Controlled Blending of Procedural Implicit Surfaces

Transcription

1 236 Contolled Blending of Pocedual Implicit Sufaces Zoan KaCic-Alesic Bian Wyvill Depatment of Compute Science Univesity of Calgay Calgay, Albeta, Canada T2N IN4 Abstact Implicit sufaces ae becoming inceasingly popula in geometic modeling. Pocedually defined implicit sufaces, in paticula sufaces built aound skeletons, povide an intuitive epesentation fo many natual objects, and objects commonly used in geometic modeling. This pape pesents seveal techniques that povide good contol ove the way diffeent skeletally defined implicit sufaces blend togethe. Some extensions to these techniques povide a simple and convenient epesentation fo "soft" sufaces of evolution, andomly defomed sufaces, and othe inteesting shapes that would othewise be difficult to model. Keywods: Implicit Sufaces, Blending Functions, Skeletons. Intoduction Paametic cuves and sufaces have taditionally been favoite modeling pimitives in compute aided geometic design and gaphics. A ich liteatue exists on techniques fo fee-fom modeling with these pimitives. In ecent yeas, howeve, implicit sufaces have become inceasingly impotant. The majo advantage of implicit sufaces that makes them vey attactive fo modeling and animation is the blending popety. Sepaate implicit sufaces, unlike paametic sufaces, can blend togethe smoothly and fom vey complex, non-intesecting shapes. Implicit sufaces also conveniently define volumes. It is easy to detemine whethe a point is above, below, o on the suface, simply by evaluating the implicit function at the given point. Computing the suface nomal is also an easie task than it is fo paametic sufaces. A fequent objection to the use of implicit sufaces is that the available modeling techniques do not povide sufficient contol of the shape of the suface. Also, the shape of the suface is not vey intuitive fom its implicit fomulation. The effects of changing coefficients in the algebaic fomulation of the suface ae usually unpedictable and vey difficult to contol. The contol flexibility necessay in a design envionment is, thus, hadly achievable by diect manipulation of the coefficient in the algebaic fomulation of the suface. The goal of ou eseach is to develop a modeling technique which will enable the designe to specify an implicit epesentation of an abitay shape. The goal is ambitious and many poblems still have to be solved. In this pape we pesent some new techniques that povide the designe with bette contol ove the way in which pocedual implicit sufaces blend togethe. Seveal new blending functions that can be customized inteactively to the need of the designe ae intoduced. These blending functions have a consideable influence on the oveall shape of the suface and can povide additional contol in the design pocess. Skeletal Implicit Sufaces In ode to povide a moe intuitive epesentation we use skeletally defined implicit sufaces [Bloomenthal 90b]. In geneal, any thee dimensional object can be a pat of the skeleton, as long as it is possible to detemine the distance fom a given point in space to the skeleton. Skeletons ae useful fo seveal easons: skeletons povide intuitive epesentation fo many natual objects, skeletons themselves ae simple and easy to manipulate and display, complex shapes can be modeled with few elements. In this pape we limit ou discussion to skeletons that consist of elements such as points, lines, polygons, cicles, spline cuves and spline patches. Each skeletal element is suounded with an imaginay foce field F(), with the intensity of the field being the highest at the skeleton, and deceasing with distance fom the skeleton. The function F() that elates the field value (intensity) to distance fom the skeleton has an impact on the shape of the suface, and moe impotantly, detemines how

