Lecture 11 - Spline curves (slides)

Graphics lecture 11 slide 21 a typical bezier curve a

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Unformatted text preview: ezier Curves" •  Bezier curves were developed as a method for CAD design. They give very predictable results for small sets of knots, and so are useful as spline patches.! •  The main characteristics of Bezier curves are! –  They interpolate the end points! –  The slope at an end is the same as the line joining the end point to its neighbour! Graphics Lecture 11: Slide 21! A typical Bezier curve A typical Bezier Curve" Graphics Lecture 11: Slide 22! Casteljau’s Algorithm" •  Bezier curves may be computed and visualised using a geometric construction introduced by Paul de Casteljau. ! •  Like a cubic patch, we need a parameter µ which is ! –  0 at the start of the curve ! –  1 at the end. ! •  The construction ! –  is recursive! –  can be made for any value of µ ! Graphics Lecture 11: Slide 23! Casteljau’s Algorithm Casteljau’shere is approximatelyµ = 0.25" Construction 0.25 The value of µ Graphics Lecture 11: Slide 24! 24 / 38 Casteljau’s Algorithm Casteljau’s Construction µ = 0.6" µ ⇡ 0.6 Graphics Lecture 11: Slide 25! 25 / 38 Casteljau’s Algorithm Casteljau.’µ ⇡ 0.9 s Construction µ = 0.9" One more . . Graphics Lecture 11: Slide 26! 26 / 38 Bernstein Blending Function Bernstein Blending Function" •  Splines (including curves) can be can be formulated of Splines (including BezierBezier curves)formulated as a blendas a the blend of the knots.! knots. Consider the vectorvector line equation! •  Consider the line equation P = (1 µ)P0 + µP1 •  It is a linear ‘blend’ of two points, and could also be It is considered the two-point Bezier curve!! a linear blend of two points. Can also be considered as the 2 point Bezier curve!‡ Graphics Lecture 11: Slide 27! ‡ One level of recursion 27 / 38 Blending Equation Blending Equation Blending Equation" Any  Any pointthesplinespline is simply a ofall the other points. Anypoint on the splineisissimply aablend ofblend of other points. simply blend all the all the other • point on on the For Noints. knotsNwehave: we have:! For p + 11knots we have: N + For +1 knots N N X X P(µµ)= P( ) = W (N, i,i, µ) Pi W (N, µ) Pi i=0 i=0 where W isisW is Bernsteinblending functionfunction! where W the Bernstein blending function •  where the the Bernstein blending ✓◆ ✓◆ Ni i W (N, i,i, µ)= N µµ(1 µµNN i i W (N, µ) = i (1 ) ) i ✓◆ ✓◆ N N! ! N= N i i =(N i)!i!i! (N i)! Graphics Lecture 11: Slide 28! 28 / 38 28 / 38 Blending Equation: Expansions for di↵erent N Blending Equation: Expansions for different N " N Expansion 1 (1 µ)P0 + µP1 2 (1 µ)2 P0 + 2µ(1 µ)P1 + µ2 P2 3 . . . (1 . . . µ)3 P0 + 3µ(1 µ)2 P1 + 3µ2 (1 µ)P2 + µ3 P3 Graphics Lecture 11: Slide 29! 29 / 38 Bezier Curves lack local control " •  Since all the knots of the Bezier curve all appear in the blend they cannot be used for curves with ﬁne detail. ! •  However they are very effective as spline patches. ! ! Gra...
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