Digital Intuition Cluster, Smart Geometry 2013, Stylianos Dritsas, Mirco Becker, David Kosdruy, Juan Subercaseaux Search Basics – Templates Tutorial Overview 1. License Information 2. Rhino Integration 3. Generative Components Integration 4. Grasshopper Integration

Digital Intuition Cluster, Smart Geometry 2013, Stylianos Dritsas, Mirco Becker, David Kosdruy, Juan Subercaseaux License Information Digital Intuition and Prediction Workshop Code Base & Notes The MIT License (MIT) Copyright (c) 2013 Stylianos Dritsas Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Digital Intuition Cluster, Smart Geometry 2013, Stylianos Dritsas, Mirco Becker, David Kosdruy, Juan Subercaseaux Search Basics – Overview Work Flow 1. Almost Brute Force Search - Incrementally looking for a solution 2. Bisection / Divide and Conquer Strategy - Looking in between boundary conditions 3. Newton / Raphson Method - Slopes and Gradients 4. Evolutionary Methods - In another episode

Digital Intuition Cluster, Smart Geometry 2013, Stylianos Dritsas, Mirco Becker, David Kosdruy, Juan Subercaseaux Search Basics – Tutorial Example Segment a smooth curve into a number of chords. A fairly trivial example with many applications (see cladding setting-out). -Smooth curves (NURBS/BSplines) can be parameterized by arc-length, thus getting points at a certain arc distance from a parameter is trivial. Rhino does this by default (instead of using normalized parameter space). -Chord length is a bit trickier because it involved Euclidian distances. So how can we find a point at a chord distance (without geometrically looking for the intersection of a curve and a circle)?

Digital Intuition Cluster, Smart Geometry 2013, Stylianos Dritsas, Mirco Becker, David Kosdruy, Juan Subercaseaux 1.Start from a point at some parameter 2.Move forward by a fixed step size 3.Evaluate the current position and measure the distance from start 4.If we are spot-on (within tolerance) return the parameter found 5.Otherwise, if we have overshoot, then half the step size and change direction, or if we are still far away keep marching forward. QnA How fast is this method? Will this always work? How do we pick the fixed step? Grasshopper Physics

Digital Intuition Cluster, Smart Geometry 2013, Stylianos Dritsas, Mirco Becker, David Kosdruy, Juan Subercaseaux Grasshopper Physics 1.Set a min and a max parameter value the point at chord distance may take 2.Select the parameter half way between min and max 3.Evaluate the position and measure the distance from start 4.If we are spot-on (within tolerance) return the parameter found 5.If the distance is too far set the minimum equal to middle (solution must be greater than half way) otherwise set maximum to middle. QnA How fast is this method? Will this always work?

Digital Intuition Cluster, Smart Geometry 2013, Stylianos Dritsas, Mirco Becker, David Kosdruy, Juan Subercaseaux 1. Define Equation: f( t ) = | start - Curve( t ) | - chord |... | is the distance between the start point and a point at parameter t We are really looking for t such that f( t ) = 0 aka we are looking for the roots 2. Define Derivative (approximation): f’( t ) = ( f( t + h ) – f( t ) ) / 2h or in stereo f’( t ) = ( f( t + h ) – f( t – h ) ) / 2h The derivative is just the value of the function a notch away really 3. Apply Newton-Raphson step till success t_next = t_prev – f( t_prev ) / f’(t_prev) QnA Makes any sense? How fast is this method? Will this always work? What is the step factor h? Grasshopper Physics