Curve fitting: least square vs. spline?

Question

What is the difference between the two? Which curve fitting method will give a more "good looking" fit? If I will apply it to a road lane boundary with 3 to 5 known points, which method will give a better fit in terms of how we see it with our eyes.

Answer 1

Apples and oranges.

Least squares is a method for finding the "best" approximating curve. You can apply least squares to almost any kind of curve including splines, and this is often done, but you can also apply least squares to other curves: straight lines, exponential, etc.

http://en.wikipedia.org/wiki/Least_squar...

A spline is a member of a particular family of related curves. There are the traditional splines, quadratic and cubic splines, B-splines, Non-Uniform Rational B-splines, etc.

http://en.wikipedia.org/wiki/Spline_(mat...

Splines, like many other curves, can be used to interpolate the data or to smooth the data. If you are using it to smooth the data, then you'll apply least squares to find the "best" smoothing spline. If you are using it to interpolate the data, then you'll use the splines directly.

But splines can also be used to create curves with direct manipulation. B-splines and NURBS are frequently used geometric construction (such as the shape of a car, etc.)  in this mode, not fitting a predefined set of data points at all.

http://en.wikipedia.org/wiki/NURBS

Your term "fit"  together with "3 to 5 known points" could apply to either interpolation or smoothing. But a road designer would probably go the construction route rather than the fitting route.

As for what looks better, do you, your parents, and everyone else you know always agree on matters of what looks better or sounds better? What would you think that what I claim looks better would be the same as what you are sure looks better?

(As it happens, way back when, Toyota and Ford both used bi-cubic splines to design their car shapes, but they used different parameters to get the different "looks" the two groups of designers were looking for.)