pst-bspline (31jul11)

git-svn-id: svn://tug.org/texlive/trunk@23294 c570f23f-e606-0410-a88d-b1316a301751

1 files changed, 130 insertions, 35 deletions

diff --git a/Master/texmf-dist/doc/generic/pst-bspline/pst-bspline-doc.tex b/Master/texmf-dist/doc/generic/pst-bspline/pst-bspline-doc.tex
index 8e034e19af3..d88db518937 100644
--- a/Master/texmf-dist/doc/generic/pst-bspline/pst-bspline-doc.tex
+++ b/Master/texmf-dist/doc/generic/pst-bspline/pst-bspline-doc.tex
@@ -7,6 +7,17 @@
 \usepackage{multido,pst-node,pst-bspline,pstricks-add}
 \usepackage{amssymb}
 \usepackage[parfill]{parskip}
+\usepackage{lmodern}
+\usepackage[scaled=.82]{luximono}%requires T1+textcomp
+\usepackage[T1]{fontenc}
+\newcommand\textSMC[1]{{\SMC #1}}
+\newcommand\acro[1]{\textSMC{#1}\@}
+\DeclareRobustCommand\cs[1]{\texttt{\char`\\#1}}
+\usepackage[cal=boondoxo]{mathalfa}
+\usepackage{textcomp}
+%\usepackage{url}
+%\def\url@ttstyle{%
+% \@ifundefined{selectfont}{\def\UrlFont{\tt}}{\def\UrlFont{\ttfamily\small}}}
 \usepackage{hyperref}
 \hyphenation{Post-Script}
 %\date{}                                           % Activate to display a given date or no date
@@ -15,17 +26,40 @@
 \large Michael Sharpe\\[10pt]
 msharpe@ucsd.edu}
 \end{center}
-A cubic uniform B-spline curve with control points $B_0 \ldots B_n$ is a curve parametrized by the interval $[0,n]$, which is, except in degenerate cases, $C^2$-continuous (that is, has continuous curvature) and is on each interval $[k-1,k]$ given by a cubic B\'ezier curve whose control points are derived from the $(B_k)$. These curves are discussed in any reasonably modern text on Numerical Analysis. One easily accessible source is the UCLA lecture notes of Kirby Baker:
+A cubic, uniform B-spline curve with control points $B_0 \ldots B_n$ ($n \ge 2$) is a curve parametrized by the interval $[0,n]$, which is, except in degenerate cases, $C^2$-continuous (that is, has continuous curvature) and is on each interval $[k-1,k]$ ($0<k\le n$ an integer) given by a cubic B\'ezier curve whose control points are derived from the $(B_k)$. These curves are discussed in any reasonably modern text on Numerical Analysis. One easily accessible source is the UCLA lecture notes of Kirby Baker:
 
