Representing Curves. Representing Curves. 3D Objects Representation. Objects Representation. General Techniques. Curves Representation
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1 Reresentng Crves Fole & n Dm, Chter Reresentng Crves otvtons ehnqes for Ojet Reresentton Crves Reresentton Free Form Reresentton Aromton n Interolton Prmetr Polnomls Prmetr n eometr Contnt Polnoml Slnes ermte Interolton D Ojets Reresentton Sol oelng ttemts to evelo methos n lgorthms to moel n reresent rel ojets omters Ojets Reresentton hree tes of ojets n D: D rves D srfes D ojets We nee to reresent ojets when: oelng of estng ojets D sn - moelng s not rese oelng new ojet from srth CAD - moelng s rese - ntertve sltng ltes enerl ehnqes Crves Reresentton Prmtve Bse: A omoston of smle omonents Not rese Effent n smle Free Form: lol reresentton, rve mnfols Prese Comlte Sttstl: oelng of ojets generte sttstl henomen, sh s fog, trees, ros
2 Prmtve Bse Reresentton Lne segments: A rve s romte olleton of onnete lne segments Free Form Reresenttons Elt form: = f, f, mst e fnton Not rotton nvrnt reresentton Dfflt to reresent vertl tngents Imlt form: f,, = Dfflt to onnet two rves n smooth mnner Not effent for rwng sefl for testng ojet nse/otse Prmetr: t, t, t A mng from [,] R er ommon n moelng Free Form Reresenttons Aromte vs. Interolte Crves Emle: A Crle of rs R Imlt: + + -R = & = Prmetr: = R os = R sn = ven set of ontrol onts P nown to e on the rve, fn rmetr rve tht nteroltes/romtes the onts P P P P P 4 P P P Interoltng Crve Aromtng Crve P P 4 Prmetr Polnomls For nteroltng n onts we nee olnoml of egree n- Emle: Lner olnoml. For nteroltng onts we nee olnoml of egree 4 Emle: Lner Polnoml he geometrl onstrnts for re: P ; P Solvng the oeffents for we get: P ; P P P P P Solvng for [ ] we get: P P P = =
3 Emle: Lner Polnoml = = P P P 4 4 Prmetr Polnomls Polnoml nterolton hs severl svntges: Polnoml oeffents re geometrll menngless Polnomls of hgh egree ntroe nwnte wggles Polnomls of low egree gve lttle flelt f [, ] where otherwse Solton: Polnoml Slnes 4 Polnoml Slnes Peewse, low egree, olnoml rves, wth ontnos jonts C C C f [, ] where C otherwse Avntges: Rh reresentton eometrll menng oeffents Lol effets Intertve sltng ltes 4 4 ngent etor Let =[,, ], [,] e ontnos nvrte rmetr rve n R he tngent vetor t,, s: m e thoght of s the trjetor of ont n tme In ths se, s the nstntneos velot vetor t tme = = Prmetr Contnt Let n, [,], e two rmetr rves Level of rmetr ontnt of the rves t the jont etween n : C - : he jont s sontnos, C : Postonl ontnos, = C : ngent ontnos, C & = C, >: Contnos to the -th ervtve, j = j, j eometr Contnt In omter e geometr esgn, we lso onser the noton of geometr ontnt: -, : Sme s C - n C : Sme tngent reton: = : All ervtves to the -th orer re roortonl ven set of onts {}: A eewse onstnt nterolnt s C - A eewse lner nterolnt s C C - C C
4 Prmetr n eometr Contnt - oth C n - s t not C S-C s C S-C s C S-C s C In generl, C mles not ve vers Eeton when the tngents re ero Prmetr C Crves C olnomls efnng rve n R hve the form: Where s n [,]. Defnng: he rve n e rewrtten s: n Prmetr C Crves he oeffents re nnown n shol e etermne For ths rose we hve to sl 4 geometrl onstrnts Dfferent tes of onstrnts efne fferent tes of Slnes ermte Crves Assme we hve n ontrol onts { } wth ther tngents { } W.L.O.. reresents rmetr fnton for the seton etween n + For we hve the followng geometr onstrnts: = ; = + '= ; '= + P + + ermte Crves Sne we hve tht We n wrte the onstrnts n mtr form: An ths Where ermte Crves [ ] eometr mtr Blenng fntons
5 ermte Crves ermte Crves..8 = = Chnge n gnte of.6.4. = = ermte Blenng Fntons P P Chnge n Dreton of Proertes: ermte Crves he ermte rve s omose of lner omntons of tngents n lotons for eh Alterntvel, the rve s lner omnton of ermte ss fntons the mtr n Drw: ermte Crves Reqres the sefton of the tngents hs nformton s not lws vlle It n e se to rete geometrll nttve rves he eewse nterolton sheme s C ontnos he lenng fntons hve lol sort; hngng ontrol ont or tngent vetor, hnges ts lol neghorhoo whle levng the rest nhnge
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