlabels=0pt]
+  \psclip{\psframe(-8,-4)(4,4)}
+    \psRQBCmasse[autoTrace,linewidth=1.5pt](0,-1)(1,0)(0,1){1,1,\rA}
+    \uput[u](P2){$P_2$}\uput[l](P1){$P_1$}\uput[d](P0){$P_0$}
+    \psRQBCmasse[linecolor=red,linewidth=1.5pt](0,-1)(1,0)(0,1){1,-1,\rA}
+    \psaxes[labelFontSize=\scriptscriptstyle,linewidth=0.01]{->}(0,0)(-8,-4)(4,4)
+    \rput[rb](3.5,3){$w_2=\rA$}%
+  \endpsclip
+}
+\end{animateinline}
+\end{lstlisting}
+
+
+\clearpage
+
 
 \section{List of all optional arguments for \texttt{pst-bezier}}
 
@@ -355,11 +734,13 @@ there are many things that do not work.
 \bgroup
 \raggedright
 \nocite{*}
-\bibliographystyle{plain}
-\bibliography{pst-bezier-doc}
+\printbibliography
 \egroup
 
 \printindex
 \end{document}
 
 
+
+
+Moreover, we can choose a non Euclidean metric. The use of mass points, Bézier curves, conics and the space of spheres in the Minkowski-Lorentz space permits to realise G1-continous blend between Dupin cyclides : to blend surfaces in R3, we blend Bézier curves in R5. For example, we can build a seahorse (see 09_LorentzHippocampeComplet.png), the article (in French) is here: