There are many other questions that arise when performing analysis of solids, depending upon the application, some of which we shall see later on.

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1 CAD: Ces ad Sfaces: Pat I of III We saw ealie how CSG o BREP ca be sed as schemes to epeset solids i a compte. Usig eithe a CSG o a BREP model, oe ca, potetially aswe ay qestio abot the solid that is epeseted. Some of these qestios cold be: What is the weight of the solid? If oe is to iew the solid fom a gie diectio, what wold that iew be like (that is, which edges will be isible, ad which oes will be hidde)? As a extesio of this qestio, coside this: If we defie the locatio of a set of soces of light, ad thei itesity, ad the optical popeties (sch as eflectiity, colo etc.) of the solid, what will the solid look like? Sch images ae omally daw by most off-the-shelf solid modelig packages, ad ae called edeed images. The method of figig ot what this image is like is called edeig. What ae the coodiates of the cete of mass of the solid? What is its momet of ietia abot a gie axis? Thee ae may othe qestios that aise whe pefomig aalysis of solids, depedig po the applicatio, some of which we shall see late o. Poblem: I CSG, the solid is bilt sig a set of pimities. Thee may be some shapes that ca ot be bilt sig ay combiatio of a gie set of pimities. A example of this is shapes that ae kow as sclpted (o fee-fom) shapes. Fee-fom shapes, as fa as we ae coceed, epesets shapes that ae ot liea/plaa, o based o coic sectios. May objects i life hae shapes that ca ot be descibed by opeatios o simple pimities. This meas that moe complex mathematical epesetatios ae eeded to descibe sch shapes. Whe a CSG epesetatio is bilt, the se specifies the ales of the paametes fo the pimities. Fo istace, chagig the legth, width o height of a block; chagig the diamete o height of a cylide etc. I all these opeatios, the basic shape of the pimitie emais the same, i some sese.. fo istace, a block is scaled diffeetly alog the X-Y-Z axes, bt it is still a ectagla paallelepiped. Howee, i case of sclpted sfaces, each object we stdy has a diffeet basic shape. How the ca we specify a pimitie shape fo it i a CSG system?

2 This poblem was stdied i depth i the ealy days of CAD de to its impotace to the atomobile (ad late, aicaft) idsty. Basically, we still eed a "ice" mathematical epesetatio of a sface i ode to se it i a compteized system. I tems of "ice", oe of the best behaed set of fctios (ad also most stdied) ae polyomials. (Why?) Reiew of ecto algeba A poit i 3-D Eclidea space ca be epeseted by the liea sm of its coodiates alog thee mtally idepedet ectos (called the basis). The most commo basis is thee mtally othogoal it ectos deoted (i, j, k). Ths a poit, P, is deoted by its positio ecto: P = pxi pyj pzk; px, py, pz ae the compoets of P alog the basis ectos. The om of P, deoted P, is the Eclidea distace of P fom the oigi; P = px py pz. The scala podct of two ectos is gie by: P Q = pxqx pyqy pzqz = P Q cosθ; θ is the agle betwee P ad Q. What is the physical sigificace of the dot podct? The coss podct of two ectos is gie by: P x Q = i j k p q x x p q y y p q z z = P Q siθ ; whee is a it ecto pepedicla to P ad Q, ad oieted by the ight-had le, measig θ fom P to Q. What is the physical sigificace of the coss podct?

3 Ce Geomety: Thee ae thee ways of epesetig a ce: a implicit ce; a explicit (o o-paametic) ce; a paametic ce; Coside a cicle as show below: y R P( x, y) O θ x The cicle ca be epeseted as: x y = R. This is called the implicit eqatio of the cicle, sice the ale of x is ot diectly epeseted as a fctio of y (o ice esa). The geeal fom of a implicit eqatio is: g( x, y, z) = 0. We ca ewite the cicle eqatio as: y = (R x ) 1/ Does the fctio y(x) epeset the cicle? The fom y = y(x) is said to be i explicit fom. Eqialetly, we ca epeset the cicle i the followig way: P = (px, py) = ( Rcosθ, Rsiθ).

