The application of piecewise polynomials to problems of curve and surface approximation

The application of piecewise polynomials to problems of curve and surface approximation

Kubik, K.

Bosman, E.R.
Eckhart, D.

Rijkswaterstaat

KWP-collection

1971-05-01

For many years, long, thin strips of wood or some other material have been used by draftsmen to fair in a smooth curve between specified points. These strips or mechanical splines are kept in place by attaching lead weights called "ducks " at points along the strip. In order to represent similar smooth curves mathematically, the concept of piecewise polynomial functions was introduced. To this purpose, the individual curve sections of the mechanical spline between two adjacent ducks are replaced by different polynomials, which join smoothly at the locations of the ducks. The piecewise polynomial is then a smooth function consisting of different polynomials between each pair of adjacent junction points (ducks). Various conditions may now be imposed upon the shape ofthe piecewise polynomial function. Possible conditions are, that either the slope or the curvature of the curve is in the average as small as possible. The idea of piecewise polynomial functions can also be extended to functions of two or more variables. For two variables the piecewise polynomial function, which may be visualized as a surface, consists of parts of polynomial surfaces, which join continuously with the adjacent surface sections. Piecewise polynomial functions are found to have highly desirable characteristics as approximating, interpolating and curve fitting functions.

polynomials
curve fitting

http://resolver.tudelft.nl/uuid:4499749a-b628-4f93-9cc5-0c1b7af4119b

Rijkswaterstaat

Rijkswaterstaat communications 12

Hydraulic Engineering Reports

report

© Authors