The Geometry of Bending

Image trouble

2011-12-07T22:41:00.004+01:00

The images that were hosted on Picasa seem to have disappeared unfortunately. I'll try to re-upload them. Videos from Vimeo are still working.

Experiments by Joe Magri

2011-11-17T10:17:00.006+01:00

Joe Magri is making some really nice plywood models to investigate bending geometry. In his own words: "Through a series of analogue experiments I have been applying several patterns to plywood to explore the deformation of the material to create a potential sculpture or pavilion. The cuts are to allow for bending and openings to occur."

Hinge Force in Kangaroo

2011-08-22T22:47:00.011+02:00

The new Hinge Force in Kangaroo is perfect for simulating bent developable surfaces!

In these examples, Kangaroo wants to flatten the meshes by trying to set the angle between each neighbouring triangles to zero. Springs make sure the surfaces don't deform. Thanks to Daniel Piker for this great force! More reading about the Hinge concept here.

Exhibition opened!

2011-04-14T23:51:00.013+02:00

Opening of the exhibition 'function', showing works by Ulrika Karlsson & Marcelyn Gow, Pablo Miranda Carranza & Åsmund Gamlesæter and Mårten Nettelbladt (me). Exhibition hosted, curated and designed by Fritz Halvorsen.

From the catalog:

Searching for the obvious
Mårten Nettelbladt

When you bend a thin strip of plywood you get a beautifully shaped curve. What geometry does this curve follow? There is a peaceful simplicity to the shape, and yet, it doesn't fall into the normal categories of basic geometric shapes as we know them. The exhibition shows two different ways to approach this challenge. Part one: A plywood strip, twelve meters long, curled and twisted into a double loop shape. This geometry is a result of the material trying to resist, and thereby minimize, the forces of bending and torsion. Part two: A computer generated surface, curling and twisting according to user input. Two lists of values control the curvature and the direction of the surface. The resulting single-curved surface will always be developable and unroll to a straight strip. Question: Is there a simple mathematical solution that will produce the same geometry as in the plywood loop? The search continues.

Special thanks toDavid Rutten, McNeelAndy Payne & Jason K. Johnson, Firefly ExperimentsGrasshopper Forum

Download Rhino + Grasshopper files:

The firefly stuff has been omitted in this version since not everyone have their hardware sliders...

Exhibition 14th April - 19th May 2011, Stockholm

2011-04-08T00:08:00.006+02:00

Welcome to the opening of the exhibition 'function' at FFAR forum for architecture 6.00-10.00 p.m. April 14, 2011. The exhibition presents works by servo, Omkrets arkitektur and gran on the theme mathematics and architecture. I'll be showing two models (one plywood + one grasshopper).

R.S.P.V to info@ffar.se before April 10, 2011

Information on upcoming events on www.ffar.se and Facebook

Ringvägen 141, Stockholm, Sweden.

Thanks to Fritz Halvorsen from FFAR for this opportunity!

Work by Joel Letkemann

2011-04-07T22:50:00.008+02:00

Joel Letkemann made some very nice and thorough studies of plywood bending for his project "New Prosthesis: Bent Wood Exoskeletons". He also studied various methods for assembly and developed some very interesting structural concepts.

Images: Joel Letkemann

smartgeometry 2011 Copenhagen

2011-04-02T23:51:00.004+02:00

I made a brief visit to the 2011 smartgeometry workshop in Copenhagen. Fun!

Thanks to everyone who took their time to show me their work.

Kangaroo Curvature Graphs

2011-03-24T23:10:00.002+01:00

I used Kangaroo to generate these bending curves and plotted their Curvature Graphs above.

Comparing 5 curves

2011-03-12T00:32:00.014+01:00

This diagram shows five different versions of a bending curve.

1. Elastica curve, curvature varies with Sine (0-180°)

2. Clothoid curve, linear increase / linear decrease of curvature

3. The original traced saw blade (it was difficult to meassure the curvature of the scanned line)

4. Curve from Daniel Pikers Kangaroo (see previous post)

5. A curve with curvature made up of two different Sine curves.

The last curve is based on a diagram by Maarten Kuijvenhoven, see image below from his thesis. The curve is a combination of two sine waves, the first one with aplitude 1 (0-180°) and the other one with amplitude -0.1 (0-540°). The amplitude relation was something I had to experiment with, but the one used above seemed to work quite well. In the illustration below, both sine waves have amplitude 1.