 \noindent\url{http://www.math.ucla.edu/~baker/149.1.02w/handouts/dd_splines.pdf}
 
-I'll focus on two special cases: (i) relaxed, uniform B-splines; (ii) periodic, uniform B-splines. Uniform refers to the condition mentioned in the first paragraph: each B\'ezier sub-curve is parametrized by an interval of  length~1. Relaxed means that the curvature at the endpoints $t=0, t=n$ is zero. Periodic means in effect that the $B_i$ repeat periodically, and the curve generated is a closed curve. 
-
-\section{Relaxed, Open B-spline} The algorithm has the following steps.
+I'll focus on two special cases: (i) relaxed, uniform B-splines; (ii) periodic, uniform B-splines. `Uniform' refers to the condition mentioned in the first paragraph: each B\'ezier sub-curve is parametrized by an interval of  length~1. `Relaxed' means that the curvature vanishes at the endpoints $t=0, t=n$. `Periodic' means in effect that the $B_i$ repeat periodically, and the curve generated is a closed curve. 
+
+\section{Quick summary of the macros}
+\begin{description}
+\item[\cs{psbspline}(1,1)(3,0)(5,2)(4,5)]
+ draws the relaxed, uniform B-spline interpolating the specified points.
+\item[\cs{psBspline}(1,1)(3,0)(5,2)(4,5)] draws the relaxed, uniform B-spline with specified control points.
+\item[\cs{psBspline}\{B\}(1,1)(3,0)(5,2)(4,5)] draws the relaxed, uniform B-spline with specified control points, using \texttt{B} as basename for the constructed points.
+\item[\cs{psBsplineE}(1,1)(3,0)(5,2)(4,5)] is the same as \cs{psBspline}(1,1)(3,0)(5,2)(4,5) except that it omits the first and last segments.
+\item[\cs{psBsplineC}(1,1)(3,0)(5,2)(4,5)] extends the specified points periodically, drawing a closed curve with the specified points as control points.
+\item[\cs{psBsplineNodes}\{B\}\{4\}] draws the relaxed, uniform B-spline with control points \texttt{B0}..{\tt B4}.
+\item[\cs{psBsplineNodesE}\{B\}\{4\}] is the same as \cs{psBsplineNodes}\{B\}\{4\} except that it omits the first and last segments.
+\item[\cs{psBsplineNodesC}\{B\}\{4\}] extends the node sequence periodically, drawing a closed curve with them as control points.
+\item[\cs{beztobsp}(1,2)(-3,-4)(5,6)(-7,-8)\{B\}] creates nodes {\tt B0}..{\tt B3} for which the curve \cs{psBsplineNodesE}\{B\}\{3\} is identical to the B\'ezier curve determined by the specified points. (It does not draw the curve.)
+\item[\cs{bspcurvepoints\{B\}\{5\}\{P\}}] creates PostScript arrays to describe a sequence of points along the curve that would be the result of the command  \cs{psBsplineNodes}\{B\}\{5\}, naming those arrays {\tt P.X}, {\tt P.Y} (for position), {\tt PNormal.X}, {\tt PNormal.Y}, {\tt PDelta.X} and {\tt PDelta.Y}. (Nothing is drawn.)
+\item[\cs{bspcurvepointsE\{B\}\{5\}\{P\}}] does the same as \cs{bspcurvepoints}, but omits the first and last segments. (Nothing is drawn.)
+\item[\cs{bspNode\{P\}\{5\}\{1.3\}\{Q\}}] requires that you first run \cs{bspcurvepoints}{\tt[E]} to create PostScript arrays with basename {\tt P}. It then sets a node {\tt Q} at position $t=1.3$ on the curve. (Nothing is drawn.) 
+\item[\cs{bspFnNode\{P\}\{5\}\{2.3\}\{Q\}}] requires that you first run \cs{bspcurvepoints}{\tt[E]} to create PostScript arrays with basename {\tt P}. It then sets a node {\tt Q} at position $x=1.3$ on the curve. (Nothing is drawn.) The result is meaningful only for a B-spline curve that is the graph of a function of $x$ and where $x_0<x_1<\cdots$.
+\item[\cs{psBsplineInterp\{S\}\{4\}}] will construct a sequence {\tt SB0}..{\tt SB4} for which the associated B-spline curve interpolates {\tt S0}..{\tt S4}. (Nothing is drawn---you have to then issue the command \cs{psBsplineNodes\{SB\}\{4\}}.)
+\item[\cs{psBsplineInterpC\{S\}\{4\}}] will construct a sequence {\tt SB0}..{\tt SB5} for which the associated closed B-spline curve interpolates {\tt S0}..{\tt S4}. (Nothing is drawn---you have to then issue the command \cs{psBsplineNodesC\{SB\}\{5\}}.)
+\item[\cs{thickBspline\{B\}\{5\}\{12pt\}\{<graphic to clip>\}}] defines a clipping path 12{\tt pt} wide around the B-spline curve with control points {\tt B0}..{\tt B5}, then draws the {\tt <graphic>} clipped to that path.
+\item[\cs{bspcurvenodes\{P\}\{Q\}}] creates a node sequence {\tt Q0} {\tt Q1},... from the position data in the arrays {\tt P.X}, {\tt P.Y}.
+\end{description}
+Details and examples are provided below.
+
+\section{Relaxed, Open B-spline} The algorithm to generate such a curve from a sequence of control points $B_0$, $\cdots$, $B_n$ is as follows:
 
 \begin{itemize}
-\item The curve starts at $B_0$ and ends at $B_n$.
-\item Divide each line $B_{k-1}B_k$ into equal thirds, with subdivision points labeled $R_{k-1}$, $L_k$ respectively, so that $B_k$ has $L_k$ as its immediate neighbor to the left, and $R_k$ as its immediate neighbor to the right.
+\item The curve starts at $B_0$ and ends at $B_n$. (Important: $n \ge 2$.)
+\item Divide each line segment $B_{k-1}B_k$ into equal thirds, with subdivision points labeled $R_{k-1}$, $L_k$ respectively, so that $B_k$ has $L_k$ as its immediate neighbor to the left, and $R_k$ as its immediate neighbor to the right.
 \item For $0<k<n$, divide the line segment $L_kR_k$ in half, letting $S_k$ denote the midpoint. In effect, for $0< k<n$, $S_k=(B_{k-1}+4B_k+B_{k+1})/6$.
 \item Let $S_0=B_0$ and $S_n=B_n$.
 \item For $0<k\le n$, construct the cubic B\'ezier curve with control points $S_{k-1}$, $R_{k-1}$, $L_k$, $S_k$, parametrized by $k-1\le t\le k$.
@@ -45,9 +79,9 @@ There is another optional argument that can be applied if you wish to be able to
 
