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

1987; Society for Industrial and Applied Mathematics; Volume: 18; Issue: 4 Linguagem: Inglês

10.1137/0518074

1095-7154

John Clements, L.J. Leon,

Advanced Numerical Analysis Techniques

This work is concerned with the derivation of a fast, accurate algorithm for the isometric mapping of a developable surface onto the plane .$\mathcal{M}:\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} \to \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {R} $. The algorithm is based on the relationship between the ruling lines $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} $ generating the developable surface $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {S} (s,t)$ and one additional geodesic $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {g} (s)$ constructed within the surface as the solution of the geodesic curvature equation ${\operatorname{det}}(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {g} '\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {g} ''\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {n} ) = 0$, where $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {n} $ is the unit normal to the surface $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {S} $ at $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {g} $ and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {R} $ is the image of $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} $ in the plane. Since $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {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 $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {g} $ and the angles of intersection of $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} $ and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {g} $.