1. Chapter 10-2:
Curves

3. Cubic Spline Interpolation: Spline Functions
A thin, flexible strip used to draw a smooth
curve through a given series of points
Fig – 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.

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

7. 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
Eq. (10.45) ~ Eq. (10.50)

10. “[Pi-1(1)  Pi(0)]”

11. 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

19. Interpolation summary

21. 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.

23. 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

25. 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.

26. Bezier Curve of Degree n

27. Bernstein Polynomial

28. Cubic Bezier Curve

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

38. Matrix Approach

42. 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.

43. Uniform Cubic B-splines
Li(t): a cubic C2 basis function in Fig
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.

48. Matrix Approach
i = 1, 2, 3, 4

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

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

53. 예제 10.8

54. Conversion Between Representations

56. Table 10.3