3 useful cones, part 2

Related the post "3 useful cones" (part 1). When unrolled, these cones make up 1/4, 2/4 and 3/4 of a circle.

If you look closely, you can see all the lines/squares line up perfectly...

Geodesics on a Cone

Mark L. Irons did some thinking about Geodesics on a Cone.
Geodesics on a Cone is probaly the same thing as a Conic plank line.
He explains very clearly why there can be more than one geodesic line that connects two given points on a cone (or a sphere).
(Images: Mark L. Irons)

3 useful cones

These 3 cones unroll to 270, 180 and 90 degrees, which can be useful if you want to line up an orthogonal pattern. For example rectangular plywood sheets.

Explaining the five cases of elastic bending

At the moment, this is how I would explain the geometry of the 5 cases of elastic bending (see previous post):

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.

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.

Five cases of elastic bending

I have so far identified five cases of elastic bending in a thin strip:

Case #1
Two loose ends pushed together (2d).

Case #2

One clamped end, the loose end is pushed (2d).

 Case #3
Both ends clamped together to form a loop (2d).

Case #4Both ends clamped to form a loop, but with a distance sideways between endpoints (3d).

Case #5Both ends clamped and twisted to form a loop (3d).

"Conic plank line"

I would like to introduce the term "Conic plank line". (See these images).
The meaning being what I have earlier described as "Cone strip" or "Wrapping a cone with a straight strip".

Loop comparison. Investigation completed?

I hesitated for some time before I compared the new Grasshopper-generated Cone Strip with the original digitized saw blade surface. It took some fine tuning of the "Cone Angle" parameter and some scaling+rotating to find the corresponding shape.
After having compared the two surfaces closely I have found that they are almost identical!
Please download the 3d-model (rhino .3dm-file) and have a look for yourself.

Well, this pretty much concludes my investigation I think! You are welcome to prove me wrong!

Thanks for your interest in this blog!

Mårten

Single curved Cone Strip

This time it's single curved...

Download GHX-file.

More cone testing

Generating the cone plank line directly in Grasshopper (no ToyCar). The strips turn out slightly double curved. Why? Probably because the surface is a loft between lines that are perpendicular to the plank line curve (they shouldn't be, they should all point towards the tip of the cone and vary in length).

ToyCar plug-in for Rhino by David Rutten

David Rutten at Robert McNeel was kind enough to revive his plug-in named "ToyCar" for me. The ToyCar runs along a surface and finds a plank line path on it. PLEASE NOTE that the plug-in is not yet finished!


ToyCar + Grasshopper helped me create this:

Thanks for your help David!

Wrapping cones

Some more testing with wrapping cones.

This is what the unrolled strip should look like. The yellow lines represent a fixed change of direction.

Collar on the cone

Today I discovered that the loop I have been trying so hard to understand may in fact be a part of a cone!

The loop sits like a collar on the cone.

The loop photographed from the cone focal point.

This is a bit of a breakthrough! It's interesting that such a simple paper model can be so useful. In CAD, how would you constrain a straight strip to follow the shape of a (developable) cone?