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Computer Graphics. Representing Curves and Surfaces. Review eq(11.5). Review eq(11.6/11.7). Review eq(11.8). Review eq(11.8). Review eq(11.9). Review eq(11.10). Review. Blending function (also called ‘Basis’ function). Hermite Curves. 以曲線端點 P 1 . P 4 以及端點斜率 R 1 . R 4 求曲線方程式.

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Computer graphics

Computer Graphics

Representing Curves and Surfaces


Review eq 11 5

Revieweq(11.5)


Review eq 11 6 11 7

Review eq(11.6/11.7)


Review eq 11 8

Review eq(11.8)


Review eq 11 81

Review eq(11.8)


Review eq 11 9

Review eq(11.9)


Review eq 11 10

Review eq(11.10)


Review

Review

  • Blending function

    (also called ‘Basis’ function)


Hermite curves

Hermite Curves

  • 以曲線端點P1.P4以及端點斜率R1.R4求曲線方程式


Hermite curves eq 11 12

Hermite Curveseq(11.12)


Hermite curves eq 11 13

Hermite Curveseq(11.13)


Hermite curves eq 11 14

Hermite Curveseq(11.14)


Hermite curves eq 11 15

Hermite Curveseq(11.15)


Hermite curves1

Hermite Curves


Hermite curves eq 11 16

Hermite Curveseq(11.16)


Hermite curves eq 11 17

Hermite Curveseq(11.17)


Hermite curves eq 11 18

Hermite Curveseq(11.18)


Hermite curves eq 11 19

Hermite Curveseq(11.19)


Hermite curves eq 11 20

Hermite Curveseq(11.20)


Hermite curves eq 11 21

Hermite Curveseq(11.21)


Hermite curve

Hermite Curve


Hermite curve eq 11 22

Hermite Curveeq(11.22)


Hermite curve eq 11 23

Hermite Curveeq(11.23)

  • Reduce 6 multiplies and 3 additions to 3 multiplies and 3 additions.


Bezier curves

Bezier Curves

  • 以曲線端點P1.P4以及控制點P2.P3求曲線方程式

  • 曲線端點斜率為


Bezier curves eq 11 24

Bezier Curveseq(11.24)


Bezier curves eq 11 25

Bezier Curveseq(11.25)


Bezier curves eq 11 26

Bezier Curveseq(11.26)


Bezier curves eq 11 27 11 28

Bezier Curveseq(11.27/11.28)


Bezier curves eq 11 29

Bezier Curveseq(11.29)


Bezier curves1

Bezier Curves


Bezier curves2

Bezier Curves

  • Define n as the order of Bezier curves.

  • Define i as control point.


Bezier curves3

Bezier Curves


Bezier curves4

Bezier Curves


Bezier curves5

Bezier Curves

  • De Casteljau iterations


Bezier curves6

Bezier Curves

  • Linear Bezier splinesControl points: P0, P1


Bezier curves7

Bezier Curves

  • Quadratic Bezier splinesControl points: P0, P1, P2


Bezier curves8

Bezier Curves

  • Quadratic Bezier splinesControl points: P0, P1, P2


Bezier curves9

Bezier Curves

  • Cubic Bezier splinesControl points: P0, P1, P2, P3


Bezier curves10

Bezier Curves

  • Cubic Bezier splinesControl points: P0, P1, P2, P3


Bezier curves11

Bezier Curves

  • Cubic Bezier splinesControl points: P0, P1, P2, P3


Bezier curves12

Bezier Curves


Bezier curves13

Bezier Curves

  • http://www.ibiblio.org/e-notes/Splines/Bezier.htm


Spline

Spline

  • Natural cubic spline

    • C0, C1, C2 continuous.

    • Interpolates(passes through) the control points.

    • Moving any one control point affects the entire curve.


Spline1

Spline

  • B-spline

    • Local control.

    • Moving a control point affects only a small part of a curve.

    • Do not interpolate their control points.

    • Sharing control points between segments.


B spline

B-spline

  • m+1 control points P0, …, Pm, m≥3

  • m-2 curve segments Q3, Q4, …, Qm

  • For each i≥4, there is a join point or knot between Qi-1 and Qi at the parameter value ti.


B spline eq 11 43 11 44

B-splineeq(11.43/11.44)


B spline eq 11 44

B-splineeq(11.44)


Uniform nonrational b spline

Uniform Nonrational B-spline

  • ‘Uniform’ means that the knots are spaced at equal intervals of the parameter t.

