User:Imlikewhoa/sandbox/Mowitz crease

Mowitz crease A type of curved crease in an infinite, unstretchable sheet which has a finite length and does not extend to the boundary.

needs picture a.png from Mowitz file, with caption. Photo of a Mowitz crease in a paper sheet to be included. [wiki refused my a.png picture, as not being a photo that I took.] The Mowitz crease is a mathematical idealization intended to explain the sharply-curved, localized regions that appear in real crumpled sheets like Fig 2.

The two idealizations used in defining the Mowitz crease are the same as those used by Dias and Santangelo [ref] to define general curved creases. First the sheet is a deformation of a flat two-dimensional sheet formed without material strain. That is, the distance between any two points on the sheet, as measured through the material is unchanged by the deformation. Such deformations are thus termed isometric"[ref]. Second, the crease is defined as a line in the deformed sheet at which the otherwise smoothly varying surface normal is discontinuous. That is, the normal vector points in different directions on the two sides of the crease line.  Knowing the magnitude of the discontinuity, the shape of the crease line, and its curvature in the deformed sheet dictates the shape of the sheet extending from the crease.  In particular, it determines the direction of the generator lines'' extending from each point of the crease into the sheet. A non-flat, isometric sheet necessarily has a single direction with no curvature.  This direction extends continuously from the crease to the boundary for any smooth isometric deformation, forming a straight line through each point in the material.  These lines are the generators of the sheet. The generator lines determine the shape of the sheet. Dias and Santangelo [ref] determined how the curvature of the crease in the material and in space combine with the opening angle of the crease to dictate the directions of the generators extending on both sides of the crease. When the crease line is indefinitely shorter than the size of the sheet, the resulting surface shape is further constrained, compared to the general crease above. In general, the generators of the entire surface terminate on the crease. As one approaches either end of the crease line, the two sheets extending from either side of the crease must join smoothly together. Thus the generator lines must become tangent to the crease at its end points. This condition of tangency forces the crease to be nonplanar in a specified. The shape is further constrained by the condition that the generators on both sides of the crease must splay outward. This implies for example that the crease line in the material must change direction by no more than 180 degrees [verify].

[to do. define outer categories, isometric deformations, thin sheet deformations, defected thin sheets.] Lots of references.