Wednesday, January 31, 2007

070131_Unroll Surfaces

Non-nested pattern (with obvious overlaps) of unrolled surfaces
("le champ" project display panels)

Rhinoscript "Tips'n trick" :
Call Rhino.Command (CStr("_UnrollSrf Explode=No Labels=No _Enter"), vbFalse)

In mathematics, a developable surface is a surface with zero Gaussian curvature. That is, it is surface that can be flattened onto a plane without distortion (i.e. stretching, compressing, tearing). Inversely, it is a surface that can be made by transforming a plane (i.e. folding, bending, rolling, cutting, and gluing).

The developable surfaces that can be realized in 3D space are:

cylinders and, more generally, the generalized cylinder: the cross-section can be any smooth curve
cones and, more generally, conical surfaces, away from the apex
(trivially:) planes, which can be viewed as a cylinder whose cross-section is a line
Spheres are not developable surfaces under any metric as they cannot be unrolled into a plane. The torus has a metric under which it is developable, but such a torus does not embed into 3D space. It can be realized in four dimensions.

Formally, in mathematics, a developable surface is a surface with zero Gaussian curvature. One consequence of this is that all developable surfaces embedded in 3D space are ruled surfaces (though hyperboloids are examples of ruled surfaces that are not developable). Because of this, many developable surfaces can be visualised as the surface formed by moving a straight line in space. For example, a cone is formed by keeping one end of a line fixed while moving the other end in a circle.

A surface on which the Gaussian curvature is everywhere positive. When is everywhere negative, a surface is called anticlastic. A point at which the Gaussian curvature is positive is called an elliptic point.

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