2 237 sepaate sufaces blend togethe. Fo that eason we call such functions blending functions. The suface is defined by the set of points in space fo which the intensity of the field has some chosen constant value (thus the name iso-suface). We call this value a contou value. The suface so defined is an offset suface of the skeleton. Fields fom the individual elements of the skeleton add togethe (o subtact. in the case of negative foce fields). what in tun has the blending effect on the sufaces defined by sepaate elements of the skeleton: whee Ftotal(P) = L CiFi(T;).,=1 Ti = fi(p) = dist(p.qi). Ftotal(P) = the intensity of the field at the point p. Ci = a scala value (scale facto) that epesents the fi eld magnitude due to the ith skeleton (it can be negative). Fi = the blending function fo the ith skeleton. Ti = distance fom the point P to the skeleton i. P(xP. YP. zp) = point in!r 3 at which the field function is evaluated. Q(xQ.YQ.zQ) = point in!r 3 - the neaest point on the skeleton i to the point P. The evaluation of the field function has two steps. The fist step involves finding the neaest point on the skeleton to the given quey point and calculating the distance between them. This pocedue depends on the geomety of the skeleton and can be vey simple (tivial in the case of a point skeleton). o quite complex in the case of spline cuves and patches. when an iteative o numeical method has to be used. The second step involves evaluation of the blending function. The shape of a skeletally defined implicit suface is detemined by: the geomety of the skeleton. a blending function to weight the contibution of individual skeletal elements. modifications to the blending function that esult fom geomety. oientation. size. o othe popeties of the skeleton. andom defomations poduced by petubation of the blending function with some thee dimensional noise function. The suface is contolled by applying local o global tansfomations. such as scaling. tanslation. and otation. to the elements of the skeleton. and by changing the blending functions. Many skeletally defined sufaces can be expessed analytically. but in ode to keep the epesentation intuitive and simple these implicit functions ae usually teated as pocedual. i.e. defined by pocedues that etun a scala field value fo any point in 3-D space. Pocedual implicit functions can be used to model sufaces fo which analytical epesentation is difficult o impossible to fomulate [Bloomenthal 90b]. Suface Smoothness The notion of suface smoothness is usually elated to the concept of tangent plane (suface nomal) continuity. A suface is said to be smooth at a paticula point if thee exists a unique plane tangent to the suface at that point. The idea of tangent plane continuity can be genealized to highe odes of continuity. A moe in depth discussion of geometic continuity fo algebaic sufaces can be found in [Waen 89]. The gadient of the implicit function is the suface nomal at a given point on the implicit suface. Pocedual implicit sufaces may not have an analytical epesentation. o it may be vey difficult to obtain. so the gadient must be appoximated by evaluating the implicit function at thee neaby points [Bloomenthal 90b]. It is vey difficult. if not impossible. to pove analytically that a given pocedual implicit suface is smooth. The smoothness of a skeletally defined implicit suface depends on: the continuity of distance to the skeleton. the continuity of blending function. Examples pesented in this pape show objects whose skeletons consist of vey simple elements fo which the distance function is continuous. It should be noted. though. that distance functions may be discontinuous if the elements of the skeleton ae themselves discontinuous. self-intesecting. o of such shape that fo a given point within the adius of influence thee is not a unique neaest point on the skeleton. This can easily happen when spline cuves o patches ae used fo skeletons. o when skeletons ae gouped togethe (see the section on selective blending). A blending function is defined in this pape as ode k continuous if the fist k deivatives ae continuous in the inteval [0. RJ, and they vanish at the end points: d i d i -d. F(O) = O. -d. F(R) = O. i = O... k. ' T' It is impotant to notice that non zeo deivatives at the end points esult in discontinuities in the implicit field function at distance R fom the skeleton and that the esulting suface is discontinuous (see Figues 7. 8 and 9). An implicit suface is said to be ode k continuous if all the blending functions used ae at least ode k continuous (assuming that distance functions along a staight path in space ae also continuous fo all the elements of the skeleton).