  • ‘Nonrational’ is used to distinguish these splines from rational cubic polynomial curves, see Section 11.2.5

  • We assume that t3=0 and the interval ti+1-ti=1


Uniform nonrational b spline eq 11 32 11 33 11 34

Uniform Nonrational B-splineeq(11.32/11.33/11.34)


Uniform nonrational b spline eq 11 35

Uniform Nonrational B-splineeq(11.35)


Uniform nonrational b spline eq 11 36

Uniform Nonrational B-splineeq(11.36)


Uniform nonrational b spline eq 11 37

Uniform Nonrational B-splineeq(11.37)


Uniform nonrational b spline eq 11 38

Uniform Nonrational B-splineeq(11.38)


Uniform nonrational b spline1

Uniform Nonrational B-spline


Uniform nonrational b spline2

Uniform Nonrational B-spline


Uniform nonrational b spline eq 11 39 11 40 11 41

Uniform Nonrational B-splineeq(11.39/11.40/11.41)


Uniform nonrational b spline eq 11 42

Uniform Nonrational B-splineeq(11.42)

  • The curve can be forced to be interpolate specific points by replicating control points.


Nonuniform nonrational b spline

Nonuniform Nonrational B-spline

  • Parameter interval between successive knot values need not be uniform.

  • Blending functions are no longer the same for each interval.

  • Continuity at selected join points can be reduced from C2 to C1 to C0 to none.

    • When the curve is C0, the curve interpolates a control point.


Nonuniform nonrational b spline1

Nonuniform NonrationalB-spline

  • If the continuity is reduced to C0, then the curve interpolates a control point, but without the undesirable effect of uniform B-splines, where the curve segments on either side of the interpolated control point are straight lines.


Nonuniform nonrational b spline2

Nonuniform NonrationalB-spline

  • m+1 control points P0, P1, …, Pm

  • nondecreasing sequence of knot valuest0, t1, …, tm+4


Nonuniform nonrational b spline3

Nonuniform NonrationalB-spline


Nonuniform rational cubic polynomial curve

Nonuniform Rational Cubic Polynomial Curve


Nonuniform rational cubic polynomial curve1

Nonuniform Rational Cubic Polynomial Curve

  • Advantages

    • They are invariant under rotation, scaling, translation and perspective transformations of the control points.

    • They can define precisely and of the conic sections.


Catmull rom spline

Catmull-Rom spline


Catmull rom spline1

Catmull-Rom spline


Catmull rom spline eq 11 47

Catmull-Rom splineeq(11.47)


Subdividing curves

Subdividing Curves

  • Bezier Curve


Subdividing curves1

Subdividing Curves


Subdividing curves2

Subdividing Curves

  • Uniform B-spline Curve

    • Doubling:

    • Do k subdivision steps (for k-degree B-spline)


Subdividing curves3

Subdividing Curves


Conversion eq 11 56

Conversioneq(11.56)


Conversion eq 11 57 11 58

Conversioneq(11.57/11.58)


Drawing curves eq 11 59 11 60 11 61

Drawing Curveseq(11.59/11.60/11.61)


Drawing curves eq 11 62 11 63

Drawing Curveseq(11.62/11.63)


Drawing curves eq 11 64 11 65 11 66 11 67

Drawing Curveseq(11.64/11.65/11.66/11.67)


Drawing curves eq 11 68 11 69 11 70

Drawing Curveseq(11.68/11.69/11.70)


Drawing curves eq 11 71 11 72

Drawing Curveseq(11.71/11.72)


Comparison

Comparison


Parametric bicubic surfaces eq 11 73

Parametric Bicubic Surfaceseq(11.73)


Parametric bicubic surfaces

Parametric Bicubic Surfaces


Parametric bicubic surfaces eq 11 74 11 75

Parametric Bicubic Surfaceseq(11.74/11.75)


Parametric bicubic surfaces eq 11 76

Parametric Bicubic Surfaceseq(11.76)


Hermite surfaces eq 11 77

Hermite Surfaceseq(11.77)


Hermite surfaces

Hermite Surfaces


Hermite surfaces eq 11 78

Hermite Surfaceseq(11.78)


Hermite surfaces eq 11 79 11 80 11 81

Hermite Surfaceseq(11.79/11.80/11.81)


Hermite surfaces eq 11 82 11 83

Hermite Surfaceseq(11.82/11.83)


Hermite surfaces eq 11 84

Hermite Surfaceseq(11.84)


Hermite surfaces1

Hermite Surfaces


Normals to surfaces eq 11 88 11 89

Normals to Surfaceseq(11.88/11.89)


Normals to surfaces eq 11 90

Normals to Surfaceseq(11.90)


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