Lecture 11 - Spline curves (slides)

Lecture 11 spline curves slides

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Unformatted text preview: Interactive Computer Graphics:   Lecture 11 " Introduction to Spline Curves ! Splines Splines" Graphics Lecture 11: Slide 2! 2 / 38 Splines" Splines •  The word spline comes from the ship building trade where planks were originally shaped by bending them The word spline comes fromthe ground.! round pegs ﬁxed in the ship building trade where planks were originally shaped by bending them round pegs ﬁxed in the ground. •  Originally it was the pegs that were referred to as splines.! ! Originally it was the pegs that were referred to as splines. Now it is the smooth curve that is called a spline. Graphics Lecture 11: Slide 3! Interpolating Splines Interpolating Splines" Modern   splines aresplines are smooth curves a small set of a small • Modern smooth curves deﬁned from deﬁned from points often called knots. set of points often called knots. ! •  In one main class of splines, the curve must pass In one main class of splines, the curve must pass through each through each point of the set.! point of the set. •  These are called interpolating splines! These are called ! interpolating splines Graphics Lecture 11: Slide 4! 4 / 38 Approximating Splines Approximating Splines"have to pass through all the In other cases the curves do not points. •  In other cases the curves do not pass through the points.! Points can act as control pointspoints which the user can move •  The points act as control which the user can move to adjust the tshape of the curve interactively o adjust the shape of the curve interactively! Graphics Lecture 11: Slide 5! 5 / 38 Non-Parametric Spline" •  The simplest splines are just equations in x and y (for two dimensions)! •  The most common is the polynomial spline:! y = a2x2 + a1x + a0 •  Given three points we can calculate a2, a1 and a0 Graphics Lecture 11: Slide 6! A Non-Parametric (Parabolic) Spline" Non Parametric Splines •  Example ofdegree 2 (parabolic) non-parametric spline: Example of a a degree 2 (parabolic) non-parametric spline:! •  There is no control using non parametric splines. Only There is no control using non parametric splines. Only one curve one curve ﬁts parabola) ﬁts the data. ! (a the data. (a parabola) Graphics Lecture 11: Slide 7! 7 / 38 Parametric Splines Parametric Splines" If write our spline in a in a vector form we If• we we write our spline vector form we get: get:! P = a2 µ2 + a1 µ + a0 •  which a parameter µ by µ! which has has a parameter convention, as µ ranges from 0 to 1 the point P traces out a curve⇤ . •  By convention, as µ ranges from 0 to 1 the point P traces out a curve.! ⇤ Note that P, a0 , a1 and a2 are all vectors and that µ is a scalar 8 / 38 Graphics Lecture 11: Slide 8! Calculating simple parametric splines Calculating simple parametric splines Calculating simple parametric splines" We can now solve for the vector constants a0 , a1 andaa2 nd a as We can now solve for the vector constants...
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• Spring '14
• µ, cubic spline patch, Interpolating Splines

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