Page 10: Comparing Interpolation Methods

Before we learn how to use cubics to interpolate points, we will first motivate why we like to use cubics by comparing them with their alternatives.

The Graphics in 5 Minutes Video introduces this topic:

• Splines in 5 minutes: Part 1 – cubic curves

Interpolation

We say a curve interpolates a value if it takes that value given a certain value of its parameter. We call that value of the parameter the site. So, when we specify that a curve $\mathbf{f}(u)$ has value ${\mathbf{p}}_0$ at its beginning (i.e., $\mathbf{f}(0)={\mathbf{p}}_0$), we would say “the curve interpolates ${\mathbf{p}}_0$ at site 0.”

When we specify interpolation, we might not specify the site - we might just specify that the curve has the value at some point in its range. Also, interpolation might specify the value of the derivative, not just the position.

However, the most common form of interpolation problem is to “connect the dots.” We are given a list of points (the sites), and we need to make a curve through it. The problem might specify the t value for the curve when it reaches each of those sites. But it is common just to assume uniform spacing at the integers.

There are an infinite number of ways to connect the dots. An implementation method effectively chooses one.

Comparing Interpolation Types

Here, we look at three ways to do interpolation to compare them and bring out some of the pros and cons of the approaches. All of the demos are linked. Move the points in any one of them to see how the others behave.

The simplest way to connect the dots is with straight lines. Besides being simple and easy to describe, it has the property of being the shortest curve that interpolates the points.

A problem with the connected line segments is that the resulting curve is not smooth. It is just C(0).

But notice two positives that you might take for granted. First, when you move one point, only the line segments close to that point move. If you move something at the beginning, things at the end don’t change. We call this property locality.

And second, notice how there are no extra wiggles. It’s pretty easy to predict and control the curve. We just cannot get smoothness.

Our next example is the opposite: we will fit a single eighth-degree polynomial through the nine points. In general, we can fit an n-1 degree polynomial through n points. The particular version we are using here is called a Lagrange Polynomial. You can look the equation up in the curve notes or some other book. It’s a neat demo for a graphics class.

Because this is a single polynomial, we can take the derivative as many times as we want. It will never have a discontinuity. It is as “smooth” (in the continuity sense) as it can be.

However, this curve does not have locality: notice how if you move the first point, the entire curve changes - including things near the end. The changes to the end are subtle if you move the beginning; but if you change the middle notice how big the changes are everywhere. Also, notice how it tends to wiggle all over the place; often overshooting its target so it can smoothly return after taking a long route.

In practice, higher-degree polynomials are rarely used in graphics. In addition to these issues that make them hard to control, there are numerical issues when you raise small numbers to large powers.

In the final example, we consider something that tries to balance the two. This is a curve made of a series of cubic segments. A cubic polynomial is used between every pair of points. The neighboring points are used to determine the tangents at the interpolation points. This particular curve is called a Cardinal Spline, and we will learn more about it on the upcoming pages.

But, for now, notice that it does have local control. Moving one point only affects a few parts of the curve close to it. It is pretty smooth (it is C(1)), and doesn’t overshoot points too much.

In computer graphics, we prefer these kinds of piecewise polynomial representations (like the Cardinal Spline) because they balance smoothness and control. Because the pieces are cubics, the math is also well behaved.

Choosing Cubics

In Computer Graphics, we often use piecewise polynomial curves, but rarely use curves of more than degree 3 (cubics). Cubics provide a good balance of smoothness and control. We can achieve C(1) or C(2) relatively easily, which is sufficient for most graphics applications.

The term for a piecewise polynomial curve is a spline.