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:
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.
Searching for the obviousMårten NettelbladtWhen 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
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
including some very nice tests with physical models:
Of course, this investigation is not only about mimicking bending geometry, but also about understanding it. Kangaroo brings my understanding to a new level!
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!
Raw grasshoppers should be eaten with caution, as they may contain tapeworms. http://en.wikipedia.org/wiki/Grasshopper
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.”
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.
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)
Thanks to Ola Jaensson for finding and sharing this!