Introduction
Elastic rulers that were bent to pass through a number of predefined points (the "knots") were used for making technical drawings for shipbuilding and construction by hand, as illustrated by Figure 1.
The approach to mathematically model the shape of such elastic rulers fixed by n+1 "knots" is to interpolate between all the pairs of "knots" and with polynomials .
The curvature of a curve
is
As the elastic ruler will take a shape that minimizes the bending under the constraint of passing through all "knots" both and will be continuous everywhere, also at the "knots". To achieve this one must have that
and that
for all i, . This can only be achieved if polynomials of degree 3 or higher are used. The classical approach is to use polynomials of degree 3, this is the case of "Cubic splines".
Read more about this topic: Spline Interpolation
Famous quotes containing the word introduction:
“Do you suppose I could buy back my introduction to you?”
—S.J. Perelman, U.S. screenwriter, Arthur Sheekman, Will Johnstone, and Norman Z. McLeod. Groucho Marx, Monkey Business, a wisecrack made to his fellow stowaway Chico Marx (1931)
“We used chamber-pots a good deal.... My mother ... loved to repeat: “When did the queen reign over China?” This whimsical and harmless scatological pun was my first introduction to the wonderful world of verbal transformations, and also a first perception that a joke need not be funny to give pleasure.”
—Angela Carter (1940–1992)
“For better or worse, stepparenting is self-conscious parenting. You’re damned if you do, and damned if you don’t.”
—Anonymous Parent. Making It as a Stepparent, by Claire Berman, introduction (1980, repr. 1986)