4 I this case, θ is a paamete of the ce, ad ay poit o the cicle ca be iqely idetified by the coespodig ale of θ. Coside the followig e-paamtizatio: φ = θ/ px = Rcosθ = R cos( φ) = R( cos φ si φ) = R cos φ ( 1 ta φ) = R( 1 ta φ) / sec φ = R ( 1 ta φ) / ( 1 ta φ) ad, py = Rsiθ = R si( φ) = R si φ cos φ = R ta φ cos φ = R ta φ / sec φ = R ta φ /( 1 ta φ) Sbstittig t = ta φ, we get: px = R ( 1 - t) / ( 1 t); py = R t / ( 1 t) The aboe is a paametic eqatio of the cicle, ad is said to be i atioal polyomial fom, sice each eqatio is a atio of polyomial fctios of the paamete t. The best way to epeset space ces (o sfaces) fo se i comptatioal geomety ae paametic foms, sig eqatios of the fom: x = x(t), y = y(t), z = z(t). O, if we epeset the ce as the locs of the positio ecto R, the paametic eqatio is: (t) = ( x(t), y(t), z(t)). Remembe that a ce may be plaa, o 3-Dimesioal. Fo example, what does the followig ce epeset: (t) = a cost i a sit j ct k? Some popeties of ces: Ac Legth: We hae see that oe paamete, t, aies i its ale oe eal iteal [a,b], with (a) beig the statig poit of the ce, ad (b) the ed poit. It is theefoe possible to

5 descibe a ce by a paamete, s, i the age [0, L] whee L is the legth of the ce, ad the paamete ale at ay poit eqals the legth of the ce fom the stat poit to (s). Fo most ces, it is ot easy (o coeiet to fid sch a paametic fom). We will distigish the ac-legth paamete (s) fom all othe types of paametes (t) that we shall se. Fo ce (s), ac legth betwee poits (a) ad (b) = b a. Fo a ce (t), the ac legth betwee two poits coespodig to t = a ad t = b is gie by: b s =. dt whee = d / dt a If we expess the ac legth fom t = a to a abitay poit, t, the we get the followig impotat fom: ac legth = t a s( t) =. dt Taget: s P δs δ Q (t) (t δt) A ce, (t) is show i the fige aboe. As δt 0, the ecto d/dt will become paallel to the taget. Ths the it taget ecto, T = (d/dt) / d/dt. I paticla, sice s is the ac legth, as ds 0, the chod δ will appoach the legth of the ac, δs, ad d/ds 1. Ths, if the ce is epeseted i tems of the ac legth as (s), the it taget ecto is gie by d/ds. The ecto d/dt is the taget to the ac (t). The it taget ecto, T = (d/dt) / d/dt is the it taget ecto to the ce.

6 [Notatios: We will se bold-face to epeset ectos, omal fot to epeset scalas, a dot aboe a fctio to show its deiatie with espect to t, ad the pime, o apostophe sig, to deote its deiatie with espect to s.] T = = d / ds. What is the paametic eqatio of the taget to a ce? The fige aboe shows a geeal 3D ce. Notice that the ce is chaacteized at ay poit by two thigs: how mch it is cig, ad how mch it is twistig. Fo sch 3D ces, the plae pepedicla to T is a omal plae, ad ay ecto i the omal plae is a omal ecto. The ecto dt/dt lies i the Nomal plae (sice it is pepedicla to T) The it ecto paallel to dt/dt is called the picipal omal, N. I paticla, if = (s), the dt/ds idicates the magitde by which the taget ecto is cig i othe wods: dt/ds = κn Whee κ is the cate of the ce, ad the adis of cate of the ce is defied: ρ = 1 / κ. x It ca be show that cate = κ ( s) = ( s) = "( s) = 3 A ecto B, called the it biomal, pepedicla to T ad N, foms a ight-haded system as follows: B = T x N.

