Abstract
Laguerre geometry provides a simple approach to the design of rational curves and surfaces with rational offsets. These so-called PH curves and PN surfaces can be constructed from arbitrary rational curves or surfaces with help of a geometric transformation which describes a change between two models of Laguerre geometry. Closely related to that is their optical interpretation as anticaustics of arbitrary rational curves/surfaces for parallel illumination. A theorem on rational parametrizations for envelopes of natural quadrics leads to algorithms for the computation of rational parametrizations of surfaces; those include canal surfaces with rational spine curve and rational radius function, offsets of rational ruled surfaces or quadrics, and surfaces generated by peripheral milling with a cylindrical or conical cutter. Laguerre geometry is also useful for the construction of PN surfaces with rational principal curvature lines. New families of such principal PN surfaces are determined.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 223-249 |
| Number of pages | 27 |
| Journal | Computer Aided Geometric Design |
| Volume | 15 |
| Issue number | 3 |
| DOIs | |
| State | Published - Mar 1998 |
| Externally published | Yes |
Keywords
- Geometrical optics
- Laguerre geometry
- NC milling
- Offset
- Principal curvature line
- Principal patch
- Rational curve
- Rational offset: Pythagorean-hodograph curve
- Rational surface
ASJC Scopus subject areas
- Aerospace Engineering
- Automotive Engineering
- Modeling and Simulation
- Computer Graphics and Computer-Aided Design