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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 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.

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

- Define n as the order of Bezier curves.
- Define i as control point.

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

- 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-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 splineeq(11.47)

Subdividing Curves

- Bezier Curve

Subdividing Curves

- 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)

Parametric Bicubic Surfaceseq(11.73)

Parametric Bicubic Surfaceseq(11.74/11.75)

Parametric Bicubic Surfaceseq(11.76)

Hermite Surfaceseq(11.77)

Hermite Surfaceseq(11.78)

Hermite Surfaceseq(11.79/11.80/11.81)

Hermite Surfaceseq(11.82/11.83)

Hermite Surfaceseq(11.84)

Normals to Surfaceseq(11.88/11.89)

Normals to Surfaceseq(11.90)

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