Image: Maarten Kuijvenhoven

Kangaroo Curvature Analysis

2011-03-10T12:38:00.000+01:00

Work by Maarten Kuijvenhoven and Matthijs Toussaint

2011-03-09T23:53:00.009+01:00

I received an interesting email from Maarten Kuijvenhoven, structural engineer at DHV in The Netherlands. He studied bending geometry in his thesis work at TU Delft (February 2009) http://www.mechanics.citg.tudelft.nl/~pierre/MSc_projects/reportKuijvenhoven.pdf

"About three years ago I wrote my thesis at TU Delft about timber grid shells and also tried to answer what geometry an elastically bent beam will have. The problem was that standard engineering formulas for deformation of beams exist, but are only valid as long as deformations remain small. Therefore I had to work it out in a more elaborate way using the concept of minimal potential energy."

Image: Maarten Kuijvenhoven

One of Maartens colleagues, Matthijs Toussaint, wrote his thesis on Timber grid shells as well (May 2007): http://www.mechanics.citg.tudelft.nl/~pierre/MSc_projects/reportToussaint.pdf including some very nice tests with physical models:

Image: Matthijs Toussaint

Daniel Pikers KANGAROO

2011-02-27T22:17:00.009+01:00

(As reported earlier in this previous post) Daniel Piker is developing ways to accurately simulate physical behavior in his Grasshopper component "Kangaroo".

In the latest release there are tools for simulating bending geometry.

This is nothing but a revolution for this investigation!

Never before have I been able to recreate bending geometry so accurately in an "artificial way". At the moment Kangaroo works very accurately for 2d-bending with both fixed and hinged ends and also in 3d for rods.

3d-bending of developable surfaces is a little more tricky and requires a careful setup to make sure the surfaces stay developable.

Kangaroo works in an iterative way by letting some predefined forces (like springs, bending resistance, pressure and gravity) affect the geometry, step by step, until (usually) a stable solution is reached. The beautiful thing is that all this is done in real-time, so you can play around with different constraints and setting and see the result instantly.
Of course, this investigation is not only about mimicking bending geometry, but also about understanding it. Kangaroo brings my understanding to a new level!

For more reading on how Kangaroo deals with bending, Daniel Piker has recommended a paper written in 1998 by S.M.L. Adriaenssens and M.R. Barnes called Tensegrity spline beam and grid shell structures, published in Engineering Structures 23 (2001), pages 29–36.

Many thanks to Daniel Piker for making and sharing the Kangaroo!

Kangaroo Bending + Reactivision

2011-02-26T23:57:00.008+01:00

A video response to these photos by Amir Gazit. Thanks to Daniel Piker for this setup: Also thanks to Andy Payne and Jason K Johnson for including the Reactivision stuff in the latest FireFly. It's a lot of fun!

Realtime curvature analysis of a Kangaroo bending curve

2011-02-24T10:47:00.003+01:00

Some more testing with bending in Daniel Pikers Kangaroo.
Kangaroo is a component for Grasshopper (Rhino).

Kangaroo Physics "Drop shape"

2011-02-23T14:42:00.001+01:00

This is the closest approximation I have found so far in my quest for finding the Geometry of Bending. It's a fairly simple setup in Kangaroo. Pretty amazing how well it works! Thanks Daniel...

Bending simulation in Kangaroo

2011-02-23T00:20:00.002+01:00

A very simple test in Kangaroo (only 8 control points) turned out to be very realistic in the way it moves. View it at Vimeo.

Try the Tapeworm

2011-02-13T23:50:00.006+01:00

Some people have shown interest in my Tapeworm script, so I'm posting it here for anyone to try.

Download Tapeworm v002

There are some notations inside the VB script explaining what is going on.

