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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

Representing Curves and Surfaces


Review eq(11.6/11.7)

Review eq(11.8)

Review eq(11.8)

Review eq(11.9)

Review eq(11.10)


  • 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



  • Natural cubic spline

    • C0, C1, C2 continuous.

    • Interpolates(passes through) the control points.

    • Moving any one control point affects the entire curve.


  • 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.


  • 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.



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



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)


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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