X-Git-Url: https://git.ucc.asn.au/?a=blobdiff_plain;f=chapters%2FBackground_Bezier.tex;h=2ec2a2f00d72593ca06495e0900dcb5c2aa11a14;hb=3cc6f72b6bbdde973827f4f3cd47563d240cc345;hp=974f06454fe5a224c3648b0e1a5d80c6395607ad;hpb=b1c5fe49ec552755fd19073c3f91c8e9866d6938;p=ipdf%2Fsam.git

diff --git a/chapters/Background_Bezier.tex b/chapters/Background_Bezier.tex
index 974f064..2ec2a2f 100644
--- a/chapters/Background_Bezier.tex
+++ b/chapters/Background_Bezier.tex
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+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}
@@ -10,5 +12,7 @@ Figure \ref{bezier_3} shows a Bezier Curve defined by the points $\left\{(0,0),
 
 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. The cost of this modification is that the algorithm is no longer $O(n)$ but $O(n^2)$; although it is not explicitly stated by Goldman it seems clear that the modified algorithm is mainly of theoretical interest.
+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.
+
+