Calculating the curves of bending

These saw blade curves (seen in two previous posts: here and here) are arranged to have identical midpoint and when viewed like this, the endpoints of the curves all lie on two circles. The radius of each circle is 2/5 of the curve length, and the two circles are 1/5 of the curve length apart.
Based on this fact, I could use the Pythagorean theorem to formulate a mathematical relationship between the original length L, distance d between end points and curve height h:


New book!

I have transformed the blog into a book!
ISBN 978916373329-1. 48 pages. 176 x 250 mm

You can order it here.
The price is 119 SEK + packaging and postage


Book project

I'm planning to publish a booklet with selected investigations from this blog


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...


Experiments by Joe Magri

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

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!

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

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

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

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

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


Comparing 5 curves

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


Work by Maarten Kuijvenhoven and Matthijs Toussaint

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://homepage.tudelft.nl/p3r3s/IASSpaperKuivenhovenHoogenboom.pdf (reworked into a paper)
"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):
including some very nice tests with physical models:

Image: Matthijs Toussaint


Daniel Pikers KANGAROO

(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

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

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


Kangaroo Physics "Drop shape"

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

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

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

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

"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

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

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.



Guitar string + Grasshopper scripting

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

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


Spline definition

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

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

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

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. To 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

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

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

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

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!