Properties of splines

　　A number of properties have emerged as good criteria for evaluating interpolating splines.

Fairness

　　In almost all applications, the most important property of a spline is fairness, also known as smoothness. There are a number of metrics that correlate reasonably well with perception of fairness. Knowledge of the human visual system and empirical evidence show that fairness is intimately tied to curvature, which of course is an objective mathematical property. Curves with regions of extremely high curvature are clearly not fair. In addition, while more subtle, discontinuities in curvature and regions with large curvature variation are also perceived as lack of fairness. Curvature is central in many other application domains as well. There is compelling evidence that the human visual system perceives curves in terms of curvature features, motivating a substantial body of literature on shape completion, or inferring missing segments of curves when shapes are occluded in the visual field. A recent example, advocating the Euler spiral spline.

Continuity

　　A concrete property related to fairness is the degree of continuity, essentially the number of higher derivatives that exist. All splines worthy of the name have at least G1-continuity, meaning that the second derivative exists everywhere, but relatively fewer have G2-continuity, corresponding to continuous curvature.

Roundness

　　One specific aspect of fairness is roundness. A round spline always yields a circular arc when the control points are co-circular. It seems intuitively obvious that a circular arc is the fairest (or smoothest) curve. Roundness is not an all-or-nothing property. Even the splines that do not generate perfect circular arcs produce curves that are fairly close. Exactly how close can be quantified, as we will see in the discussions of individual splines.

Extensionality

　　An extensional interpolating spline is one in which adding a new point to the control polygon, lying on the curve generated from the original control polygon, does not change the resulting curve. Unlike fairness, extensionality is not so much a property of the curve segments used to generate the spline as of the effect of changes to the control polygon (adding and deleting points) on the resulting curve. Splines defined in terms of minimizing a functional tend to be extensional. Extensionality is a powerful filter for weeding out crude approximations, but it does admit a variety of interesting splines in addition to those defined in terms of optimization. Roundness is a very weak extensionality property. However, even though these two properties are related, there are round splines that are not extensional , and extensional splines that are not round.

Existence and uniqueness

　　One highly desirable property of a spline is that all control polygons yield interpolating curves. In general, splines defined in terms of meeting continuity constraints at the control points tend to have better existence properties. Another desirable property is uniqueness, the principle that only one curve exists that satisfies the definition of the spline, for a fixed input. It is possible to fix the uniqueness problems of a spline by adding an additional (global) criterion, to prefer one solution to the other. Then, the revised spline can be defined as one that both satisfies the local constraints and also optimizes the global criterion.

Locality

When a control point moves, how much of the resulting curve changes? A small section in the neighborhood of the control point, perhaps, or does the entire spline change? This is the property of locality. However, finite support conflicts directly with extensionality, so in general we will measure locality in terms of how quickly the effect of moving a control point dies out as we move away from it.

Transformational invariance

　　Another basic property is invariance under various transformations and symmetries. All functions worthy of being considered splines are invariant under rigid body transformations (rotation and translation) as well as uniform scaling. If the control polygon undergoes a rigid body transformation, then the resulting curve undergoes exactly the same transformation. A sometimes desirable property of splines is affine invariance, meaning that the shape of the spline is preserved through affine transformation of the control points.

Counting parameters

　　In an arbitrary spline, the family of curve segments between any two control points is potentially drawn from an infinite-dimensional space. In practice, most splines use curve segments chosen from a finite-dimensional manifold. Generally, there is a vector of real parameters that uniquely determines the shape of the curve between any two endpoints. In counting these parameters, we hold the endpoints fixed, and apply the rotation, scaling, and translation to make the curve coincide with these endpoints.

Monotone curvature

　　This property suggests that all curvature extrema should coincide with the given control points, and that the spline segments in between should exhibit monotone curvature. In interactive curve design it is reasonable to expect designers to place these points first, then refine the curves with more control points only as needed. A good spline should accommodate such a designer by preserving the curvature extrema as marked.

Summary

　　The most important property for interpolating splines is fairness. Yet, other properties such as robustness and locality are also important.