7 Sfaces: All physical objects ae bod by sfaces. Sfaces ca be descibed mathematically i implicit, explicit (o-paametic) o paametic fom. These epesetatios ae extesios of defiitios of ces. Implicit foms: Coside a sphee of adis R, ceteed aod poit (x, y, z). What is the epesetatio of all poit o the sface of the sphee? (x - x) (y - y) (z - z) - R = 0. What is the epesetatio of all poits o o iside this sphee? No-paametic foms: Reaagig the sphee eqatio fom aboe, we get the coespodig explicit fom: z = [R - (x - x) - (y - y) ]1/ zx. Paametic foms: The paametic epesetatio of a ce was a set of fctios, x(t), y(t) ad z(t), i tems of oe paamete, t. Sfaces ae epeseted similaly, sig two paametes, ad. Ay positio ecto o the sface, (, ), is epeseted: (, ) = ( x(, ), y(, ), z(, )). Fo istace, a it sphee, ceteed o the oigi, ca be epeseted i tems of paametes ad as:

8 (, ) = ( cos cos, cos si, si ), with 0 π, 0 π/. A atioal paametic fom of the it sphee ca be deied to be: x(, ) = (1 - ) ( 1 - ) / [( 1 ) ( 1 )], y(, ) = ( 1 - ) / [( 1 ) ( 1 )], z(, ) = ( 1 ) / [( 1 ) ( 1 )]. Taget ad Nomal of Sfaces: Taget plae (, 1). (t) (t) c( t) ( 0, ) (, ) ( 1, ) (, 0) - domai Fo a sface defied by (, ), coside a ce, c( t), lyig o the sface. Ths, c( t) = ( ( t), ( t)). Sice c( t) lies o the sface (, ), theefoe c(t) foms a ce, ( t), o the sface. ( t) = ( ( t), ( t)) = ( x( (t), (t)), y( (t), (t)), z( (t), (t))). Of cose, a taget to the ce will be a taget to the sface. Diffeetiatig the aboe expessio, d d d = = = dt dt dt

9 Whee d/dt is the taget ecto of the ce (t); ad lie i the taget plae, ad ae the tagets to isopaametic ces (ces with oe of the paametes, o, emaiig costat). The omal ecto lies alog the coss podct of ad. Uit omal ecto = = ( x ) / x The omal ecto is extemely impotat i machiig, sice it is sed i comptig the offset sface fo (, ). Fo istace, a offset sface with offset d, is gie by: (, ) = (, ) d (, ). Remembe fom yo egieeig maths that the gadiet of a implicit eqatio of a sface defies the omal to the sface f( x, y, z) = 0: f f f gad f = x y z The gadiet has a impotat popety that it is iaiat with espect to the coodiate fame. It also has the popety that it lies alog the omal to the sface. This ca be poed as follows. Coside a scala epesetatio of a sface, f( x, y, z) = 0, ad a ce, ( t) = x(t) i y(t) j z(t) k, that lies o this sface. Sice all poits o this ce mst satisfy the eqatio of the sface, we hae: f( x(t), y(t), z(t)) = 0. diffeetiatig with espect to the ce paamate t, ad sig the chai le, we get: f f f x y z = ( gad f ). = 0 x y z This idicates that (gad f) is pepedicla to the diectio of the taget that is, it lies alog the omal to the sface. This is a easy way to compte the omal of sfaces descibed by a implicit eqatio. Howee, most of the time, we will deal with sfaces that ae defied i a paametic fom. Cate of the sface: Aothe popety of a sface we ofte eed to compte is its cate. Fo example, if we eed to machie a sface, ad o tool, say a ball-ed mill, toches the sface at some poit. If the adis of the sface (alog ay diectio) is less tha the adis of the tool, the the tool will ct ito the pat. This sitatio, called gogig, is desiable. Ths, if we hae a sface that eeds to be machied, we mst fist compte, fo each poit o it, the adis of cate. The miimm ale fo the adis (at the poit o the