 \noindent sets the root of the naming scheme to {\tt B}, the effect of which is that the B-spline control points will be nodes of type \verb|\pnode| with names {\tt B0}, {\tt B1} and so on, the other points being similarly named {\tt BL0}, {\tt BL1}, ... , {\tt BR0}, {\tt BR1}, ... , {\tt BS0}, {\tt BS1}, ... . For example, to draw a line between {\tt BL1} and {\tt BS4}, just use \verb|\ncline(BL1)(BS4)|. 
 
-The  algorithm is implemented entirely in PSTricks code with PostScript specials, but no PostScript header file, depending for the most part on the flexibility of nodes, and above all the \verb|\multido| macro, which allows one to construct with relative ease items that look and feel like arrays. Use of \verb|\SpecialCoor| is essential.
+The  algorithm  depends for the most part on the flexibility of nodes, and above all the \verb|\multido| macro, which allows one to construct with relative ease items that look and feel like arrays. Use of \verb|\SpecialCoor| is essential.
 
-There is a closely related macro \verb|\psBsplineE| which removes the first and last B\'ezier segments, much as \verb|\psecurve| acts in relation to \verb|\pscurve|, allowing one one to draw B-splines with non-zero curvature at the endpoints.
+There is a closely related macro \verb|\psBsplineE| which removes the first and last B\'ezier segments, much as \verb|\psecurve| acts in relation to \verb|\pscurve|, allowing one to draw B-splines with non-zero curvature at the endpoints.
 
 \begin{verbatim}
 \documentclass{article}
@@ -60,17 +94,12 @@ There is a closely related macro \verb|\psBsplineE| which removes the first and
 \begin{pspicture}[showgrid=true](-.5,-.5)(6,5)
 \psBspline[showframe=true]{B}(.5,.5)(2,0)(5,2)(6,4)(4,5)(2,4)
 \multido{\i=0+1}{5}{\uput[20](B\i){B\i}}
-\uput[90](B5){B5}
-\uput[90](BS1){S1}
-\uput[90](BS2){S2}
-\uput[180](BS3){S3}
-\uput[270](BS4){S4}
-\uput[-45](BR1){R1}
+\uput[90](B5){B5}\uput[90](BS1){S1}\uput[90](BS2){S2}
+\uput[180](BS3){S3}\uput[270](BS4){S4}\uput[-45](BR1){R1}
 \uput[-45](BL2){L2}
 \end{pspicture}
 \end{document}
 \end{verbatim} 
-
 \vspace{1pc}
 \begin{center}
 \psset{unit=.6in}
@@ -87,6 +116,7 @@ There is a closely related macro \verb|\psBsplineE| which removes the first and
 \end{pspicture}
 \end{center}
 
+\newpage
 \begin{verbatim}
 \documentclass{article}
 \usepackage{pstricks}
@@ -124,7 +154,19 @@ There is a closely related macro \verb|\psBsplineE| which removes the first and
 \uput[-45](BL2){L2}
 \end{pspicture}
 \end{center}
-
+\subsection{B\'ezier curves as B-spline curves}
+Consider a B\'ezier curve $\mathcal{C}$ with control points $S_1$, $C_1$, $C_2$, $S_2$. To identify $\mathcal{C}$ as a B-spline curve of the type discussed above, consider the problem of finding the B-spline control points $B_0$, $B_1$, $B_2$, $B_3$ for which \verb|\psBsplineE| yields $\mathcal{C}$. This is a simple problem in linear algebra whose solution is:
+\begin{verbatim}
+B0=6S1-7C1+2C2
+B1=2C1-C2
+B2=-C1+2C2
+B3=6S2+2C1-7C2
+\end{verbatim}
+In other words, any cubic B\'ezier curve may be considered to be the output of \verb|\psBsplineE| for the $B_i$ described above. In this way, all macros described for B-spline curves may be applied to an arbitrary B\'ezier curve as a special case. The macro
+\begin{verbatim}
+\beztobsp(S1)(C1)(C2)(S2){B}
+\end{verbatim}
+results in defining $B_0$,$\dots$,$B_3$ exactly as above.
 \section{Periodic B-spline} The result here is a closed curve. The algorithm is essentially the same as in the preceding case, except:
 
