X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fsam.git;a=blobdiff_plain;f=chapters%2FBackground_Bezier.tex;h=2ec2a2f00d72593ca06495e0900dcb5c2aa11a14;hp=2d49de986d4b2ad2b658da65864d78e63f8bed27;hb=20f882fefa7e17840ddec6ce1c5c8e15764bb0fa;hpb=0d77e99e89699e2ffc8ed002d2c2abfd6f665172

diff --git a/chapters/Background_Bezier.tex b/chapters/Background_Bezier.tex
index 2d49de9..2ec2a2f 100644
--- a/chapters/Background_Bezier.tex
+++ b/chapters/Background_Bezier.tex
@@ -1,3 +1,5 @@
+Cubic beziers form all curves in the PostScript\cite{plrm}, PDF\cite{pdfref17} and SVG\cite{svg2011-1.1} standards which we will discuss in Section \ref{Document Representations}. One of the shapes in Figure \ref{SVG} is a region defined by a cubic bezier spline. Beziers are also used to construct vector fonts for rendering text in these standards.
+
 A Bezier Curve of degree $n$ is defined by $n$ ``control points'' $\left\{P_0, ... P_n\right\}$. 
 Points $P(t)$ along the curve are defined by:
 \begin{align}
@@ -7,9 +9,10 @@ Points $P(t)$ along the curve are defined by:
 From this definition it should be apparent $P(t)$ for a Bezier Curve of degree $0$ maps to a single point, whilst $P(t)$ for a Bezier of degree $1$ is a straight line between $P_0$ and $P_1$. $P(t)$ always begins at $P_0$ for $t = 0$ and ends at $P_n$ when $t = 1$.
 
 Figure \ref{bezier_3} shows a Bezier Curve defined by the points $\left\{(0,0), (1,0), (1,1)\right\}$.
-Figure \ref{SVG} shows a more complex spline defined by Bezier curves.
 
 A straightforward algorithm for rendering Bezier's is to simply sample $P(t)$ for some number of values of $t$ and connect the resulting points with straight lines using Bresenham or Wu's algorithm (See Section \ref{Straight Lines}). Whilst the performance of this algorithm is linear, in ???? De Casteljau derived a more efficient means of sub dividing beziers into line segments.
 
 Recently, Goldman presented an argument that Bezier's could be considered as fractal in nature, a fractal being the fixed point of an iterated function system\cite{goldman_thefractal}. Goldman's proof depends upon a modification to the De Casteljau Subdivision algorithm which expresses the subdivisions as an iterated function system.
 
+
+