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Computer Graphics - PowerPoint PPT Presentation

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

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)

• Blending function

(also called ‘Basis’ function)

• 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 Curveseq(11.16)

Hermite Curveseq(11.17)

Hermite Curveseq(11.18)

Hermite Curveseq(11.19)

Hermite Curveseq(11.20)

Hermite Curveseq(11.21)

Hermite Curveeq(11.22)

Hermite Curveeq(11.23)

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

• 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)

• Define n as the order of Bezier curves.

• Define i as control point.

• De Casteljau iterations

• Linear Bezier splinesControl points: P0, P1

• Quadratic Bezier splinesControl points: P0, P1, P2

• Quadratic Bezier splinesControl points: P0, P1, P2

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

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

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

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

• 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.

B-splineeq(11.43/11.44)

B-splineeq(11.44)

• ‘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(11.39/11.40/11.41)

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 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.

• Bezier Curve

• Uniform B-spline Curve

• Doubling:

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

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)

Hermite Surfaceseq(11.79/11.80/11.81)

Hermite Surfaceseq(11.82/11.83)

Normals to Surfaceseq(11.88/11.89)