Geometric Modeling
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1 Geometrc Modelg Crves Morteso Chater -5 ad Agel Chater 9
2 Crve Bascs Crve: Locs of a ot movg wth degree of freedom. Some tyes of eqatos to descrbe crves: Itrsc o relace o exteral frame of referece Exlct Vale of deedet varable terms of deedet varables e.g. y = f x Imlct e.g. f x,y = 0 Parametrc Exress vale of each satal varable terms of a deedet varable the arameter e.g. for arameter D: x = x y = y z = z sorce: Morteso, Agel
3 Itrsc Defto o relace o exteral frame of referece Reqres eqatos as fctos of arc legth* s: Crvatre: Torso: f s gs Torso D measres how mch crve devates from a lae crve. or lae crves, alteratvely: d ds *legth measred alog the crve Treated more detal Chater of Morteso ad Chaters 0, 9 of ar. sorce: Morteso
4 Exlct orm Vale of deedet varable terms of deedet varables e.g. D: oe y vale for each x vale: y = f x e.g. D: y = f x ad z = g x Axs-deedet ot garateed to exst e.g. for D crcle oly oe half s descrbed by: Secal cases ca be roblematc e.g. y = mx + h s arorate for vertcal les y r x sorce: Morteso, Agel
5 Imlct orm Geeral form: f x,y = 0 Examles: Straght le: Ax + By + C=0 Coc crve: Ax + Bxy + Cy + Dx + Ey + =0 Axs-deedet Ca rereset all les ad crcles ad more Some mlct forms are hard to arameterze. sorce: Morteso, Agel
6 Parametrc orm Exress vale of each satal varable terms of a deedet varable the arameter e.g. for arameter D: x = x y = y z = z or a crve segmet, tycally [0,] Crve segmets ca be joed to form comoste sle* bedable crves. Parametrc form of a gve crve s ot ecessarly qe. Every arametrc form has a mlct form. *see Morteso. 05 or ar. 67 for sle defto as a mmm eergy crve. sorce: Morteso, Agel
7 Parametrc orm taget vector* vector Vector comoets of : x dx d y dy d z dz d *less otherwse oted, taget vectors ot draw to scale. sorce: Morteso
8 Parametrc orm Advatages Allow searato of varables ad drect comtato of ot coordates. Accommodate all sloes. Boded whe arameter les a boded terval. More degrees of freedom to cotrol crve shae. Varos forms of cotty ca be eforced at crve segmet jo ots. See Morteso.46 for more sorce: Morteso
9 Cotty at Jo Pots Dscotos: hyscal searato Parametrc Cotty Postoal C 0 : o hyscal searato C : C 0 ad matchg frst dervatves C : C ad matchg secod dervatves Geometrc Cotty Postoal G 0 = C 0 Tagetal G : G 0 ad tagets are roortoal, ot same drecto, bt magtdes may dffer Crvatre G : G ad taget legths are the same ad rate of legth chage s the same sorce: Morteso, Agel, Wk
10 4 Tycal Tyes of Parametrc Crves Cotrol ots flece crve shae. Iterolatg Crve asses throgh all cotrol ots. Hermte Defed by ts edots ad taget vectors at edots. Iterolates all ts cotrol ots. ot varat der affe trasformatos. Secal case of Bezer ad B-Sle. Bezer Iterolates frst ad last cotrol ots. Crve s taget to frst ad last segmets of cotrol olygo. Easy to sbdvde. Crve segmet les wth covex hll of cotrol olygo. Varato-dmshg. Secal case of B-sle. B-Sle ot garateed to terolate cotrol ots. Ivarat der affe trasformatos. Crve segmet les wth covex hll of cotrol olygo. Varato-dmshg. Greater local cotrol tha Bezer. sorce: Morteso, Agel
11 Iterolatg Iterolates all cotrol ots. Geometrc form: b 0 Rarely sed de to lack of dervatve cotty at crve segmet jo ots. cbc case wth eqally saced arameter vales sorce: Agel
12 Hermte Geometrc form cbc case: Hermte crves ca rovde C cotty at crve segmet jo ots. 4 sorce: Morteso 4 0 0
13 Bezer Geometrc form cbc case: Bezer crves ca rovde C cotty at crve segmet jo ots. B 0, B, Berste olyomals. + = mber of cotrol ots = degree + Addg a cotrol ot elevates degree by. B, 0 Covex combato, so Bezer crve ots all le wth covex hll of cotrol olygo. Ratoal form s varat der ersectve trasformato: where h are rojectve sace coordates weghts h B 0 0, h B, sorce: Morteso
14 B-Sle Geometrc form o-form, o-ratoal case, where K cotrols degree K - of bass fctos:,, 0 f t otherwse t 0, K, k t t k, k t t k t k, k t t are kot vales that relate to the cotrol ots. Uform case: sace kots at eqal tervals of. Reeated kots move crve closer to cotrol ots. Cbc B-sles ca rovde C cotty at crve segmet jo ots. Ratoal form URBS s varat der ersectve trasformato, where h are rojectve sace coordates weghts. 0, K Covex combato, so B-sle crve ots all le wth covex hll of cotrol olygo. 0 h 0 h, K, K sorce: Morteso
15 Geometrc Modelg OeGL Demo to accomay HW#
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