GRASSHOPPER TUTORIAL 02 PERFORATED CURVATURE www.exlab.org
IDEA PERFORATED CURVATURE THIS TUTORIAL EXTENDS UPON TUTORIAL 01 BY CREATING A SIMPLE DEFINITION THAT ANALYSES THE CURVATURE OF A DOUBLY CURVED SURFACE AND INDICATES HOW THIS COULD BE FURTHER EXTENDED TO MANIPULATE MATERIAL QUALITIES TO ACHIEVE COMPLEX DOUBLE CURVATURE FROM PERFORATIONS IN A FLAT SHEET. FURTHER INFORMATION REGARDING THE FURTHER EXTENSION OF THIS DEFINITION IS PROVIDED AT THE END OF THE TUTORIAL.
EXERCISE CREATE A DOUBLY CURVED SURFACE IN RHINO THAT WILL BE ANALYSED BY THE GRASSHOPPER DEFINITION. REFERENCE THIS SURFACE FROM RHINO INTO GRASSHOPPER USING THE SURFACE COMPONENT. RIGHT CLICK ON THE COMPONENT AND SELECT SET ONE SURFACE (PARAMS/GEOMETRY/SURFACE) TO EVALUATE CURVES ONTO THE SURFACE POINTS MUST BE DEFINED ACROSS THE SURFACE. THIS IS ACHIEVED BY DIVIDING THE SURFACE INTO U&V TOPOLOGICAL CO-ORDINATES USING THE DIVIDE SURFACE COMPONENT. (SURFACE/UTILITY/SDIVIDE) SURFACES ARE DEFINED BY TWO DIRECTIONS IN PARAMETRIC SPACE, REFERRED TO AS U AND V DIRECTIONS. THE DOMAIN OF EACH OF THESE DIRECTIONS ARE DIVIDED BY THIS COMPONENT, RETURNING POINTS AND U,V PARAMETERS OF THE DIVISION POINTS, AND THE VECTOR NORMALS AT THESE POINTS. SPECIFY THE NUMBER OF DIVISIONS IN EACH OF THE U-DIRECTION AND V-DIRECTION USING NUMBER SLIDERS. (PARAMS/SPECIAL/NUMBER SLIDER) THE NUMERIC DOMAIN (THE MINIMUM, MAXIMUM AND RANGE) OF THE SLIDER AND THE TYPE OF NUMBER ROUNDING CAN BE ALTERED BY RIGHT CLICKING ON THE SLIDER AND SELECTING EDIT. WE ARE GOING TO MAKE A PERFORATION AT EACH OF THESE DIVISION POINTS INITIALLY BY CREATING A CIRCLE USING THE DIVISION POINTS AND NORMAL VECTOR AT THAT POINT. (CURVES/PRIMITIVES/CIRCLE CNR)
EXERCISE N.B. YOU CAN SEE THAT THE GH WIRES ARE NOW DOTTED AND HOLLOW. THIS INDICATES THAT THE OBJECT DATA HAS BEEN SEPARATED INTO BRANCHES. THE DATA STRUCTURE CAN BE VIEWED BY USING THE PARAM VIEWER COMPONENT. (PARAMS/SPECIAL/PARAM VIEWER) THE RADIUS OF THE CIRCLE CAN BE ALTERED BY ADDING A NUMBER SLIDER TO THE RADIUS INPUT. THIS RESULTS IN UNIFORM PERFORATION CURVES OVER OUR SURFACE, BUT NOTHING THAT RELATES TO SURFACE CURVATURE. WE WILL NOW ANALYSE THE SURFACE AT THE DIVISION POINTS AND USE THESE LOCAL CONDITIONS TO INFORM THE PERFORATIONS. TO ANALYSE THE SURFACE CURVATURE AT EACH OF THE UV COORDINATES INSERT THE SURFACE CURVATURE COMPONENT. (SURFACE/ANALYSIS/SURFACE CURVATURE) FOR FURTHER INFORMATION ON SURFACE CURVATURE SEE REFERENCE MATERIAL. INPUT THE LOCAL GAUSSIAN CURVATURE TO INFORM THE RADII OF THE CIRCLES. INPUT THE LOCAL MEAN CURVATURE TO INFORM THE RADII OF THE CIRCLES AND EVALUATE THE DIFFERENCE.