 \begin{itemize}
@@ -187,6 +229,43 @@ defines a sequence of \verb|\pnode|s with the node root {\tt P}: {\tt P0}=(2,1.5
 \psBsplineNodesE{<node root>}{<top index>}
 \end{verbatim}
 corresponding to the macros \verb|\psBspline|, \verb|\psBsplineC| and \verb|\psBsplineE|. The difference is that the macros with {\tt Nodes} in the name have as arguments  the root node name and the  last index, rather than the list of points. For example, with the above definition of {\tt P} in force, \verb|\psBsplineNodes{P}{2}| has exactly the same effect as  \verb|\psBspline(2,1.5)(3,4)(5,1).|
+\subsection{The \cs{bspcurvepoints} macros}
+There are two macros that provide for B-spline curves essentially the same functionality as the \verb|\pscurvepoints| macro from {\tt pstricks-add}. (That macro takes as input a parametric curve and constructs as output (at the PostScript level) arrays of data associated with the curve: the positions of points along the curve, the increment from the previous point and a normal vector to the curve. The principal uses for such data are (i) the \verb|\pspolylineticks| macro from {\tt pstricks-add}, which allows placement of ticks and other marks along a curve that has been approximated by a polyline; (ii) the \cs{polyIntersections} macro from \textsf{pst-node}, which allows you to find the points of intersection of the curve (approximated by  a polyline) and an arbitrary line.) The macros
+\begin{verbatim}
+\bspcurvepoints{<source name>}{<source max index>}{<dest. name>}
+\bspcurvepointsE{<source name>}{<source max index>}{<dest. name>}
+\end{verbatim}
+work, in the first case, for a relaxed, uniform B-spline curve, and in the second, for such a curve with its initial and final segments removed, corresponding to the output from \verb|\psBsplineE| rather than \verb|\psBspline|. In both cases, you may set the keyword {\tt plotpoints} (default value: $50$) to change the number of sample points on each  B\'ezier component. This will result in the construction of PostScript arrays with indices from $0$ to $n=$\textsf{num of segments}$\times$\textsf{(plotpoints-1)}. After running
+\begin{verbatim}
+\pnodes{B}(1,2)(3,-1)(4,1)(6,2)% define B0..B3
+\bspcurvepoints[plotpoint=11]{B}{3}{P}
+\end{verbatim}
+the following PostScript arrays are created, each indexed from 0 to 30:
+\begin{verbatim}
+P.X, P.Y (position)
+PNormal.X, PNormal.Y (normal vector)
+PDelta.X, PDelta.Y (increment from previous position)
+\end{verbatim}
+and these may be used in the usual way to create nodes. For example,
+\begin{verbatim}
+\pnode(! P.X 8 get P.Y 8 get){Q}
+\pnode(! PNormal.X 8 get PNormal.Y get){Dir}
+\psrline(Q)(1cm;{(Dir)})
+\end{verbatim}
+places {\tt Q} at the position on the curve with index 8, defines {\tt Dir} to be a normal vector at that point, then draws a line from {\tt Q} of length {\tt 1cm} in the direction of that normal.
+\subsection{Setting nodes on a B-spline curve}
+To set a node at parameter value $t$ on a B-spline curve after running \cs{bspcurvepoints}{\tt[E]}, call the macro
+\begin{verbatim}
+\bspNode{<control point root>}{<top index>}{<t>}{<node name>}
+\end{verbatim}
+For example, if I have constructed a B-spline curve using control points $B_0$,$\dots$,$B_5$, then \verb|\bspNode{B}{5}{2.1}{Q}| defines a node named $Q$ at $t=2.1$.
+
+The macro \cs{bspcurvenodes\{P\}\{R\}} creates a node sequence {\tt R0}..{\tt Rn} at the locations specified by the arrays {\tt P.X}, {\tt P.Y}. (Those arrays must first have been created with one of the \cs{bspcurvepoints} macros.)
+\subsection{B-spline function curves}
+By this we mean an open B-spline curve which is the graph of a function $y=f(x)$ and whose orientation is toward the right. It is not analytically simple to specify a formula for $f$ in most cases, and to compute $y$ from $x$ involves (a) finding the index of the B\'ezier segment containing $x$; (b) solving the cubic $x(t)=x$ for $t$; (c)  substituting in $y(t)$. The package provides a macro to perform these calculations after generating the data using \cs{bspcurvepoints}{\tt[E]}:
+\begin{verbatim}
+\bspfnNode{<control point root>}{<top index>}{<x0>}{<node name>}
+\end{verbatim}
 
