AML710 CAD LECTURE 16 SURFACES. 1. Analytical Surfaces 2. Synthetic Surfaces

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1 AML7 CAD LECTURE 6 SURFACES. Anlticl Surfces. Snthetic Surfces Surfce Representtion From CAD/CAM point of view surfces re s importnt s curves nd solids. We need to hve n ide of curves for surfce cretion. In the sme w surfces form the boundries of the solids. Tpes Anlticl surfces Eg. Plne surfces, sphere, ellipsoid Snthetic surfces Eg. Bicubic surfce, Beier surfce Appliction Modeling prts in CAD/CAM, representtion of dt surfces like isotherml plnes, stress surfces/contours

2 Definition A Jordn surfce is defined b prmetrition tht estblishes homomorphism with the surfce of unit sphere. A hole-free -mnifold is clled hole-free surfce. This is topologicl definition of surfce. A surfce is either hole-free surfce or surfce with frontiers. A simple hole-free surfce is homomorphic to sphere i.e Jordon surfce. Surfce Representtion It is just n etension of representtion of curves. We cn represent surfce s series of grid points inside its bounding curves. Surfces cn be in two-dimensionl spce (plnr) or in three-dimensionl spce (generl surfces). Surfce cn be described using non-prmetric or prmetric equtions Surfces cn be represented b equtions to pss through ll the dt points (fitting) or hve ptches of them connected t the dt points (pproimtions)

3 NON-PARAMETRIC REPRESENTAION In generl surfce or surfce ptch is represented nlticll b n eqution of the form P (,, ) [ ] [ f (, )] Where P is the position vector. The nturl choice for f(,) is polnomil. Thus for nlticl representtion of surfces we cn use equtions of tpe f (, ) p q m n mn m n P(,,) PARAMETRIC REPRESENTAION In prmetric surfces vector vlued function P(u,v) of two vribles is used s follows: P( u, v) [ ] [ ( u, v) ( u, v) ( u, v)] u min u um ; m A surfce m be one ptch or constructed using severl ptches. All comple surfces re represented using mn ptches v min v v v v P(u,v) u Prmetric spce u Crtesin spce 3

4 ANALYTICAL SURFACES IN PARAMETRIC FORM.Surfce of Revolution r l l l r Revolving line Clindricl surfce ANALYTICAL SURFACES IN PARAMETRIC FORM.Surfce of Revolution l Line perpendiculr to -is Revolving line tht mke n ngle to -is Revolving closed polgon 4

5 Surfce of Revolution The plne curve P ( [ ( ( ] The Biprmetric surfce of revolution Q ( t, ) ( ( cos ( sin [ ] Note tht t,) is vector vlued function t, ) ( i + ( cosj + ( sink Surfce of Revolution Sphere r cos; r sin π The eqution of the surfce, ( ) ( )cosφ ( )sinφ [ ] [ r cos r sin cosφ r sin sinφ] π; φ π Here is clled the ltitude ngle nd φ longitude ngle 5

6 Surfce of Revolution Ellipsoid cos; bsin π The eqution of the surfce OR [ cos bsin cosφ sin sinφ], b π; φ π [ bsin sinφ bsin cosφ cos ], π; φ π Surfce of Revolution Torus: When the is of rottion does not pss through the centre of the circle or ellipse we get torus h + cos; k + bsin π Where h,k re the coordintes of the centre of the torus The eqution of the surfce, h + cos ( k + bsin )cosφ ( k + bsin )sinφ [ ] π ; φ π 6

7 Surfce of Revolution Prboloid ; m The eqution of the surfce [ cosφ sinφ], m ; φ π Surfce of Revolution Hperboloid sec; tn m The eqution of the surfce [ sec b tn cosφ tn sinφ], b m ; φ π 7

8 Surfce of Revolution An Spce curve In generl n spce curve cn be used to generte surfce of revolution P ( [ T ][ N][ G] [T] Prmeter vector [N]- Blending function mtri (normlised) [G]-Geometr informtion mtri Now surfce of revolution is defined s Q ( t, [ T ][ N][ G][ S] cosφ sinφ ] tmin t t [ S m ; φ π Sweep Surfce A 3D surfce lso cn be obtined b trversing geometricl entit like line, polgon or curve long pth in spce Recll the eqution of line P( P t t t t + ( P P ) min [ T ( s)] ns If the line of length n is prllel to -is If the sweep trnsformtion contins onl trnsltion nd scling the resulting surfce is plnr If it includes rottions lso with trnsltions then non-plnr surfce results m t, s) [ P( ][ T ( s)] t s s s 8

9 Sweep Surfce We m use n other prmetric curve like cubic spline, Beier, B-spline curve cn be used which is denoted s P( in the sweep surfce below: t, s) [ P( ][ T ( s)] t t min t tm; s s s [ T ( s)] ns If the curve on plne is swept long b n units One needs to tke the curve in such w to void surfce degenercies in full or in prt. Sweep Surfce In ddition to open curves, closed curves nd polgons cn be used to crete sweep surfces Such surfces enclose finite volume (with end cps) A squre or rectngle swept long stright pth results in prllelepiped A tringle swept long stright pth ields wedge A circle long stright pth results in clinder A circle of decresing rdius cone Rottion long with sweep cn be combined to give twist to the generted surfce. 9

10 P ( u) P ( u) Lofted Surfce cos u sin u [ π π ]; [ cos π u sin π u 4] u %Lofted Surfce [u,w]meshgrid(:.5:6.3); q(+w).*cos(*pi*u); q(+w).*sin(*pi*u); q4*w; surfc(q,q,q); u, v) ( v) P ( u) + vp ( u) v Equivlence of Surfces Two surfces S nd S re equivlent if S cn be mpped onto S b continuous mpping without tering nd dupliction nd S cn be similrl be mpped onto S. This notion of equivlence is somewht like the notion of homeomorphism tht is, n invertible mp, f : S S, such tht both f nd its inverse, f, re continuous. From Gllier nd Xu

11 Clssifiction of Surfces Orientble nd Non-orientble surfces In generl, n orientble surfce with g holes ( surfce of genus g) cn be opened up using g cuts nd cn be represented b regulr 4g-gon with edges pirwise identified, where the boundr of this 4g-gon is of the form b - b b b g b g g b g Tringultion of Surfces Tringultion (tiling) nd Orientbilit Definition b P. S. Aleksndrov (956): A tringultion surfce is orientble iff it is possible to orient ll of the tringles in such w tht ever two tringles tht hve common side re coherentl oriented, otherwise it is clled nonorientble. If Z nd Z re tringultions of the sme surfce, Z is orientble iff Z is orientble.