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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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 Spring '14

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