 \section{B-spline Interpolation}
 This is the inverse problem. Being given points $(S_k)_{0\le k\le n}$, the goal is to produce the B-spline control points $B_k$ leading to the points $S_k$, so that the associated B-spline curve interpolates the $S_k$. 
@@ -206,7 +285,7 @@ for the $B_k$. In matrix form, this becomes the tridiagonal system
 &1&4&1\\
 &&\cdots&&1\\
 &&&1&4\end{pmatrix}
-\begin{pmatrix}B_1\\B_2\\B_3\\ \cdots\\ B_{n-1}\end{pmatrix}=ß
+\begin{pmatrix}B_1\\B_2\\B_3\\ \cdots\\ B_{n-1}\end{pmatrix}=
 \begin{pmatrix}6S_1-S_0\\6S_2\\6S_3\\ \cdots\\6S_{n-1}-S_{n}\end{pmatrix}
 \]
 The LU decomposition of the tridiagonal matrix may be seen to take the form
@@ -263,7 +342,7 @@ for the $B_k$, $1\le k\le n$. In matrix form, this becomes the system
 &1&4&1\\
 &&\cdots&&1\\
 1&&&1&4\end{pmatrix}
-\begin{pmatrix}B_1\\B_2\\B_3\\ \cdots\\ B_{n}\end{pmatrix}=ß
+\begin{pmatrix}B_1\\B_2\\B_3\\ \cdots\\ B_{n}\end{pmatrix}=
 \begin{pmatrix}6S_1\\6S_2\\6S_3\\ \cdots\\6S_{n}\end{pmatrix}
 \]
 Let $(x_k,y_k)=6S_k$. We perform Gaussian elimination on the matrix
@@ -366,19 +445,30 @@ Slight difference between psccurve and B-spline interpolation\\
 \end{center}
 A B-spline curve can in many cases provide a good function interpolation mechanism, but the result is not guaranteed to be the graph of a function.
 \begin{verbatim}
-\begin{center}
-\begin{pspicture}[showgrid=true](-.5,-.5)(6,4)
-\psdots(0,3.5)(1,.5)(3,2.5)(4,0)(5,2)(6,.5)
+\begin{pspicture}(-.5,-.5)(6,4)
+\psdots(0,3.5)(1,.5)(3,2.5)(4,0)(5,2)(6,.5)%
+\pnodes{S}(0,3.5)(1,.5)(3,2.5)(4,0)(5,2)(6,.5)% S0..S5
+\psBsplineInterp{S}{5}% construct SB0..SB5
+\psBsplineNodes{SB}{5}% draw B-spline with control pts SB0..SB5
+\bspcurvepoints[plotpoints=10]{SB}{5}{P}
+% construct the PS arrays
+\bspFnNode{SB}{5}{4.5}{QQ}% node QQ on curve at x=4.5
+\psdot[linecolor=red](QQ)%
 \psaxes(0,0)(-.5,-.5)(6,4)
-\psbspline(0,3.5)(1,.5)(3,2.5)(4,0)(5,2)(6,.5)
 \end{pspicture}
-\end{center}
 \end{verbatim}
 
+\vspace{1pc}
 \begin{center}
 \begin{pspicture}(-.5,-.5)(6,4)
-\psdots(0,3.5)(1,.5)(3,2.5)(4,0)(5,2)(6,.5)
-\psbspline(0,3.5)(1,.5)(3,2.5)(4,0)(5,2)(6,.5)
+\psdots(0,3.5)(1,.5)(3,2.5)(4,0)(5,2)(6,.5)%
+\pnodes{S}(0,3.5)(1,.5)(3,2.5)(4,0)(5,2)(6,.5)% S0..S5
+\psBsplineInterp{S}{5}%SB0..SB5
+\psBsplineNodes{SB}{5}% draw B-spline with control pts SB0..SB5
+\bspcurvepoints[plotpoints=10]{SB}{5}{P}
+% construct the PS arrays
+\bspFnNode{SB}{5}{4.5}{QQ}% node QQ on curve at x=4.5
+\psdot[linecolor=red](QQ)%
 \psaxes(0,0)(-.5,-.5)(6,4)
 \end{pspicture}
 \end{center}
@@ -386,7 +476,6 @@ A B-spline curve can in many cases provide a good function interpolation mechani
 