3 238 Blending Functions In geneal, any function can be used fo blending. Blinn used supeimposed exponential density distibution functions to model atoms and molecules [Blinn 82]. A vaiation of the above sufaces, called soft objects, uses a blending function which is a cubic polynomial [Wyvill 86b, Bloomenthal 90b]. Soft object have been successfully used fo modeling of objects and figues commonly found in natue, and fo animation of objects that change shape in time [Wyvill 86a]. Blending sufaces based on low degee polynomial functions [Midlleditch 85] and supe-elliptic blends [Rockwood 87] ae used in solid modeling. Fom this ealie wok and ou own expeience thee ae seveal impotant popeties of blending functions: each skeleton should have a limited adius of influence R, i.e. the value of the blending function should be zeo beyond some adius of influence. This enables consideable computational savings because implicit functions of distant skeletons need not be evaluated. blending functions have a maximum at the skeleton (zeo distance), and they dop smoothly and monotonic1y to zeo at the adius of influence. Without loss of geneality, we assume that the maximum value is 1.0 and that the contou value is 0.5. the fist, and if possible second deivative of the blending function should be continuous, and they should vanish at zeo and at the adius of influence. This will ensue that sufaces blend smoothly (assuming that the distance function is continuous). 1 F() 0.5 F() 0.5 O~ =~ o Rj2 R Figue 1: A "60ft" blending function _ ~1~~==-----1~ o Rj2 R Figue 2 : A "hanf' blending function The shape of the blending function affects the "amount of blending." Blending functions that dop to zeo shotly afte they fall bellow the contou value will esult in vey little blending (Figues 2 and 4). On the othe hand, functions that dop slowly to zeo will esult in "soft" blending (Figues 1 and 3). Too much blending is often undesiable. Fo example, when modeling a human body it is not desiable that the ams blend with the body anywhee but at the shouldes. As a esult of using soft blending functions the blended suface is bulgy (Figue 3). "Had" blending functions can educe the bulge (Figue 4). Low degee polynomial functions ae most commonly used fo blending [Wyvill 86b, Bloomenthal 90b]. The lowest degee polynomial that satisfies the fist ode continuity equiements is a cubic: whee = distance fom the skeleton R = adius of influence. Figue 3: A soft blend of cylindes using the blending function fom Figue 1. Note the bulge aound the intesection.

4 239 A cubic in 2 (Figue 5) is moe commonly used [Wyvill 86b, Bloomenthal 90b]: Since the fist deivative Figue 4 : A had blend of cylindes using the blending function fom Figue 2. The bulge is gone. is zeo at = 0 egadless of the choice of coefficients, additional constaint can be put on the cuve Fcub2(). In [Wyvill 86b] an additional point on the cuve is specified such that Fcub2(0.5) = 0.5. This esults in a cuve (Figue 5) vey simila to the odinay cubic in. Some sot of contol ove the shape of the blending function can be achieved by moving this point. Howeve, vey little vaiation fom the shape in Figue 5 is achievable this way. The blending function can be defined as an intepolating cuve though a numbe of contol points. Any numeical analysis book eadily offes seveal intepolating schemes. Natual cubic spline intepolation is pobably the most popula intepolating scheme, and it poduces excellent esults fo a limited ange of cuves in ou example. Howeve, intepolating cuves do not posses the convex hull popety and often fail to emain monotonic when the contol points ae moved aound. Nonpaametic cuves in Benstein-Bezie fom [Fain 88] do not, in geneal, pass though the contol points (except the end points), have a convex hull popety and thus povide a vey good solution to ou poblem: n F() = ~b j Bj(jR) whee j=o Bj(jR) 0.5 F - F' " F" F () ;< , /, / ',,',./, /, /, /, '.-...,,"... _----_... /. bj E [0, 1] = the odinate of j-th contol point (a scala value). The k-th deivative (k :::; n) is also a nonpaametic cuve in Benstein-Bezie fom: -l~~ _, ~~ o R/2 R Figue 5 : Cubic blending function and it's deivatives. (Ft = O.5~~,F" = O.2~:~).6. 1 bj = bj+l - bj,.6. k bj =.6. k - 1 bj +l _.6. k - 1 bj = t ( ~ ) (_l)k-ibi+j.,=0 The n + 1 contol points (jrjn,bj) jj = O,...,n ae equally spaced, with inceasing abscissae, along the axis in the inteval [0, R]. The cuve is thus guaanteed to be a functional cuve. Setting bo,..., bk = 1, and

