What is "twist" in a developable surfaces program?

[Ian]

I use ProBasic, a sub-set of ProSurf. I thought I knew what twist was. I did developable surfaces at design school. But the way the term is used in this program baffles me. To me, a conic section is inherently twisted, and unless a surface is "compounded" or curved in every direction it should be developable. This program says that twisted surfaces are not actually developable but allows you to set 'acceptable' parameters for twist that may be buildable. But they seem also to say conic surfaces are developable.

Somebody help me! What the hell are they talking about? 'Compounded--I understand. 'Tortured'--I understand. 'Twisted'--I thought I understood......?

 

[cestes]

The actual definition of a developable surface is one where any number of rulings across the surface can be found such that, when a plane is placed on the surface at each of the rulings, it is tangent to the surface.

Additionally, no two rulings can cross.

It can readily be seen that if a surface is twisted, sort of like a potato chip, you can't place the plane anywhere along the surface and have it come tangent to the surface.

Conversely, if you look at a cylinder as an example of a conic surface, it will be easily recognized that a surface can be placed anywhere along one edge of the surface and a point can be found along the other edge where the plane will be tangent. Nowhere on the surface will there be a ruling that crosses any other ruling.

The other conics have the same property, making them all developable.

'zat help?

 

[Ian]

It can readily be seen that if a surface is twisted, sort of like a potato chip, you can't place the plane anywhere along the surface and have it come tangent to the surface.[unquote]

This is where I get lost. My dictionary has fifty-one definitions of 'twist' but the two definitions that concern alteration of a plane surface are essentialy the same-- 'To alter in shape, as by turning the ends in opposite directions, so that parts previously in the same straight line and plane are located in a spiral curve'. To me, this is also how you turn a plane into a cone. I see these shapes as having rulling lines that are tangent to a plane.
A potato chip is not twisted, as it is a plane that is bent on two different axes that are ninety degrees to each other and is a compounded surface that has had to undergo stretching to reach that shape. Is this yet another example of the sciences high-jacking a english word and changing its meaning? Or am I really missing something?

 

[Ted Andresen]

Developable surfaces and curvature

As I understand it, a developable surface is one that can be obtained from another surface without changing the intrinsic curvature of that surface.

For example, cylinders and cones can be developed from a flat surface. All three surfaces have zero intrinsic curvature. Geodesics (sail seam centers) on all three surfaces are the same. The broadseams to build surfaces with zero curvature are parallel as the approach the edge of the surface.

If by a potato chip, you mean a saddle, you are writing about a surface that has negative intrinsic curvature. A spiral, like a twisted headsail (stack for horizontal sail chords at successively higher angles of attack) has negative curvature. The broadseams to build surfaces with negative curvature are get wider as the approach the edge of the surface

A sphere and ellipsoid have positive curvature. These surfaces cannot be developed from a surface like a plane with zero curvature. The broadseams to build surfaces with positive curvature are get narrower as the approach the edge of the surface

Hope that helps,

Ted Andresen

 

[Andrew Mason]

A developable surface is ususally generated between two lines or curves. If there are points on the generating curves that touched by a common tangent plane a surface ruling can be created between those points. This process is repeated until you have rulings covering most of the generating curve's lengths.

If the two curves are straight but not parallel, or if the two generating curves do not have any points on them that share a common tangent plane, no developable surface can be created. It is the generating curves of the developable surface that have been "twisted".

If simple rulings are generated between the twisted generating curves the resulting surface will typically have negative Gaussian curvature and will not be developable.

 

[Ted Andresen]

Okay, a developable surface is one with zero intrinsic Gaussian curvature. It's a flat surface. Cylinders and cones have zero curvature.

A headsail made from a stack of twisted horizontal airfoils has negative curvature.

Initially parallel geodesics on a surface with negative curvature diverge, so the panels that are used to make a headsail should be narrower at the luff than at the leech. That argues for luff broadseaming in headsails.

Ted Andresen