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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

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Computer Graphics

Representing Curves and Surfaces


Revieweq(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P1.P4以及端點斜率R1.R4求曲線方程式


Hermite Curveseq(11.12)


Hermite Curveseq(11.13)


Hermite Curveseq(11.14)


Hermite Curveseq(11.15)


Hermite Curves


Hermite Curveseq(11.16)


Hermite Curveseq(11.17)


Hermite Curveseq(11.18)


Hermite Curveseq(11.19)


Hermite Curveseq(11.20)


Hermite Curveseq(11.21)


Hermite Curve


Hermite Curveeq(11.22)


Hermite Curveeq(11.23)

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


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 Curveseq(11.24)


Bezier Curveseq(11.25)


Bezier Curveseq(11.26)


Bezier Curveseq(11.27/11.28)


Bezier Curveseq(11.29)


Bezier Curves


Bezier Curves

  • Define n as the order of Bezier curves.

  • Define i as control point.


Bezier Curves


Bezier Curves


Bezier Curves

  • De Casteljau iterations


Bezier Curves

  • Linear Bezier splinesControl points: P0, P1


Bezier Curves

  • Quadratic Bezier splinesControl points: P0, P1, P2


Bezier Curves

  • Quadratic Bezier splinesControl points: P0, P1, P2


Bezier Curves

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


Bezier Curves

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


Bezier Curves

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


Bezier Curves


Bezier Curves

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


Spline

  • Natural cubic spline

    • C0, C1, C2 continuous.

    • Interpolates(passes through) the control points.

    • Moving any one control point affects the entire curve.


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

  • 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-splineeq(11.43/11.44)


B-splineeq(11.44)


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-splineeq(11.32/11.33/11.34)


Uniform Nonrational B-splineeq(11.35)


Uniform Nonrational B-splineeq(11.36)


Uniform Nonrational B-splineeq(11.37)


Uniform Nonrational B-splineeq(11.38)


Uniform Nonrational B-spline


Uniform Nonrational B-spline


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


Uniform Nonrational B-splineeq(11.42)

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


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 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 NonrationalB-spline

  • m+1 control points P0, P1, …, Pm

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


Nonuniform NonrationalB-spline


Nonuniform Rational Cubic Polynomial Curve


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 splineeq(11.47)


Subdividing Curves

  • Bezier Curve


Subdividing Curves


Subdividing Curves

  • Uniform B-spline Curve

    • Doubling:

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


Subdividing Curves


Conversioneq(11.56)


Conversioneq(11.57/11.58)


Drawing Curveseq(11.59/11.60/11.61)


Drawing Curveseq(11.62/11.63)


Drawing Curveseq(11.64/11.65/11.66/11.67)


Drawing Curveseq(11.68/11.69/11.70)


Drawing Curveseq(11.71/11.72)


Comparison


Parametric Bicubic Surfaceseq(11.73)


Parametric Bicubic Surfaces


Parametric Bicubic Surfaceseq(11.74/11.75)


Parametric Bicubic Surfaceseq(11.76)


Hermite Surfaceseq(11.77)


Hermite Surfaces


Hermite Surfaceseq(11.78)


Hermite Surfaceseq(11.79/11.80/11.81)


Hermite Surfaceseq(11.82/11.83)


Hermite Surfaceseq(11.84)


Hermite Surfaces


Normals to Surfaceseq(11.88/11.89)


Normals to Surfaceseq(11.90)


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