5 F() 0 l~-_~ F' F" - -- C.Pts,,_... _ _... ;'---..._ ,,-.,,, /....,,,-..., "" " " //... ',,- ""<', ",,,/... _--", ~~ ~~ o R/2. R Figue 6: Fifth degee Benstein-Bezie blending function. it's contol points and deivatives. (F' = 0.5 ~~. F" = 0.15f.f) Note that both deivatives have zeo values at the end points. bn-k... bn = 0 esults in the fist k deivatives being equal to zeo at the end points ( = O. and = R). Fo k = 1 at least 4 contol points ae necessay to assue that the fist deivative will vanish at the end points, and the esulting cuve is cubic. Fo k = 2 we need at least 6 contol points to have both fist and second deivative vanish at the end points. at the cost of having to evaluate a fifth degee polynomial (Figue 6). Ou examples (Figues 7, 8 and 9) show that a blended suface poduced by a second ode continuous blending function appeas consideably smoothe than the suface poduced by a blending function which only has fist ode continuity. The diffeence is paticulaly noticeable on sufaces polygonized at a low level of subdivision (i.e. when fewe polygons ae used to appoximate the suface). Nonpaametic B-splines descibed in [Fain 88] povide moe contol flexibility than Bezie cuves. The abscissae of the contol points ae deived fom the knot sequence (which is vey non-intuitive fo inteactive design) and ae, in geneal, less constained. Fewe contol points ae necessay in ode to achieve simila contol flexibility and the degee of the cuve is geneally lowe, but the cuves ae not as smooth. One additional popety of blending functions is often equied. If the point with the contou value on the gaph of the blending function is moved in the diection, the size of the objects defined by individual skeletons is changed. It is often undesiable that the choice of blending function should affect the size o shape of a single object. It should only affect the blending of two o moe sepaate objects. Without loss of geneality we will additionally constain ou blending functions, so that evey function has the contou value exactly at one half the adius of influence: F;(R/2) = contou value = 0.5. Figue 7: A blend of two cylindes using an ode zeo continuous Benstein-Bezie blending function with fou contol points. Figue 8: A blend of two cylindes using a fist ode continuous Benstein-Bezie blending function with fou contol points. In the case of a single skeleton any blending function should poduce the same suface, which is an offset by

6 241 C2 function at the point of inflection (which also has the contou value) is -cl/1i'. The function will appoximate the step function as the value of Cl inceases; = value of /R fo which Fatan has the contou value (0.5). This function is used to poduce the blended suface in Figue 4 (Cl = 100). The function does not pass though points (0,1) and (R, 0), and it's deivatives do not vanish at the end points. Howeve, the discepancies ae small (Figue 10), and may safely be ignoed fo steep functions. Fo not so steep functions (Cl < 50) the following modification will assue that Fatanl,O(O) = 1, and Fatanl,O(R) = 0: Fatano() = Fatan() - Fatan(R), Figue 9: A blend of two cylindes using a second ode continuous Benstein-Bezie blending function with six contol points (Figue 6). 0.5, '., F F' F" ---, F() 0 -..:-f- :..;..;.-,;.,., :...:--.,;. -::..,..;.-::..._:. ", 1',,-.., I '\ I I l\ I I \\: 11 '\.{ 1 \ 1 -l j ~ 0.4 R R/2 0.6 R Figue 10: Actangent blending function and it's deivatives. (Cl = loo,f' = 0. 03~~,F" = 0.04f,{) R/2 fom the skeleton. As a esult of this additional equiement most of the blending functions will be symmetical in espect to the point (R/2, 0.5). The contol of the blending is achieved by changing the slope of the function at R/2, while maintaining the constaints at the end points, smoothness, and monotonicity of the function. While all the blending functions descibed so fa can easily be constained to pass though the point (R/2, 0.5), it is often difficult to consideably change the slope of the function at that point. A vey good solution fo steep blending functions is based on the actangent function (Figues 2 and 10): whee 1 Fatan() = actan(cl(/ R - C2)) 11' Cl = a numbe that contols the steepness of the function ( ). The slope of the Fatanl,o() = Fatano()/ Fatano(O). All the functions descibed so fa in this pape ae explicit, i.e. they ae expessed in tems of. It is well known that paametic cuves ae much bette suited fo shape design. We have found low degee paametic cuves useful fo design of ou blending functions: F = fi(t), =h(t),te~. Fo any given it is necessay to solve h fo t, in ode to calculate F. Fo quadatic and cubic paametic spline cuves, a closed fom solution exists. Cae has to be taken to assue that the spline cuve defined by the contol points indeed is a functional cuve, i.e. thee must not be points with vetical slope o with multiple values of F fo