-Splines are continuous curves formed from piecewise polynomial segments. A polynomial of $n$th degree is defined by $n$ constants $\{a_0, a_1, ... a_n\}$ and:
-\begin{align}
- y(x) &= \displaystyle\sum_{k=0}^n a_k x^k
-\end{align}
-
-Splines may be rasterised by sampling of $y(x)$ at a number of points $x_i$ and rendering straight lines between $(x_i, y_i)$ and $(x_{i+1}, y_{i+1})$ as discussed in Section \ref{Straight Lines}.
-
-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. Bezier 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 beziers is shown in Figure \ref{spline.pdf}
-
-\begin{figure}[H]
-\centering
-\begin{minipage}[t]{0.3\textwidth}
-\begin{figure}[H]
- \centering
- \includegraphics[width=\textwidth]{figures/spline_labelled.pdf}
-\end{figure}
-\end{minipage}
-\begin{minipage}[t]{0.3\textwidth}
-\begin{minted}{xml}
-<!-- DOM element in SVG used to construct the spline -->
-<path d="M 0,300
- C 0,300 200,210 90,140
- -20,70 200,0 200,0"
- style="stroke:#000000; stroke-width:1px;
- fill:none;"/>
-\end{minted}
-\begin{minted}{postscript}
-% PostScript commands for a similar spline
-0 300 moveto
-0 300 200 210 90 140 curveto
--20 70 200 0 200 0 curveto stroke
-\end{minted}
-\end{minipage}
-\begin{minipage}[t]{0.3\textwidth}
-\begin{figure}[H]
- \centering
- \includegraphics[width=\textwidth]{figures/spline.pdf}
-\end{figure}
-\end{minipage}
- \caption{Constructing a Spline from two cubic beziers \\ (a) Showing the Control Points (b) Representations in SVG and PostScript (c) Rendered Spline}\label{spline.pdf}
-\end{figure}
-\subsubsection{Bezier Curves}
-\input{chapters/Background_Bezier}
-
-\subsection{Font Rendering}\label{Font Rendering}