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Published byBrittney Welch Modified over 6 years ago

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1 Introduction Curve Modelling Jack van Wijk TU Eindhoven

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2 Overview curve modelling Parametric curves Requirements Concepts Lagrangian interpolation Bézier curve B-spline Cubic splines

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3 Parametric curves t p(t)p(t)p’(t)

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4 Tangent line to curve t p(t)p(t)p’(t) s q(s)q(s)

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5 Example t p(t)p(t) p’(t) s q(s)q(s)

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6 Curve modelling Problem: How to define a smooth curve? Solution: –Specify a sequence of points p i, i = 1,…, N, (control-points); –Generate a smooth curve that interpolates or approximates these control-points.

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7 Requirements Smooth –no discontinuities in direction and curvature; Local control –Change of a point should have only local effect; Intuitive and easy to use –no oscillations, variation diminishing Approximate or interpolate?

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8 Parametric interpolation

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9 Linear interpolation of two points p1p1 p2p2 t

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10 Linear interpolation of N points p1p1 p2p2 t p3p3 p4p4 p5p5 t wiwi 1 2345 1 0

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11 Linear interpolation Not Smooth Local control Intuitive and easy to use Interpolate

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12 Lagrangian interpolation - 1

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13 Lagrangian interpolation - 2

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14 Lagrangian interpolation - 3

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15 Lagrangian interpolation - 4 Smooth –no discontinuities in direction and curvature; NO local control Wild oscillations, not variation diminishing! Interpolating

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16 Bézier curve - 1 Puzzle: Define a smooth curve that interpolates the first and last point and approximates the others. p1p1 p2p2 p3p3 p4p4

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17 Bézier curve - 2 Solution for N=3: p1p1 p2p2 p3p3 q2q2 q1q1 p(t)

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18 Bézier curve - 3 Solution for N=4: p1p1 p2p2 p3p3 p(t) p4p4

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19 Bézier curve - 4

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20 Bézier curve - 5

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21 Bézier curve - 6 General Bézier curve: Degree = #points-1 Smooth –no discontinuities in direction and curvature; NO local control Variation diminishing, convex hull property Interpolates first and last, further approximating

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22 Convex hull property

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23 Convex hull property example Curve outside convex hull Curve inside convex hull

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24 B-splines Piecewise polynomial, locally non-zero Degree: user definable Continuity: degree-1 –first degree: continuous in position –second degree: continuous in tangent –third degree: continuous in curvature

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25 Quadratic B-spline

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26 Cubic B-spline

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27 B-splines - 3 General B-spline: Degree: from 1 to N-1 Smooth (if degree > 1) Local control Variation diminishing, convex hull property Approximating

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28 Cubic splines - 1 Most popular in Computer Graphics: powerful –inflection points, continuity simple –low degree polynomials local control Many versions: –Bézier, B-spline, Catmull-Rom,...

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29 Cubic splines - 2

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30 Cubic splines - 2

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31 Cubic splines - 3

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32 Tangent vector t p’(t) p(t)

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33 Significant values p(0) : startpoint segment p’(0) : tangent through startpoint p(1) : endpoint segment p’(1) : tangent through endpoint t p(0) p(1) p’(0) p’(1)

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34 Cubic Bézier curve p0p0 p1p1 p2p2 p3p3

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35 Joining two Bezier segments p0p0 p2p2 p3p3 q0q0 q2q2 q3q3 Positional continuity: p 3 = q 0 Tangential continuity: p 3 - p 2 // q 1 - q 0 p1p1 q1q1

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36 Cubic B-spline curve p0p0 p3p3 p1p1 p2p2

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37 Conversion - 1

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38 Conversion - 2

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39 Puzzle 1: Hermite spline

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40 Puzzle 2: Limit on interpolation Find out why a curve that –interpolates the control points, –stays within the convex hull, –and is smooth cannot exist, both graphically and mathematically.

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41 Puzzle 3: Split Bezier curve Find a recipe to split a cubic Bezier curve segment into two segments: p0p0 p2p2 p3p3 q0q0 q1q1 q6q6 p1p1 q2q2 q3q3 q4q4 q5q5

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42 Finally... Curves: interpolation of points –Interpolation is generally applicable, f.i. surfaces: interpolation of curves Demo program: www.win.tue.nl/~vanwijk/2M050/spline.exe

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43 B-splines - 2

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