10 sface that is most ced ) idicates the lagest tool that ca machie the etie sface withot gogig. Thee ae two poblems: (1) The sface has a ifiite mbe of poits. This poblem is hadled by discetizatio of the sface, ad oly cosideig a mesh of poits that appoximate the sface closely. () At each poit, thee ae a ifiite mbe of diectios, ad a (possibly) diffeet cate alog each diectio. Fotately, it ca be show that at each poit, we ca compte the piciple diectios, ad the cates alog the two othogoal picipal diectios ae the local extema oe picipal cate is the miimm, ad the othe the maximm. Hece, oe oly eeds to compte the ales of the picipal cates at each poit. Coside agai, a sface (,) with a ce (t) lyig o it deoted ( (t), (t)). We shall se a x1 matix, (t) = = [(t), (t)] T to deote the fctios of the two paametes. Let s look at the ce (t) lyig o o sface. Usig the chai le fo diffeetiatio, the taget ecto of this ce is obtaied as: x / x / = = A, whee A = y / y / z / z / The legth of the taget is calclated fom: T T T T = = A A we deote A A = G = G is called the fist fdametal matix of the sface, ad the it taget ecto alog the ce (t) o the sface is gie by: T A = = T ( G ) 1/ Fo a geeal ce, (t) lyig o a sface, sig the defiitios i the peios sectio ad the chai le, we see that: d d ds d( st ) dt dt ds dt = = st ; ad = = st s = st s = st s = st s κn dt ds dt dt dt ds dt ds Ad sig the eqatio fo taget deied ealie, ad the chai le, we get the followig:

11 dt d dt d dt d = = ) / ( ) / ( Fom the aboe two expessios, takig the dot podct with a it ecto,, omal to the sface, we get the expessio fo the cate (ote: is pepedicla to T ad to the patial deiaties of with espect to ad ): N s = κ Usig the matix otatio, we ca wite this as: = =, D whee D N s T κ D is called the secod fdametal matix of the sface. Note that N is a it omal to the ce (o the sface), while is a it omal to the sface. I geeal, these two ectos may ot be paallel. Yo ca imagie this by thikig of a poit P o a sface, ad imagiig seeal ces passig thogh this poit. The omal ectos to these ces at P may ot all be paallel. Imagie a ce sch that its omal at P is paallel to the sface omal. The cate of the sface alog this ce, at poit P, is called the omal cate ad deoted κ. I this case, sice N. = 1, we get: G D s D T T T = = κ P sface: (, ) ce: = (t) = A.. omal plae thogh P, ṙ P sface: (, ) ce: = (t) = A.. = A.. omal plae thogh P, ṙ omal plae thogh P, ṙ As we otate the omal plae i the aboe fige thogh the poit P, the ale of κ chages. The miimm ad maximm ales of the cate ca theefoe be deied by diffeetiatig the expessio of κ with espect to the taget diectio [d/dt, d/dt].

12 These diectios ae called the picipal diectios, ad the coespodig ales of cate ae called the picipal cates. The ales of the piciple cates ae gie by the followig expessio: ( g κ = 11 d d 11 g g 1 d 1 ) ± ( g 11 d G d 11 g g 1 d 1 ) 4 G D whee g ij ad d ij ae the (i,j) elemets of the G ad D matices, ad G, D ae the detemiats of the G ad D matices espectiely. We eed sfaces fo epesetig the geomety of sefl pats. These sfaces ae complex by ate. What mst we be able to do with o sface eqatios? (a) We mst be able to geeate the Catesia coodiates of the poits o the sface. (b) We mst be able to geeate the eqatios of taget plaes, ad omal ectos at ay poit o the sface (c) We mst be able to fid the cate (if it is defied) at ay poit o the sface. (d) We mst be able to appoximate the sface by a gid of small (plaa) patches. I paticla, we mst be able to appoximate the etie sface by a set of small, wellbehaed tiagles. Likewise, we mst be able to coet a solid cotaied by oe o moe sfaces by a set of small, well-behaed tetahedos. Nomals ad cates hae pactical applicatios i geeatig the machiig plas fo sfaces. Tiaglatio of sfaces is sefl i ceatig a plaa epesetatio called STL, which is commoly sed fo layeed mafactig. Tetahedal tessellatio is sefl fo fiite elemet aalysis. I the ext sectio, we stdy the followig poblem: amog the diffeet methods of epesetig ces ad sfaces (implicit, explicit, paametic), which fomat is best fo what we eed to do?