a given. If that is povided, a single eal solution to h exists fo any given. The method is pone to floating point impecisions when coefficients, paticulaly the coefficient with the highest powe of t, become vey small. That is usually the case when the contol points ae almost collinea, o when they ae equally spaced along the axis. Special cae has to be taken to detect such situations. Figue 11 shows the useful ange of a single segment cubic Bezie spline. The Cadano's fomula used to solve the cubic equation in t, is numeically stable in that ange. The soft blending function in Figue 11 also poduces the blended suface in Figue 3. The had blending function poduces a blended suface that is slightly softe than the suface in Figue 4. Moe segments of a piece wise quadatic o cubic spline cuve can be used if additional flexibility is desied, at an inceased cost of computation. Selective Blending As peviously mentioned, suface blending is not always a desiable effect. Thee ae two pactical solutions to this poblem. One is to goup skeletons of objects which should not blend togethe [Beie 90]. The goup is then

7 242 p F() 0.5 O~ ~~~~ o R/2 R Figue 11: Single segment cubic Bezie blending functions (paametic). The fou contol points ae (0, I), (c, I), (R-c, 0), (I, 0), whee 1/3 < c < 1. The soft (thick) cuve is identical to the cubic cuve Fcub. t = 1 Figue 12: Paametes that can be used to modify the blending function of a line skeleton (Ot = a). teated as a single skeleton, and the distance to the skeleton is the minimum distance to any of the elements of the goup. Anothe solution is to use diffeent blending functions fo diffeent skeletons. Sufaces that should not blend togethe can use "had" blending functions, while othe sufaces can use "soft" blending functions. Inteactive Contol of Blending Functions An additional benefit of using spline cuves, both nonpaametic and paametic, is the ability to inteactively design blending functions, simply by moving the contol points. Similaly, the shape of the actangent function is affected by inteactively changing the value of Cl. A geat deal of infomation about the blending function is hidden in the shape of it's deivatives. Displaying the fist two deivatives togethe with the function itself povides vey valuable infomation fo the inteactive design of blending functions. Depending on the blending function used, the polygonization of a simple suface, such as that defined by two intesecting cylindes, o by two sphees, is a elatively fast pocess. The effects of changing the blending function can thus be visualized sufficiently quickly fo inteactive design, although we cannot do this in eal time with ou cuent implementation. Modifications to Blending Functions Blending functions need not be defined in tems of distance only. Othe paametes based on geomety and oientation of the skeleton can be used to modify the shape of the suface: position of the neaest point on the skeleton. oientation of the skeleton in space. Additional paamete (o paametes) need be specified fo the efeence oientation. size (defomation) of the skeleton. This is useful in animation when skeletons change size, eithe by being stetched o squashed, in ode to simulate the pesevation of volume. Stetched objectss would, thus, appea thinne, and squashed objects would be thicke. Skeletons that have natual paametization, such as lines, spline cuves and patches, ae paticulaly suited. Functions of the paamete(s) of the neaest point on the skeleton can be used to modify the field function. A few ideas fo possible modifications to blending functions ae discussed in the next thee sections. Sufaces of Revolution and Genealized Cylindes The offset suface of a line is a cylinde. The thickness of the cylinde is detemined by the adius of influence R, and the blending function. If the adius of influence R is not teated as a constant, but athe as a function of the paamete of the neaest point on the line, R=!t(tp), F = h(, R). the esult is a cylinde of vaying thickness, which is actually a suface of evolution of the cuve!t aound the line. The cup in Figue 15 is an offset of a line skeleton with the thickness being contolled by a sine function. If a cylinde is intesected with a plane pependicula to the axis, the coss section is a cicle. N oncicula coss sections can be achieved by applying yet anothe modification to the blending function. If an additional efeence vecto that is pependicula to the line is defined, the thickness of the cylinde can be vaied with a function of the angle (l' between the efeence vecto V ej and the vecto fom the neaest point on the line to the point on the suface (Figue 12): R =!t(tp).!j(a), F = h(, R).