Basically, the script deals with the surface as if it were a long series of connected planar quads and then outputs the vertices of those flat surfaces as two lists of points. These points can be made into either polylines or interpolated curves, resulting in either a faceted or a smooth surface..

The two most important inputs for the script are "bend" and "twist". They should be lists of values that determine the curvature for each segment and the direction of this curvature. Please note that both lists should be equal in length. Also, the more values (steps), the more accurate the surface will be. The resulting surface should become single-curved (developable) and unroll to a straight strip.

Some photos and videos.

Good luck! Feedback is welcome.

Raw grasshoppers should be eaten with caution, as they may contain tapeworms. http://en.wikipedia.org/wiki/Grasshopper

Tapeworm script with sliders

2011-01-15T22:41:00.008+01:00

MD-slider controlling Bending on Y-axes and twisting (left/right) on X-axes. Surface is Baked to Rhino and then unrolled to become flat and straight. Other versions here:

Tapeworm script

2010-12-18T22:23:00.007+01:00

"Tapeworm" is the working title for a little script I'm developing. It will produce a long thin developable surface that unrolls to a straight strip. The curvature (bending + twisting) is controlled by two lists of values.

Developable curved strips in Grasshopper

2010-12-12T00:00:00.010+01:00

A script in Grasshopper that will produce rather complex surfaces that are perfectly developable and they will also unroll to straight strips. Thanks to Graph Mappers (one for bending and one for twisting) the shape is relatively easy and intuitive to adjust.

(The surface in the top left corner of the image is quite similar to this one.)

I'm very pleased with these results!

Scripted 3d Bending

2010-12-10T00:27:00.008+01:00

Fueled by a question from Jesper Thøger Christensen, I've continued to work on scripted 3d bending, something I started a while ago. At the moment it will just produce a helix (constant bend + constant twist), but I'm hoping to get some more elasticity in there.

Elastica curves video

2010-12-09T08:05:00.000+01:00

Download .ghx.

Guitar string + Grasshopper scripting

2010-11-09T23:59:00.005+01:00

Trying to mimic the loop shape of a curved guitar string. The scripted curve has linear increasing/decreasing curvature (like a Clothoid or Cornu Spiral). Pretty good fit?! (Gray line is a shadow)

(Some people may recognize the photo from this post)

Bending curves inside two circles

2010-10-01T23:29:00.010+02:00

This is a strange relationship that I discovered already in these tests. Aligned at the curve centre points, the endpoints of bending curves lay on two circles! Circle diameter = 4/5 of curve length.

UPDATE! 31 oct 2010: The link was wrong.Download corrrect .ghx file

Download .ghx-file (wrong)

Spline definition

2010-08-30T20:45:00.004+02:00

This is something I found a while back:

SPLINE: “A curve that closely approximates the shape of a strip of material that is gently bent; originally a draftsman's tool for drawing curves that represented the shapes taken by wooden and metal members of a ship's hull structure bent over fixed points or frames and, later, representing similar shapes in auto bodies and aircraft structures. A spline is the shape taken by bending material objects, like beams, that minimizes the elastic energy (or internal strain energy) stored in the beam. Mathematically, it is the smoothest curve that passes through a set of fixed points. In 3D modeling it is a curve defined by control points, often supplemented by interactive methods to modify tangents to the curve at these points and to adjust a local weighting factor. Bézier, B-Spline, and NURBS are commonly used types of splines.”

(unknown source)

Analyzing the Drop shape

2010-06-17T00:08:00.004+02:00

The characteristic Drop shape [that appears when the ends of the saw blade touch each other] turned out to look rather different in the elastica version and the cornu spiral version. The latter is closer to the saw blade. With the help of grasshopper I plotted the curvature of the three curves above. The curvature of the elastica curve and the cornu spiral curve look as expected, but the curvature of the saw blade is a bit ambiguous and noisy. I probably need to redo the measurements.

Cornu Spiral curves scripted in Grasshopper

2010-06-14T00:59:00.005+02:00

I also tried scripting the Cornu Spiral curves (described earlier) in Grasshopper VB:

When superimposed with the saw blade curves it shows a much better fit, even though the overlap is not 100%:

(I remember that the last drop shape was a bit hard to orient correctly when drawing it, because it had both supports in one point.)

