# “In the automobile industry, a discontinuous second derivative is known as a dent”

The above quote comes from a read I thoroughly enjoyed, a blog post by Cleve Moler at MATLAB Central on Splines and Pchips, wherein he discusses “MATLAB has two different functions for piecewise cubic interpolation, spline and pchip. Why are there two? How do they compare?”

He nods to Carl de Boor, who developed much of the theory and software needed to calculate splines while working for General Motors Research. According to Cleve,

GM was just starting to use numerically controlled machine tools. It is essential that automobile parts have smooth edges and surfaces. If the hood of a car, say, does not have continuously varying curvature, you can see wrinkles in the reflections in the show room. In the automobile industry, a discontinuous second derivative is known as a “dent”.

Some time spent on de Boor’s UW-Madison homepage led to more delights. Some highlights include:

-This also excellent quote, seen in Ronald DeVore and Amos Ron’s 2005 writeup on de Boor in SIAM News (Society for Industrial and Applied Mathematics), from Iso Schoenberg, “the father of splines”:

Polynomials are wonderful even after they are cut into pieces, but the cutting must be done with care. One way of doing the cutting leads to the so-called spline functions.

-This amazing photograph of a spline used by draftsmen (Google: “a person who makes detailed technical plans or drawings”) in the age before computers did work for us:

A draftman’s spline

I’ll also include a link to the wonderful photograph of a Boeing draftman using this, while respecting Dr. de Boor’s request to “view, don’t copy! I had to pay Boeing \$100 for the right to display it”:

http://pages.cs.wisc.edu/~deboor/draftspline.html

-A link to this Random Mathematical Quotations Generator, on Mark Woodard’s Furman University Mathematical Quotations Server. This excellent website dates to at least 1995, and clearly garnered some 90s internet accolades: “It was also picked as the ‘Geek Site of the Day’ on July 12, 1995. It was also picked by the Canadian Mathematical Society as the cool site of the week in March, 1996.” My quote was:

Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.

Weyl, Hermann (1885 – 1955), The American Mathematical Monthly, November, 1992 (the website’s attribution, not sure how these numbers add up)

I remain as always in awe of the mathematicians, engineers, and scientists before me who made it possible to type ‘interp1’ into my MATLAB command line and accomplish beautiful splines that took someone else many years and a lot of hard work to figure out and implement. Thanks also to Professors de Boor and Woodard for archiving their interests on the internet so I could spend some time exploring as well.