A Fast, Accurate Algorithm for the Isometric Mapping of a Developable Surface

Publisher Website
Google Scholar
Abstract
This work is concerned with the derivation of a fast, accurate algorithm for the isometric mapping of a developable surface onto the plane .$mathcal{M}:overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {r} o overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {R} $. The algorithm is based on the relationship between the ruling lines $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {r} $ generating the developable surface $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {S} (s,t)$ and one additional geodesic $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {g} (s)$ constructed within the surface as the solution of the geodesic curvature equation ${operatorname{det}}(overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {g} 'overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {g} ''overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {n} ) = 0$, where $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {n} $ is the unit normal to the surface $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {S} $ at $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {g} $ and $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {R} $ is the image of $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {r} $ in the plane. Since $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {g} $ as well as the ruling lines reduce to straight lines in the plane, the isometric mapping procedure is defined in terms of the ruling line lengths, the arclength along $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {g} $ and the angles of intersection of $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {r} $ and $overset{lower0.5emhbox{$smash{scriptscriptstyle ightharpoonup}$}} {g} $.

This publication has 1 reference indexed in Scilit: