**Case #1**This one follows the

*elastica curve*, which means the curvature varies with the sin of distance along the curve (explanation here). The curve equals half a cycle of Sin (180 degrees) which means the curvature will be zero at start point and endpoint.

[EDIT 2010-06-13] This case probably involves the Cornu spiral (clothoid), see here and here.

*(This could possibly be simply

**Case #2**This is probably* a part of a*clothoid*curve (*Cornu spiral*). Curvature is maximum at the clamped end and zero at the loose end. There is a linear change in curvature in between. (A loose end cannot store any bending energy and the curvature there must be zero).*(This could possibly be simply

*half an elastica curve*, but I find that less likely).**Case #3**This is a*circle*(cylinder). Curvature is constant along the curve.**Case #4**This is a*helix*. Curvature is constant along the curve and there is also a constant twist. This could also be called a*cylindrical plank line*, which means it has the shape of a thin (straight) strip that has been wrapped around a cylinder.**Case #5**This is a*conic plank line*, which means it has the shape of a thin (straight) strip that has been wrapped around a cone.