X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fsam.git;a=blobdiff_plain;f=chapters%2FBackground_Bezier.tex;h=3652358d48a85f38780fbc4cd7d5c965c93aaed3;hp=974f06454fe5a224c3648b0e1a5d80c6395607ad;hb=fb42dc0335ed47cbfdd9176a1cfdd37cab487ddd;hpb=b1c5fe49ec552755fd19073c3f91c8e9866d6938 diff --git a/chapters/Background_Bezier.tex b/chapters/Background_Bezier.tex index 974f064..3652358 100644 --- a/chapters/Background_Bezier.tex +++ b/chapters/Background_Bezier.tex @@ -10,5 +10,5 @@ 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.