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...

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!

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.

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.

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.