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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 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 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 curves2
Bezier Curves

  • Define n as the order of Bezier curves.

  • Define i as control point.




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 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 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 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.





Subdividing curves
Subdividing Curves

  • Bezier Curve



Subdividing curves2
Subdividing Curves

  • Uniform B-spline Curve

    • Doubling:

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



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)










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)




Normals to surfaces eq 11 88 11 89
Normals to Surfaceseq(11.88/11.89)



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