8 243 Figue 13: Paametes that can be used to modify the blending function of a cicle o tous (01 = a,f3 = b). Twisted sufaces may be modeled by otating the efeence vecto along the line. Simila modifications also apply to geneal cylindes - offset sufaces of spline cuves. Howeve, the consistency of the efeence vecto along the spline is moe difficult to maintain, as descibed in [Bloomenthal 90a]. Disc and Tous A disc is an offset suface of a plana cicula disc, and a tous is an offset of the cicula line (Figue 13) The skeleton is defined by a point C (the cente of the cicle), a nomal to the plane of the cicle, and a adius. The adius can be given by the efeence vecto Ve!, which is pependicula to the cicle nomal. The only diffeence between a disc and a tous is in the way the neaest point on the skeleton (Qd and Qt espectively) is calculated. The additional paametes that can be used to modify the blending function ae (see Figue 13): angle o between the efeence vecto V e! and the CQ vecto, distance between points C and Qd (disc only), angle f3 between the cicle nomal ij and the QtP vecto (tous only). The base of the cup in Figue 15 demonstates how the fist two of these paametes can be used. Random D efomations Blending functions can be petubed by solid noise functions, such as those descibed in [Pelin 85, Peachey 85, Lewis 89]: F(P) = h(p). (1 + CN Noise( P)). NoiseO is a scala function that etuns a value in the ange [-1, 1] fo a point in 3D space. The value of the noise function is computed by smooth intepolation between the pseudoandom values assigned to the points on a 3D intege lattice [Pelin 85]. Any blending Figue 14: Bumpy Donut. The bumps ae built into the suface by incopoating the solid noise function into the blending function. function descibed in this pape can be used fo such an intepolation. In ode to assue that the gadient of the noise function is continuous, at least a cubic intepolation function should be used. The amplitude of the defomation thus poduced is contolled by CN, and the fequency by the size of the intege lattice. Multiple noise functions with diffeent amplitudes and fequencies may be applied to the same suface. The esult of applying such a noise function (see Figue 14) is equivalent to that poduced by having a point skeleton with a andom maximum intensity and a adius of influence equal to the size of the gid, at evey point on the intege lattice. Applications to Animation Skeletally defined implicit sufaces have been used successfully in compute animation [Wyvill 86a]. They ae paticulaly suitable fo animating objects that change shape as they move. In this ealie wok models change thei shape by alteing the elative positions of skeletons. This shape change can be diven fom a key fame appoach [Wyvill 88], whee paametes goven the kinematic elationship between skeletal elements, o using dynamic simulation [Tezopoulos 89]. In this latte wok a liquid like appeaance was obtained using blended spheical paticles. One of the advantages of using spline cuves to povide ou blending functions is that the positions of the contol points can be changed ove time to alte the blending fom say soft to had. Figue 11 shows suitable extemes that can be used fo such an animation. The shape change may be used whee a soft looking object e.g. a cloud, undegoes metamophosis into a had object such as the cup shown in Figue 15. The shape of the sufaces of evolution is contolled by

9 244 Figue 16: Bumpy Donut Tophy. This is the esult of an ealy attempt to model the cup fom Figue 15. The side of the cup is too thin and the suface is polygonized at an insufficient unifom ate of sampling. The effect is easily epoducible and can be used to ceate some exciting foms. Figue 15: Cocktail a la Doughnut de Bump. The cup is a suface of evolution (cylinde) contolled by a sine function of the paamete. The inteio is taken out by a smalle negative cylinde. The base is an offset of a cicle (disc) contolled by a cosine function of the angle ex (see Figue 13) and by a linea function of the distance fom the cente. The staw is a staight line cylinde. The cup and the staw use a vey had actangent (Cl = 100) blending function, while the base and the donut use a soft cubic function. Thee is a modeate amount of blending between "had" and "soft" sufaces, and vey little blending between two "had" sufaces. a function of the paamete of the neaest point on the axis of otation. Functions of othe paametes can be used to modify the blending. These functions can also be changed ove time by alteing the positions of the contol points. This leads to a vey smooth alteation of shape with time and some exciting possibilities fo unusual metamophic opeations fo compute animation. (E.g. it would be elatively simple to change a glass into a bottle.) Conclusion Techniques that ae pesented in this pape povide good contol ove the way implicit sufaces blend togethe, and to a cetain degee, ove the shape of the entie blended suface. In tems of contol flexibility and suface fitting ability implicit sufaces, in geneal, still do not paallel paametic sufaces. In cetain cases, howeve, especially when modeling objects that have a well defined skeletal stuctue, vey good esults can be achieved. Of the vaious blending functions with which we have expeimented thee ae cetain advantages and disadvantages to each. These ae summaized in Table 1. One of the featues of ou appoach is that all the tools wok systematically so that the blending functions can be applied to any of the skeleton shape functions and any of the diffeent shapes can be blended togethe to fom complex shapes that would extemely difficult o tedious to make with othe modeling techniques.

