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:

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

Kangaroo Curvature Analysis

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.

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

Download .ghx-file (wrong)

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!

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!

Siegfried Gass, Frei Otto & Wolfgang Weidlich

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

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)