EssentialMathematicsforComputationalDesigniiEssentialMathematicsforComputationalDesignbyRobertMcNeel&AssociatesislicensedunderaCreativeCommonsAttribution-ShareAlike3.0UnitedStatesLicense.EssentialMathematicsforComputationalDesigniiiPrefaceEssentialMathematicsforComputationalDesignintroducesdesignprofessionalstofoundationmathematicalconceptsthatarenecessaryforeffectivedevelopmentofcomputationalmethodsfor3Dmodelingandcomputergraphics.Thisisnotmeanttobeacompleteandcomprehensiveresource,butratheranoverviewofthebasicandmostcommonlyusedconcepts.Thisbookisdirectedtowardsdesignerswhohavelittleornobackgroundinthesubject.ItassumesthatreadershavegoodknowledgeofRhinoceros®(Rhino),NURBSmodelingforWindows,andGrasshopper®(GH),thevisualscriptingenvironmentforRhino.Bothareusedastoolstoexplainvariousconcepts.Formoreinformation,goto:V=a1,a2Similarly,in3-Dcoordinatesystem,vectorsarerepresentedbythreerealnumbersandwouldlooklike:V=a1,a2,a3Usingacoordinatesystemandanysetofanchorpointsinthatsystem,wecanrepresentorvisualizethesevectorsusingaline-segmentrepresentation.Weusuallyputanarrowheadtoshowthedirectionofvectors.Forexample,ifwehaveavectorthathasadirectionparalleltothex-axisofagiven3-Dcoordinatesystemandamagnitudeequalto5.18units,thenwecanwritethevectorasfollows:V=5.18,0,0(Anglebracketsdifferentiateavectorfrompointcoordinates.)Torepresentthatvector,weneedananchorpointinthecoordinatesystem.Forexample,alloftheredlinesegmentsinthefollowingfigureareequalrepresentationsofthesamevector.1Grasshopperunitx-axiscomponent2Grasshoppernumberslidercomponent3GrasshopperpointcomponentsthatissettoreferencemultiplepointsinRhino(inthiscasereferencingA1,A2,A3andA4)4GrasshoppervectordisplaycomponentEssentialMathematicsforComputationalDesign2Givena3-DvectorV=a1,a2,a3,allvectorcomponentsa1,a2,a3arerealnumbers.AlsoALLlinesegmentsfromapointA(x,y,z)topointB(x+a1,y+a2,z+a3)areE