EXERCISE DEPENDING ON THE SURFACE YOU HAVE CREATED, THE SIZE OF THE GENERATED PERFORATIONS MAY BE TOO SMALL. THE SIZE OF THE RADII INPUTS CAN BE VIEWED BY USING A PANEL. (PARAMS/SPECIAL/PANEL) INSERT A FUNCTION COMPONENT TO RE SCALE THE NUMERICAL DATA TO THE TARGET SURFACE. FUNCTION COMPONENTS ALLOW YOU TO ENTER ALGORITHMS TO ALTER THE DATA USING DIFFERENT NUMBERS OF VARIABLE INPUTS. IN THIS CASE WE WANT TO INCREASE THE SIZE OF THE CURVATURE ANALYSIS DATA TO INCREASE THE RADII INPUT TO A PERCEPTIBLE AMOUNT. (MATH/UTIL/F2) RIGHT-CLICK ON THE F INPUT OF THE COMPONENT TO INPUT A MATHEMATICAL EXPRESSION. THIS CAN BE DONE IN THE WINDOW BELOW EXPRESSION EDITOR OR BY SELECTING EXPRESSION EDITOR FOR A SEPARATE WINDOW WITH EXPRESSION ICONS IN THIS CASE I HAVE ENTERED THE EXPRESSION X*(Y*1000) TO ACHIEVE AN ADEQUATE RESULT. THE EXPRESSION REQUIRED WILL DEPEND ON THE SIZE OF THE INDIVIDUAL SURFACE. EXPERIMENT WITH DIFFERENT EXPRESSIONS UNTIL AN ADEQUATE OUTCOME IS REACHED WHERE THE CIRCULAR PERFORATIONS ENLARGE AS CURVATURE INCREASES, ALTERING THE RIGIDITY OF THE MATERIAL.
MATERIAL PERFORMANCE figure 1 figure 2 BASED UPON EXPLORATION ON THE MATERIAL PROPERTIES OF A PERFROATED TIMBER LAMINATE PRESENTED AT THE ACADIA REGIONAL CONFERENCE IN 2011. FOR THE WHOLE ARTICLE, SEE THE LINK ON THE DDA\SCRIPTS PAGE: HTTP://SCRIPTS.CRIDA.NET/GH/ THE CONCEPT IS TO USE THE INHERENT STRUCTURAL PROPERTIES OF THE MATERIAL, AND THE POWER OAND ACCURACY OF COMPUTATIONAL DESIGN TOOLS TO MANIPULATE A SURFACE IN A VIRTUAL SPACE AND BE ABLE TO REPLICATE IT WITH PRECISION IN REALITY. figure 3 figure 4 THE INTIAL DOCUMENTATION PROVIDED ABOVE (FIG.1) SHOWS THE STRUCTURAL ANALYSIS OF A 10X10 SHEET OF TIMBER LAMINATE, WITH VARYING PERFORATIONS. FIGURE 2 SHOWS AN INITIAL CURVATURE ANALYSIS OF A NURBS SURFACE (FIG.2) WHICH GENERATES PERFORATIONS USING A DESIGN SCRIPT (FIG.3). THE INFORMATION REQUIRED TO INFORM A CNC CUTTING MACHINE IS THE EXTRACTED TO ENABLE THE BOTTOM FABRICATION, (FIG.4) WHICH APPEARS TO RESTRICT THE BENDING OF THE MATERIAL TO THE DESIRED DIRECTION.
REFERENCE MATERIAL Curvature As the name suggests, curvature is the amount of curve in a surface, or how much an object s geometry deviates from being flat. Curvature is measured locally but often represented via a colour gradient (see icon, right) to show the differentiation of values across a surface. Concave and convex curvature values are often represented as positive and negative values. principal and surface curvature icons within Grasshopper positive Gaussian curvature, bowl-like surface Curvature on surfaces Surfaces are defined by two directions in parametric space, often referred to as U and V directions. It is the combination of curvature in each of these directions at a point which defines the surface curvature. There are two forms of curvature you will come across in NURBS modelling and in these tutorials; mean curvature and Gaussian curvature. As the name suggests, mean curvature is the mean value of the two directional curvatures; one half the sum of the principal curvatures at a point. Surfaces with zero mean curvature across them are minimal surfaces. Gaussian curvature is the product of the two values. Therefore developable surfaces, otherwise called ruled surfaces,curve only in one isoparametric direction and have a Gaussian curvature of zero across their entirety. negative Gaussian curvature, saddle-like surface developable surface, zero Gaussian curvature Further explanations of curvature: http://en.wikipedia.org/wiki/principal_curvature Mean Curvature http://en.wikipedia.org/wiki/mean_curvature Gaussian Curvature http://en.wikipedia.org/wiki/gaussian_curvature *Gaussian curvature examples from Essential Mathematics for Computational Design