Please also note that my scripting skills are very basic, so I may very well have made errors when producing the elastica curves and cornu spiral curves.

Elastica curves scripted in Grasshopper

2010-06-13T23:18:00.013+02:00

I tried some Grasshopper scripting to produce elastica curves.

(The script is really basic, it uses Loop to copy and rotate a line a bunch of times. The Sin values between 0 and 1 Radians are used to vary the amount of rotation. The get many curves at the same time a range of values are fed into the VB script component.)

Below are the curves that I drew manually along the saw blade:

To my surprise it's quite a bad fit as can be seen in the image below, showing both sets of curves superimposed:

The typical "drop shape" (that appears when the ends of the curve meet) seems too wide and low. Also curves seem to go in the wrong direction (see at left arrow).

Kangaroo Physics by Daniel Piker

2010-05-31T21:12:00.009+02:00

The Kangaroo by Daniel Piker brings live 3D Physics into Rhino/Grasshopper (see some videos explaining what Kangaroo can do here). Kangaroo is installed like a subcomponent in Grasshopper for Rhino (a lot of animals...) Daniel kindly let me try an unofficial version which can simulate bending geometry (!) Below are images from Kangaroo in action and verification with a guitar string. Thanks Daniel!

Developable fork

2009-11-19T20:08:00.006+01:00

A developable fork is way to connect three developable surfaces with each other. They are joined by a flat triangle, tangent to the three surfaces (and tangent to the three edge curves). The developable fork is very useful when creating volumes from developable surfaces.

Some more images and info here.

Geodesics on a Cone

2009-09-29T23:33:00.009+02:00

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)

From Spiral to Spline

2009-08-29T23:26:00.007+02:00

Raph Levien will defend his thesis called From Spiral to Spline: Optimal Techniques in Interactive Curve Design on the 3rd of September 2009 at the University of California, Berkeley. It describes techniques for interpolating splines, something that can be useful in font design. The thesis also includes comprehensive sections on the history of splines, elastica curves and clothoids (Euler's spiral). Very interesting stuff! Good luck on Wednesday Raph!

Thanks to Ola Jaensson for finding and sharing this!

Siegfried Gass, Frei Otto & Wolfgang Weidlich

2009-06-12T22:49:00.014+02:00

Architect Lorenz Lachauer (who is currenty a research assistent at the ETH Zürich) sent me this scan from a book he found in the library. The book is called "Experimente: Physikalische Analogmodelle Im Architektonischen Entwerfen" by Siegfried Gass, Frei Otto and Wolfgang Weidlich, published by the Institut für leichte Flächentragwerke, Universität Stuttgart in 1990. He says it's a really amazing book and I believe him! Thanks for letting us know, Lorenz!

Update 2009-10-01:
Moritz Fleischmann at the ICD (Universität Stuttgart) kindly told me this book (also labeled IL25, ISBN: 978-3-7828-2025-7) is still available from the Verein zur Förderung des Leichtbaus e.V. (website) in Stuttgart and you can order it here. Thanks Moritz!

Email from Oliver David Krieg

2009-06-06T20:42:00.014+02:00

I was contacted by Oliver David Krieg, from the University of Stuttgart, Faculty for Architecture and Urban Planning. Oliver is one of the students in this years design studio called "Material Systems I: Reciprocities" tutored by Moritz Fleischmann and Christopher Robeller at the Institute for Computational Design (ICD) led by prof. Achim Menges.

They are investigating the properties of wood very thoroughly and of course bending geometry is one of them.

Rhino and scripting are among their tools and the goal is to develop a material system with interconnected elements that can react (passively) to surrounding influences.

It will be very interesting to see how this will develop. Thanks for getting in touch Oliver!

(Photos by Oliver David Krieg)

SurfaceTurtle and CurvatureTurtle by Lorenz Lachauer

2009-05-18T00:16:00.006+02:00

Lorenz Lachauer has a great blog called eat-a-bug. It has some very interesting topics. I used his Grasshopper file SurfaceTurtle and Rhino plugin TwistedBeams to generate these plank lines from an ellipsoid:

(The SurfaceTurtle works just like the ToyCar plugin by David Rutten).