10 245 Smoothness Computa- Blending function Flexibili ty (ode of tional Best used fo continuity) Efficiency cubic polynomial in ~ none good (1) excellent constant soft blending intepolating cubic spline fai good (1) good geneal pupose functions non paametic Bezie cuve good excellent (any) fai smooth blending non paametic B-spline vey good vey good (any) fai wide ange of blending actangent good good (2 appox.) vey good had blending paametic spline vey good good (0-1) fai wide ange of blending Table 1 : Compaison of diffeent blending functions. The flexibility efes to the ease of specifying a wide ange of blending, fom soft to had. The computational efficiency is elative to the cubic function. The ac tangent function is a few times slowe while spline cuves can easily take an ode of magnitude longe to compute, depending on the numbe of contol points and the degee of the cuve. Acknowledgements We would like to thank all those that have paticipated in the Gaphicsland poject at the Univesity of Calgay, paticulaly Caol Wang, Mike Hemann, Geoff Wyvill (Univesity of Otago) and Jules Bloomenthal (Xeox PARC). This wok is patially suppoted by the Natual Sciences and Engineeing Reseach Council of Canada. [Beie 90] [Blinn 82] Refeences Thaddeus Beie. Pactical uses fo implicit sufaces in animation. In SIGGRAPH '90 Couse Notes 23, Modeling and Animating with Implicit Sufaces, August J. F. Blinn. A genealization of algebaic suface dawing. A CM Tansactions on Gaphics, 1(1): , July [Bloomenthal 88] J. Bloomenthal. Polygonization of implicit sufaces. Compute Aided Geometic Design, 5(4): , Novembe [Bloomenthal 90a] J. Bloomenthal. Calculation of efeence fames along a space cuve. In A. S. Glassne, edito, Gaphics Gems, pages Academic Pess, [Bloomenthal 90b] J. Bloomenthal and B. Wyvill. Inteactive techniques fo implicit modeling. Compute Gaphics, 24(2): , Mach [Fain 88] Geald Fain. Cuves and Sufaces fo Compute Aided Geometic Design. Academic Pess, [Lewis 89] J. P. Lewis. Algoithms fo solid noise synthesis. Compute Gaphics SIGGRAPH '89, 23(3): , July [Midlleditch 85] A. E. Midlleditch and K. H. Seas. Blend sufaces fo set theoetic volume modeling systems. Compute Gaphics SIGGRAPH '85, 19(3): , July [Peachey 85] D. R. Peachey. Solid textuing of complex sufaces. Compute Gaphics SIGGRAPH '85, 19(3): , July [Pelin 85] K. Pelin. An image synthesize. Compute Gaphics SIGGRAPH '85, 19(3): , July [Rockwood 87] [Tezopoulos 89] [Waen 89] A. P. Rockwood and J. C. Owen. Blending sufaces in solid modeling. In G. E. Fain, edito, Geometic Mode/ing: Algoithms and New Tends, pages SIAM, Demeti Tezopoulos, John Platt, and Kut Fleische. Heating and melting defomable models (fom goop to glop). Poc. Gaphics Inteface 1989, pages , Joe Waen. Blending algebaic sufaces. A CM Tansactions on Gaphics, 8(4): , Octobe [Wyvill 86a] B. Wyvill, C. McPheetes, and G. Wyvill. Animating soft objects. Visual Compute, 2(4) : , A u gust [Wyvill 86b] G. Wyvill, C. McPheetes, and B. Wyvill. Data stuctue fo soft objects. Visual Compute, 2(4): , August [Wyvill 88] Bian Wyvill. The geat tain ubbey. SIGGRAPH 88 Electonic Theate and Video Review, Issue 26, 1988.