Chapter 5
Splines
A natural way of designing a curve (or a surface) is to first sketch a general contour of the curve (or
surface) and then make local changes in the curve to achieve the required shape. This chapter
addresses the following issues:
(1) Local modification over any segment of the curve: One should be able to change the position
of a control point in an intuitive way without changing the overall (global) shape of the entire curve.
(2) Delink the number of control points and the degree of the polynomial: One should be able to
use lower degree polynomial segments and still maintain a large number of control points to
help in shape refinement.
(3) Finer shape control by “knot” insertions: This provides additional tool for designing and local
editing of the curve shape.
In Chapter 4 we studied curve design with parametric piecewise curves using Ferguson and Bézier
segments. Composite Ferguson curves are naturally C^1 continuous at junction points. However, their
design requires specifying the first order (slope) information along with data points which most often
is non-intuitive from the designers’ perspective. For C^2 or curvature continuous composite Ferguson
curves, the slope information is reduced to specification only at the two end points. This curve has
no local control for if one changes the location of a data point, the entire curve is altered and needs
to be re-computed along with the intermediate slopes. With Bézier segments, only data points are
specified. However, individual segments have no local control. Composite Bézier curves further tend
to constrain the position of data points of the subsequent segments. For instance, slope continuity at
the junction point requires the junction point and its two immediate neighbors to be collinear. Further,
C^2 continuity requires four data points around the junction to lie on a plane that contains the junction
point itself. Choosing data points freely for composite Bézier curves with relatively lower degree
segments therefore is difficult. For this reason, Bézier segments with orders 6 or 8 (degree 5 or 7,
respectively) are employed by most CAD softwares. In addition to being parametric piecewise fits,
it is also desired for a curve to be inherently C^2 continuous everywhere with local control properties.
These design requirements are met by a class of curves called splines which are discussed in detail
in this chapter.
5.1 Definition
The term splineis derived from the analogy to a draughtsman’s approach to pass a thin metal or
wooden strip through a given set of constrained points called ducks(Figure 5.1). We can imagine any