Chapter 10-2: Curves

Cubic Spline Interpolation: Spline Functions A thin, flexible strip used to draw a smooth curve through a given series of points Fig. 10.13 – piecewise cubic polynomials Continuous up to its second derivatives at each supports – continuous w.r.t. position, tangent, and curvature Advantageous in Engineering Problems When the data values are relatively accurate and large in number.

Cubic Spline [II] Spline Parametric Continuity General piecewise parametric representation of geometry with a specified level of parametric continuity Parametric Continuity Fig. 10.14 Continuity requirements relate to the parametric formulation Cf.: Geometric continuity – parameterization independent measure of continuity

Cubic Spline [III] Normalized Cubic Spline Cubic splines are parameterized separately Interval of parameters is [0, 1] for all segments A special case of Hermite interpolation Eq. (10.43) ~ Eq. (10.44) Second Derivative Continuity at Point Pi Fig. 10.15 Eq. (10.45) ~ Eq. (10.50)

“[Pi-1(1)  Pi(0)]”

Cubic Spline [IV] All Cubic Spline Segments Case (a) Case (b) m tangent vectors are needed m – 2 equations Case (a) Known end tangent vectors, P´0 and P´m-1 Eq. (10.51) ~ Eq. (10.53) Example 10.4 Case (b) Second derivatives at the endpoints P0 and Pm-1 both made equal to zero Natural cubic spline Eq. (10.54) ~ Eq. (10.57): Example 10.5

Interpolation summary

Interpolation v.s. Approximation Mathematical Approaches to Represent Curves Interpolation: data fitting Approximation Freeform Shape: Figure 10.16 Fender of a car, transition between the wing and the fuselage of an airplane, hull of a ship, handle of a coffee mug, femur in a human leg.

Interpolation v.s. Approximation [II] Traditional Design Descriptive geometric methods – surface: planar sections and some characteristic lines Master models Final stamps and dies Approximation: Figure 10.17 Digitizing process from existing curve and surface definitions Appearance of the model be acceptable to the designer’s judgment, w/o consideration of the quality of the interpolation Smoothness of a curve or surface is the most important criteria Changes are localized. Bezier and B-Spline

Bezier Curves Features of Bezier Curve The 1st and the last points represent the start and the end of curve, respectively. No oscillation even an increase in the number of points used. The 1st two points and the last two points represent each tangent vector for both sides of a curve.

Bezier Curve of Degree n

Bernstein Polynomial

Cubic Bezier Curve

Properties of Bezier Curve Convex hull property: Fig. 10.18 Effect of moving control points: Fig. 10.19 Closed loop: first and last control points are coincide: Fig. 10.21 Lack of local control: Fig. 10.22 Piecewise Bezier curve in case of a large # of control points: Fig. 10.23 C1 continuity: 3 control points around the intersection are colinear

Matrix Approach

B-Spline Curves Major Advantage over Bezier Curve Bezier curve is global control – hard to change locally  segmentation with the price of low order continuity B-spline curve generates a single piecewise parametric polynomial curve through any number of control points. Degree of the polynomial can be selected by the designer independently. B-spline curves exhibit local control – if one control point is moved, only some curve segments are affected. Cubic or higher degree B-spline curves guarantee curvature continuity C2.

Uniform Cubic B-splines Li(t): a cubic C2 basis function in Fig. 10.25 Knot vector in non-decreasing order Four cubic polynomials: L0,3 L1,3 L2,3 L3,3 Pi(t) = L3,3 (t)Vi + L2,3 (t) Vi+1 + L1,3 (t)Vi+2 + L0,3 (t)Vi+3 Example: Figure 10.26 Why uniform ?: parametric intervals (knots), t, are equal  called uniform or periodic.

Matrix Approach i = 1, 2, 3, 4

Uniform B-splines are well suited to represent closed curves.

예제 10.8 닫힌 균일 2차 B-스플라인에 의해서 원을 근사화하기 위해 네 개의 조정점을 사용한다. 첫 세그먼트의 t=0.5에서의 근사오차를 계산하라.

예제 10.8

Conversion Between Representations

Table 10.3