Penultimate draft

[ipdf/sam.git] / chapters / Background / Rendering / BezierSplines.tex

diff --git a/chapters/Background/Rendering/BezierSplines.tex b/chapters/Background/Rendering/BezierSplines.tex
index fd0ec01..cf936ef 100644 (file)
--- a/chapters/Background/Rendering/BezierSplines.tex
+++ b/chapters/Background/Rendering/BezierSplines.tex
@@ -10,7 +10,7 @@ Splines may be rasterised by sampling of $y(x)$ at a number of points $x_i$ and
 There are many different ways to define a spline.One approach is to specify ``knots'' on the curve and choosing a fixed $n$ ($n = 3$ for ``cubic'' splines) solve for the cooefficients to generate polynomials passing through the points. Alternatively, special polynomials may be defined using ``control'' points which themselves are not part of the curve; these are convenient for graphical based editors.\end{co B{\'e}zier splines are the most straight forward way to define a curve in the standards considered in Section \ref{Document Representations}. A spline defined from two cubic B{\'e}ziers is shown in Figure \ref{spline.pdf}
 \end{comment}
 
-Cubic and Quadratic B{\'e}zier Splines are used to define curved paths in the PostScript\cite{plrm}, PDF\cite{pdfref17} and SVG\cite{svg2011-1.1} standards which we will discuss in Section \ref{Document Representations}.  Cubic B{\'e}ziers are also used to define vector fonts for rendering text in these standards and the {\TeX}  typesetting language \cite{knuth1983metafont, knuth1984texbook}. Although he did not derive the mathematics, the usefulness of B{\'e}zier curves was realised by Pierre B{\'e}zier who used them in the 1960s for the computer aided design of automobile bodies\cite{bezier1986apersonal}. 
+Cubic and Quadratic B{\'e}zier Splines are used to define curved paths in the PostScript\cite{plrm}, PDF\cite{pdfref17} and SVG\cite{svg2011-1.1} standards.  Cubic B{\'e}ziers are also used to define vector fonts for rendering text in these standards and the {\TeX}  typesetting language \cite{knuth1983metafont, knuth1984texbook}. Although he did not derive the mathematics, the usefulness of B{\'e}zier curves was realised by Pierre B{\'e}zier who used them in the 1960s for the computer aided design of automobile bodies\cite{bezier1986apersonal}. 
 
 A B{\'e}zier Curve of degree $n$ is defined by $n$ ``control points'' $\left\{P_0, ... P_n\right\}$. 
 Points $P(t) = (x(t), y(t))$ along the curve are defined by:
@@ -65,5 +65,5 @@ De Casteljau's algorithm of 1959 is often used for decomposing B{\'e}ziers into
        \includegraphics[width=0.7\textwidth]{figures/spline.pdf}
 \end{figure}
 \end{minipage}
-       \caption{Constructing a Spline from two cubic B{\'e}ziers \\ (a) Showing the Control Points (b) Representations in SVG and PostScript (c) Rendered Spline}\label{spline.pdf}
+       \caption{Constructing a Spline from two cubic B{\'e}ziers \\ (a) Showing the Control Points (b) Representations in SVG and PostScript (c) Rendered Spline} \label{spline.pdf}
 \end{figure}