Research And Application Of Boundary Conditions Of Combined Developable Surface

A ruled surface formed by the motion trajectory of a family of straight lines continuouslychanging along a space curve, when every straight generatrix of a ruled surface had only onetangent plane, the ruled surface became a developable surface. Developable surface had manyimportant properties, and one of the most common used was that there was isometric mappingbetween a developable surface and a plane, that meant a developable surface can entirelyexpand into a plane without stretching, tearing or folding, and this feature made developablesurface have a wide range of applications in the field of engineering and other fields.Developable surface only had three forms: cylinders, cones and tangent surface to aspace curve, for the wide applications, the only three separate forms of developable surfaceshad been unable to meet the demands, so combined surface spliced by the three forms ofdevelopable surface had been wildly used instead of single form of developable surface tomeet the requirements of industrial production and other aspects. However, the combinedsurface spliced by several developable surfaces was not necessarily developable surfaces, thispaper mainly studied the boundary conditions in which combined surface still wasdevelopable surfaces.This paper contributed and corrected some geometric models of common developablesurfaces products, such as the traditional single-cone lapel shaper, the multi-cone (six cones)lapel shaper, the lapel shaper without back plane, Oloid body, Sphericon body, toothpasteshell, industrial hoppers and so on, and summarized several correction methods of geometricmodel, such as cutting and splicing simple developable surface, modifying the fitting way ofinternal control points of the surface, simplifying the internal relations, modifying the internalnetwork relations and so on.This paper gave two developable surface combined ways: the intersection splicing andtangent splicing, and then gave some theoretical discussion and proof to the boundaryconditions that combined surface was still developable surface, and finally got two generalboundary conditions: when two developable surfaces spliced in the intersection way, if thetwo unit normal vectors of the two developable surfaces were symmetrical by the unit mainnormal vector of the intersection line, the combined surface was till developable surface; andwhen two developable surfaces spliced in the tangent way, if the intersection line was thestraight generatrix lines, the combined surface was till developable surface.This paper gave theory certainly to the forecast that the transitional surface of variablemulti-cone lapel shaper was a tangent surface to a space curve using the boundary conditionabove, and then proposed some new applications of lapel shaper without back plane andsimplified lapel shaper using the own material characteristics of the packaging materials toturn and roll over, and then gave a brief introduction of two special kinds of developablesurface products: Oloid and Sphericon, finally, briefly introduced the applications ofdevelopable surface in other fields.