X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fsam.git;a=blobdiff_plain;f=chapters%2FBackground_Bezier.tex;h=2ec2a2f00d72593ca06495e0900dcb5c2aa11a14;hp=3652358d48a85f38780fbc4cd7d5c965c93aaed3;hb=20f882fefa7e17840ddec6ce1c5c8e15764bb0fa;hpb=5e4b9d22c1d1077d1179b5ee20c55e8662ea723a;ds=sidebyside diff --git a/chapters/Background_Bezier.tex b/chapters/Background_Bezier.tex index 3652358..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} @@ -12,3 +14,5 @@ A straightforward algorithm for rendering Bezier's is to simply sample $P(t)$ fo 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. + +