 \begin{verbatim}
 \documentclass{article}
-\usepackage{graphicx}
 \usepackage{pstricks}
 \usepackage{pst-bspline,pstricks-add}
 \begin{document}
@@ -442,7 +531,8 @@ which creates a framework of B-spline control points {\tt SB0..SB3}.
 (The {\tt [linestyle=none]} may be omitted if you want the curve to show.)
 \item Create a clipping path of specified thickness around the interpolating curve and place graphics to be clipped:
 \begin{verbatim}
-\thickBspline[plotpoints=50,linestyle=none]{S}{3}{20pt}{\psline[linecolor=red,linestyle=solid](0,0)(6,6)}%
+\thickBspline[plotpoints=50,linestyle=none]{S}{3}{20pt}%
+{\psline[linecolor=red,linestyle=solid](0,0)(6,6)}%
 \end{verbatim}
 (The {\tt [linestyle=none]} controls whether the clipping path is rendered, and {\tt plotpoints} controls the number of subdivisions of each B\'ezier component. Its default value is 50.)
 \end{itemize}
@@ -476,18 +566,20 @@ The clipping path is drawn by default positively oriented so that objects are cl
 \end{center}
 
 \vspace{12pt}
-The \verb|\thickBspline| macro works as expected in the closed (periodic) case, taking advantage of automatic incrementing of the nodecount.
+The \verb|\thickBspline| macro works as expected in the closed (periodic) case, taking advantage of automatic incrementing of the nodecount. Note that \verb|\thickBspline| interprets thickness as visual, unaffected by possible differences between {\tt xunit} and {\tt yunit}.
 
 \begin{verbatim}
 \documentclass{article}
 \usepackage{pstricks}
 \usepackage{pst-bspline,pstricks-add}
 \begin{document}
+\psset{yunit=1.5cm}
 \begin{pspicture}[showgrid=true](-.5,-.5)(6,5)
-\pnodes{S}(1,0)(5,1)(4,4)(1,3)%
-\psBsplineInterpC{S}{\Snodecount}%
-% defines nodes SB0, SB1, SB2 etc --- the Bspline control points
+\pnodes{S}(1.5,0)(5,1)(4,4)(1,3)%
+\psBsplineInterpC{S}{3}%
+% defines nodes SB0, SB1, SB2, SB3, SB4 --- the Bspline control points
 % increments \Snodecount by 1 for future macros
+% Don't use C form of \psBsplineNodes with this new \Snodecount
 \psBsplineNodes[linestyle=none,showpoints=false]{SB}{\Snodecount}%
 % Constructs the Bezier control points SBR0, SBL1, SBR1, etc
 \thickBspline[linestyle=none]{S}{\Snodecount}{22pt}%
@@ -499,14 +591,17 @@ The \verb|\thickBspline| macro works as expected in the closed (periodic) case,
 \vspace{12pt}
 
 \begin{center}
+\psset{yunit=1.5cm}
 \begin{pspicture}[showgrid=true](-.5,-.5)(6,5)
-\pnodes{S}(1,0)(5,1)(4,4)(1,3)%
-\psBsplineInterpC{S}{\Snodecount}%
-% defines nodes SB0, SB1, SB2 etc --- the Bspline control points
+\pnodes{S}(1.5,0)(5,1)(4,4)(1,3)%
+\psBsplineInterpC{S}{3}%
+% defines nodes SB0, SB1, SB2, SB3 --- the Bspline control points
 % increments \Snodecount by 1 for future macros
 \psBsplineNodes[linestyle=none,showpoints=false]{SB}{\Snodecount}%
 % Constructs the Bezier control points SBR0, SBL1, SBR1, etc
-\thickBspline[linestyle=none]{S}{\Snodecount}{22pt}{\psframe[fillstyle=vlines](-1,-1)(6,6)}%
+\thickBspline[linestyle=none]{S}{\Snodecount}{22pt}%
+{\psframe[fillstyle=vlines](-1,-1)(6,6)}%
+%{\psframe[fillstyle=solid, fillcolor=lightgray](-1,-1)(6,6)}%
 \end{pspicture}
 \end{center}