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Слайды и текст этой презентации

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Week 7: Geometric Modeling - Parametric Representation of Synthetic Curves Spring 2018, AUA

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Planar vs. Space

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Analytic (known form) vs. Synthetic (free form)

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Interpolation vs. Approximation The curve passing through given data (control) points - interpolation curve. The curve not necessarily passing but controlled by data points - approximation curve

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Continuity

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Cubic Curves

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Hermite Cubic Splines

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Hermite Cubic Splines

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Hermite Cubic Spline – Tangent Vector

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Hermite Cubic Splines - example The Hermite curve fits the points: P0 = [1,1]T, P1 = [3,5]T and the tangent vectors: P0’ = [0,4]T, P1’ = [4,0]T. Calculate the parametric mid-point of the curve, the tangent vector on that point. Sketch the curve on the grid

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Bezier Curves - sl. 1

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Bezier Curves - sl. 3

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Bezier Curves - sl. 2

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Bezier Curves - sl. 5 Practice The coordinates of 4 control points are given: P0 = [2,2]T, P1 = [2,3]T, P3 = [3,3]T, P4 = [3,2]T Find the equation of the resulting Bezier curve, Find the points on the curve for u = 0, ¼, ½, ¾, 1, Sketch the curve.

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Bezier Curves - sl. 4

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B-spline Curves - sl. 1 See:

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B-spline Curves - sl. 2 See:

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B-spline Curves - sl. 3 Basis Functions The function Ni,k determines how strongly control point Pi influences the curve at t. Its value is a real number – 0.25, 0.5…