I also used his Grasshopper file CurvatureTurtle to generate some elastica curves:

Thanks a lot Lorenz!

3 useful cones

2009-04-16T22:44:00.004+02:00

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.

Self intersecting developable surfaces

2009-04-09T23:45:00.009+02:00

Some sketch models:

The idea is to create one continous, developable, self intersecting surface that can create a useful space (a room) and perhaps extend to become a whole building. The surface is allowed to overlap itself when unrolled.

Explaining the five cases of elastic bending

2009-03-30T21:14:00.009+02:00

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

2009-03-30T20:22:00.014+02:00

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

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

"Conic plank line"

2009-03-30T00:24:00.003+02:00

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?

2009-03-09T23:36:00.004+01:00

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

2009-03-08T02:16:00.003+01:00

This time it's single curved...

Download GHX-file.

More cone testing

2009-03-08T01:14:00.006+01:00

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).

Comment: Volker Mueller on Plank Lines

2009-03-06T22:38:00.009+01:00

A comment by Volker Mueller on the post "Wrapping cones" was so interesting that I'm making it a post of it's own. Thank you very much Volker!

Hello Marten,

A curve that might behave the way you explore with strips of paper, rubber, or metal could be what is sometimes called a plank line (GC.BSplineCurve.PlankLine).

The idea of a plank line is that it bends in one direction, twists in the second direction, and is stiff in the third, like a thin but relatively broad board of wood.

Julius Natterer did structures like that (http://ibois.epfl.ch/page12022.html; I believe also the Polydome at the Ecole Polytechnique Federale Lausanne) and Judit Leuppi did some research about plank lines published as a paper at the ACADIA 2000 Conference (Plank Lines of Ribbed Timber Shell Structures; available from http://cumincad.scix.net/data/works/att/f197.content.pdf).

Another example of such a gridshell seems to be at the Weald and Down Open Air Museum in Singleton, Sussex (images at http://www.flickr.com/search/?w=all&q=downland+gridshell&m=text).

Regards,

Volker Mueller

ToyCar plug-in for Rhino by David Rutten

2009-03-06T00:23:00.005+01:00

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

2009-03-05T22:35:00.003+01:00

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

2009-03-02T00:41:00.011+01:00

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?

Twister

2009-02-09T23:32:00.004+01:00

I discovered it was possible to replace a number with a list of numbers as an input to a VB.Net-script i Grasshopper and produced this rather weird model:

(Rendered in Maxwell)

Rubber band

2008-12-27T13:29:00.002+01:00

I found this rubber band on the table looking just like this. The shape of the bending and twisting is similar to that of the saw blade. (Christmas tree in the background...)

Analysis of helix angle in loop surface

2008-12-22T01:19:00.004+01:00

Grasshopper helped me plot the difference in angles between curve direction and curvature direction along the loop surface. Seems to start around 30° and reach 0° at surface mid point. It flips over to 180° and then decreases another 30° to 150° during the second half. Interesting! This would indicate a linear decrease/increase of the twist amount (measured as the helix angle). I think this pretty much solved the puzzle...

More helix testing

2008-12-22T00:36:00.005+01:00

Some more helix testing in Grasshopper (no scripting yet as you may notice from the messy layout). The question is now: How to make a smooth transition between helixes with different angle?
EDIT: Download ghx-file.

Developable helix surface

2008-12-21T00:01:00.003+01:00

The measurement for bending is obviously curvature, but how to measure the amount of twisting? The image above shows how the Curve Direction differs from the Curvature Direction in a developable helix surface. Perhaps this "Helix Angle" can be a useful measurent?

More elastica curves...

2008-12-03T22:44:00.005+01:00

Dr. Christopher D. Rahn at the Pennsylvania State University has developed a mine-hunting vehicle with whiskers (!) that uses the "elastica equations" to identify objects. Read more here.
Below is an image of experimentally predicted whisker shapes:

Elastica curves!

