Algebraic sums for deformation of constructive solids

‘Ilk paper ck!als wi[h an inlcrac[ivc scuipling [cchniquc based on local tfcformsil{ms I)] a cons[ruc[ivc solid by a se[ of arbitrary potnts that arc assumed (O belong [o Ihc rcsullanl surface. I<eprcsenlation of a solid by a real continuous function of three variables is used, ‘Ilc [heory of R-functions is applied for scltheoretic operatlorrs on solids. in contrast m the existing methods based on space mapping, \ve construc~ a displacement function that irmxpolatcs values t)f the defining function in given control points. “I”henan algebraic sum (difference) of [he defining function and [hc displactmcnt function describes the deformed solid. B]obby dcforrna[ion, dcformaliort with a volurnc splint, and deformation con[rollcd by a parametric curve arc (fiscusscd. f310bby deformation mmfcls pinching and pricking that resull in small bUtnpS, ‘1’() define rnorc! global dcforma[iorr we usc the volurnc spl[nc lurrctmn dcrlvcd forscattcrccf points and possessing minimal energy propcrt!. “[lis approach has been applied [o reconstruct solids from given surface points. Conlml (If dcformatiorr wilh a p:ir~mctric curve all(}vs (() produce sweep-llkc shapes placed on inltlai surlaccs. 1 Deformations and algebraic sums ‘~he sculpting metaphor of 31) shape mmfehng M inluitive and natural onc for designers, l~xwting dcf’ormali(m mclhocfs arc based (m space mappings ctmtrolled by numcncal paramclcrs [3, 23], by point Iattlces [22, 9] and arbitrary con(rol points [6, 12, 71. I;recIorm dcf’ormatiorrs propowxf by Sederberg and Parry [22] and cx(urded In [9, 12] can bc cffecu~ely applied m polygonal and parametric surfaces. On (hc other hand, the inverse mapping for implicit surfaces requires time-consuming subdivisions or an ilcrativc search. “f’he general deformation techniques [6, 7] providing forward and inverse mapping better suit to implicit surfwes, Most of the above-mentioned deformation methods are real] y global to provide a series of small bumps defined by arbitrary points. Although the method of Borrel and Rappopor[ [7] has been designed for localized space mapping, it can Icad to non intuitive results whur bounding spheres of several control points intersecl. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copyin is by permission of the Association of Computing ? Machinery. o copy otherwise, or to republish, requires a fee and/or specific permission. Solid Modeling ’95, Salt Lake City, Utah USA @ 1995 ACM 0-89791 -672 -7/95 /0005 ...$3.50 The problem slated in our work Mm c[mtr(d local dcflmnall{)ns of a solld by a sw of arbitrary polrrls and curves that arc assumuf [o belong [Oihc surface of the rcsultanl solid. NOWthat inlua] posttlon of a point on the surfi~cc M not rcqulrcd. Wc would Ilkc m support crcaling anew fcatureby addition ofjus[ onc control”pi)lnt. Pmchlng and pricking that result in small bumps can bc dcal( [n this $lay. If we assume mort smooth transition of the rcsul(an[ surface bctwccn control points, more global deformation can be obhmcd, We use representation of a solid by a continuous real function of” three variables f(x,y,z) >0. Wc call i[ a de@ring/imccion of a sol id, A zero set of such a function is usually called an implicit surface. We intend to apply algebraic operations (sum and difference) to implement deformations of functionally rcpreserrwd solids. Algebraic operations on defining functi(ms have been already applwd 10 describe solids with implicit surfaces and gCometrlC operations on them. Descriptions of implicit surfaces try bhhby model [41, mctaballs [14], sofl (Jbjccts [2!6]and dlsianw functl(]ns [5, 13] arc essentially based on algebraic sums ofdcflnlrtg f’unctlorrs. It provides a possibility LOdeform an object hy adding nc~v primitives to i[s skeleton. Deformations t)f dlslancc sur!accs in collisions by algebraic diffcrcncc of defining field functions have been proposed by Gascuel [1 1]. IIowcvcr, to construct an ohjccl of this kind one “must choose and place overlapping primitives with great skill in a most non intuit!vc way” [27], A blend surfwe can be described in terms of algebraic opttrati(ms on defining functions [25]. “Ilesc operations arc applied in practice to solid primitives but not t{)c[mstructivc solids (we, for example, [X]). ‘f’he theory of R-functions [19] provides means of functmr representation of solids constructed by the standard (nonregularizcd) set opcratiorrs (SW [21 ] for a survey). Shapiro [2 1] applies algebraic difference to construc[ ~ real function (fcfining a regular solids required in corrstructivc solid geometry (CS(;). 131cnding, offsetting and metamorphosis opera[ilms have been defined in [17,1X,15] by algebraic sums applied t{) R-function based exact descriptions of constructive solids, I’asko and Savchenko [16] describe stochastic deformation of a solid using an algebraic sum with so-called “sulid noise” functi[m, “[o model these geometric operations we use a high-level geometric language [1] supporting mixed algebraic and set operations (m dcfinlng functions. As this brief overview shows, algebraic opcratiorrs on dcf]nlng functions give a promising tool for complex transfmmatluns of geometric solids. The general idea of our approach to dcformati[)rrs is to construct a displacement furrctmn that interpolates values (JI the defining function in given control pmnts, “I”hcn Agcbralc