Computer Graphics Representing Curves and Surfaces

Review eq(11.5)

Review eq(11.6/11.7)

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

Hermite Curves eq(11.12)

Hermite Curves eq(11.13)

Hermite Curves eq(11.14)

Hermite Curves eq(11.15)

Hermite Curves

Hermite Curves eq(11.16)

Hermite Curves eq(11.17)

Hermite Curves eq(11.18)

Hermite Curves eq(11.19)

Hermite Curves eq(11.20)

Hermite Curves eq(11.21)

Hermite Curve

Hermite Curve eq(11.22)

Hermite Curve eq(11.23) Reduce 6 multiplies and 3 additions to 3 multiplies and 3 additions.

Bezier Curves 以曲線端點 P 1.P 4 以及控制點 P 2.P 3 求曲線方程式 曲線端點斜率為

Bezier Curves eq(11.24)

Bezier Curves eq(11.25)

Bezier Curves eq(11.26)

Bezier Curves eq(11.27/11.28)

Bezier Curves eq(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 splines Control points: P 0, P 1

Bezier Curves Quadratic Bezier splines Control points: P 0, P 1, P 2

Bezier Curves Quadratic Bezier splines Control points: P 0, P 1, P 2

Bezier Curves Cubic Bezier splines Control points: P 0, P 1, P 2, P 3

Bezier Curves Cubic Bezier splines Control points: P 0, P 1, P 2, P 3

Bezier Curves Cubic Bezier splines Control points: P 0, P 1, P 2, P 3

Bezier Curves

Spline Natural cubic spline C 0, C 1, C 2 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 P 0, …, P m, m≥3 m-2 curve segments Q 3, Q 4, …, Q m For each i≥4, there is a join point or knot between Q i-1 and Q i at the parameter value t i.

B-spline eq(11.43/11.44)

B-spline eq(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 We assume that t 3 =0 and the interval t i+1 -t i =1

Uniform Nonrational B-spline eq(11.32/11.33/11.34)

Uniform Nonrational B-spline eq(11.35)

Uniform Nonrational B-spline eq(11.36)

Uniform Nonrational B-spline eq(11.37)

Uniform Nonrational B-spline eq(11.38)

Uniform Nonrational B-spline

Uniform Nonrational B-spline eq(11.39/11.40/11.41)

Uniform Nonrational B-spline eq(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 C 2 to C 1 to C 0 to none. When the curve is C 0, the curve interpolates a control point.

Nonuniform Nonrational B-spline If the continuity is reduced to C 0, 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-spline m+1 control points P 0, P 1, …, P m nondecreasing sequence of knot values t 0, t 1, …, t m+4

Nonuniform Nonrational B-spline

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 eq(11.47)

Subdividing Curves Bezier Curve

Subdividing Curves

Uniform B-spline Curve Doubling: Do k subdivision steps (for k-degree B-spline)

Subdividing Curves

Conversion eq(11.56)

Conversion eq(11.57/11.58)

Drawing Curves eq(11.59/11.60/11.61)

Drawing Curves eq(11.62/11.63)

Drawing Curves eq(11.64/11.65/11.66/11.67)

Drawing Curves eq(11.68/11.69/11.70)

Drawing Curves eq(11.71/11.72)

Comparison

Parametric Bicubic Surfaces eq(11.73)

Parametric Bicubic Surfaces

Parametric Bicubic Surfaces eq(11.74/11.75)

Parametric Bicubic Surfaces eq(11.76)

Hermite Surfaces eq(11.77)

Hermite Surfaces

Hermite Surfaces eq(11.78)

Hermite Surfaces eq(11.79/11.80/11.81)

Hermite Surfaces eq(11.82/11.83)

Hermite Surfaces eq(11.84)

Hermite Surfaces

Normals to Surfaces eq(11.88/11.89)

Normals to Surfaces eq(11.90)