2008-12-03T00:28:00.011+01:00

I went back to read "The Curve of Least Energy" by B.K.P. Horn more carefully. He writes: "Unfortunately, the Cornu spiral is not optimal either...".
In this web page, prof. Albert K. Harris explains the Elastica curve: "If you compress a long thin metal rod, when it eventually kinks its shape will approximate one of the elastica. This is said to optimize the spatial distribution of bending stress..."
And finally, in "Non-linear Beam Analysis" Japaneese aircraft structures engineer Toshimi Taki uses Elastica curves to generate something very interesting:

This is the first time I've seen anything similar to my own diagram of bending curves. I'm really excited!

Maybe the Cornu Spiral isn't the answer to 2d-bending after all?

Bending Structure

2008-11-29T23:04:00.008+01:00

I used mirrored segments of the cornu spiral to create these arches.
You can download the grasshopper file here:
http://omkrets.se/grasshopper/MN-002-bending-structure.zip

This script generates the cornu spiral.

Cornu spiral in Grasshopper

2008-11-06T00:01:00.007+01:00

A test with construting a cornu spiral in Grasshopper.
Script written in VB.NET. Download .ghx-file below:

http://omkrets.se/grasshopper/MN-001-cornu-spiral.zip

Grasshopper

2008-11-03T23:10:00.005+01:00

I have used the Grasshopper plug-in for Rhino developed by David Rutten to do some more testing with creating the Cornu Spiral. First manually...

And later by using the Script function.

Generative components

2008-11-03T23:01:00.005+01:00

With GenerativeComponents (GC) I tried to mimic bending in 3d.

This shows an attempt to generate cornu spirals in GC. Not quite right - but beautiful!

Here I used scripting in GC to get a twisting strip.

Bending in 3D

2008-11-03T22:55:00.006+01:00

So far, most examples have been 2d-curves, but what happens when you bend in 3 dimensions.

I made a 3d loop with the saw blade and scanned it with a Microscribe 3d digitizer. Download 3dm-file here.

Cornu spirals in MicroStation

2008-11-03T22:34:00.003+01:00

The CAD-program MicroStation has a command for clothoids (cornu spirals) that I used to draw this figure.

When I combined these curves with the ones I had traced by hand, the match was very good!

The traced curves from the saw blade were double (mirrored) clothoid segments.

Finding the Cornu Spiral

2008-11-03T22:28:00.004+01:00

I involved a friend (Ola J.) who helped me searching. He found an article by B.K.P. Horn that was called “The Curve of Least Energy”, which contained some pages about a spiral called the “Cornu Spiral” or “Clothoid”.

This spiral is very useful when designing roads, rails and roller-coasters. Segments of the spiral are used as transitions between for example straight and curved parts and give the smoothest possible changes in curvature.

Arranging curves

2008-11-03T22:09:00.006+01:00

I digitized these curves and used a CAD-program to arrange them in different ways in order to find some kind of clue to the geometry. The curves could roughly be represented as NURBS-curves with two control points in the application Rhinoceros (Rhino).

Arranged in this way, the endpoints seemed to lie along two circles with the diameter 4/5 of the curves length. The displacement between the circles was 1/4 of the diameter.

I measured angles and distances in order to understand, but could not see any connections.

Tracing a saw blade

2008-11-03T22:04:00.004+01:00

A saw blade, 90 centimetres (3 feet) long, was traced in different positions.

The result was 35 curves with equal length but different distance between the endpoints.

First steps

2008-11-03T21:56:00.007+01:00

In order to answer these questions I started tracing strips of plywood. If the endpoints touched each other, there was almost a 90 degree angle and a drop-like shape.

Here, a strip of paper was bent and traced several times. The anchor points are fixed, but the curve length vary.

I scanned the curves and investigated them in a CAD-program. They were obviously not arcs.

Trying to understand the geometry of bending

2008-11-03T21:50:00.006+01:00

When you bend a thin strip of an elastic material you get a beautifully shaped curve. What geometry does this curve follow? Can the curve be calculated if you know the length of the material and the position of the end points? Is it possible to calculate more complex situations with several forces in different directions? Can you make similar calculations in 3d? Can this geometry be useful in design/production?