Lecture 11 - Spline curves (slides)

For the suppose we want the curve to start at point

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Unformatted text preview: a0, 1 a 2 follows:! We can now solve for the vector constants a0 , a1 and a2 Firstly, suppose we want the curve to start at point P0 . For the •  Suppose we want the curve to start at point P0 expression: Firstly, suppose we want the a µ2 + astart at point P0 . For the P0 = curve to 1 µ + a0 2 expression: P0 = a2 µ2 + a1 µ + a0 We have at 0 start start = ! we• have µ = 0 μ =theat theso P0 soa0 we have µ = 0 at the start so P0 = a0 Graphics Lecture 11: Slide 9! 9 / 38 9 / 38 Calculating simple parametric splines Calculating simple parametric splines" •  Suppose we want the spline to end at P2 Suppose we want the spline to end at P2 •  We have that at the end µ = 1 At the end, we have µ = 1 So •  Thus! P2 = a2 µ2 + a1 µ + a0 = a2 + a1 + a0 ) P2 = a2 + a1 + P0 Graphics Lecture 11: Slide 10! 10 / 38 Calculating simple parametric splines Calculating simple parametric splines" Let’s also say that at µ = 1/2 (in the middle) the curve should passAnd in the 1middle (say µ = 1/2) we want it to pass through •  through P P1 P1 = a2 µ2 + a1 µ + a0 1 1 ) P1 = a2 + a1 + P0 4 2 ! •  We have enough equations to solve for a and a2.! 1 We now have enough equations to solve for a1 and a2 . •  Notice the the method same same whether we are Notice that thatmethod is the is the whether we are working in 2 or working we or dimensions, we just for each of the 3 dimensions,in 2 just3have to solve separatelyhave to solve separately vectors a and aordinates in the vectors a1 ordinates in the for each 1of the 2 . and a2.! Graphics Lecture 11: Slide 11! 11 / 38 Possibilities using parametric splines splines" Possibilities using parametric Graphics Lecture 11: Slide 12! 12 / 38 Higher order parametric splines" •  Parametric polynomial splines must have an order to match the number of knots.! –  3 knots - quadratic polynomial! –  4 knots - cubic polynomial! –  etc.! •  Higher order polynomials are undesirable since they tend to oscillate! Graphics Lecture 11: Slide 13! Spline Patches" •  To get round the problem, we can piece together a number ofPatches Spline patches, each patch being a parametric spline.! To get round the problem, we can piece together a number of patches, each patch being a parametric spline. 14 / 38 Graphics Lecture 11: Slide 14! Cubic Spline Patches Cubic Spline Patches" •  The simplest, and most effective way to calculate Thepsimplest, and mostpatches is to to calculate parametric spline arametric spline e↵ective way use a cubic polynomial.! patches is to use a cubic polynomial. P = a3 µ3 + a2 µ2 + a1 µ + a0 •  This allows us to join the patches together smoothly! This allows us to join the patches together smoothly Graphics Lecture 11: Slide 15! 15 / 38 Choosing the the gradients" Choosing gradients Indices here are {0, 1, 2, 3} but can be any successive set Indices here are {0, 1, 2, 3} but can be any successive set of four numbers taken of four numbers taken from the available control points ! from the available contr...
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