Offset curves are important in some aspects of computeraided design. Several authors have studied, and found, approximations to offset curves of cubic B-splines. Curves made of segments of clothoid spirals, arcs of circles, and straight line segments are used as centre lines of railways and highways, and offsets of these curves are used for rail or shoulder lines. This note points out that offsets of these curves are not much more complicated than the curves themselves, and can be rendered efficiently on a plotting device.
computer-aided design, curves, offsets, clothoids
The offset curve of a given curve is the locus of points which are affixed a perpendicular distance away from the curve. Offsets are important in some aspects of computer-aided design such as highway and railway route location. Cubic B-splines are a standard tool for drawing smooth curves, but unfortunately their offsets are not cubic B-splines, and are not even polynomials. Several authors have studied, and found approximations to offset curves of cubic B-splines 1-3. A recent paper 4 gives an approximation to the offsets of piecewise conics. Clothoidal splines, or curves made of segments of clothoid spirals, arcs of circles, and straight line segments in such a way that the tangent is continously varying and the curvature is continuous, are the curves commonly used as centre lines of railways and highways s. Methods of finding and editing clothoidal splines on a microcomputer have been recently developed 6'7. Offsets of these curves are important as the offsets are used for rail or shoulder lines. This note points out that the offset curves of clothoidal splines are not much more complicated than clothoidal splines, and gives some useful properties of the offsets.