B-spline curves. 1. Properties of the B-spline curve. control of the curve shape as opposed to global control by using a special set of blending

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1 B-sple crve YZU Optmal Desg Laboratory. All rghts reserved. Last pdated: Yeh-Lag Hs (--9). ote: Ths s the corse materal for ME Geometrc modelg ad compter graphcs, Ya Ze Uversty. art of ths materal s adapted from CAD/CAM Theory ad ractce, by Ibrahm Zed, McGraw-Hll, 99. Ths materal s be sed strctly for teachg ad learg of ths corse. B-sple crves. ropertes of the B-sple crve B-sple crves are powerfl geeralzato of Bezer crves. They provde local cotrol of the crve shape as opposed to global cotrol by sg a specal set of bledg fctos that provde local flece. They also provde the ablty to add cotrol pots wthot creasg the degree of the crve. B-sple crves have the ablty to terpolate or approxmate a set of gve data pots. Iterpolato s sefl dsplayg desg or egeerg reslts sch as stress or dsplacemet dstrbto a part whle approxmato s good to desg free-form crves. Iterpolato s also sefl f the desger has measred data pots had that mst le o the resltg crve. Assgmet Use yor CAD software to draw several B-sple crves. Use the same cotrol pots to draw both approxmato crves ad terpolato crves. Report yor observatos. The theory of B-sple crves separates the degree of the resltg crve from the mber of the gve cotrol pots. The B-sple crve defed by + cotrol pots s gve by

2 B-sple crve where,, max (), are the B-sple fctos sggested by Cox ad de Boor 97. Frst, the parameter cotrols the degree (-) of the resltg B-sple crve ad s sally depedet of the mber of cotrol pots. The B-sple fctos have the followg propertes: artto of ty:, ostvty:, Local spport:, f, Cotty:, s (-) tmes cotosly dfferetable The frst property esres that the relatoshp betwee the crve ad ts defg cotrol pots s varat der affe trasformatos. The secod property garatees that the crve segmet les completely wth the covex hll of. The thrd property dcates that each segmet of a B-sple crve s fleced by oly cotrol pots or each cotrol pot affects oly crve segmets, as show Fgre. It s sefl to otce that the Berste polyomal, B,, has the same frst two propertes metoed above.

3 B-sple crve * ** Fgre. Local cotrol of B-sple crves. Assgmet Assme a set of two-dmesoal cotrol pots the x-y plae, the B-sple crve defed by ths set of cotrol pots s,. Assme a set of coeffcets the followg affe trasformato dscssed Chapter to trasform () to get ˆ : x a y a xx yx x a x a xy yy y b x y b y Use the same affe trasformato to trasform to get crve defed by ths set of cotrol pots s ˆ =, ad the B-sple,. Show that sg partto of ty dscssed above. Ths also esres that the relatoshp betwee the crve ad ts defg cotrol pots s varat der affe trasformatos. Ths s a hard qesto. Yo do ot have to prove t for geeral cases. Yo jst have to assme a set of cotrol pots ad coeffcets for the affe trasformato, ad show that ths s tre for the mbers yo assmed.

4 B-sple crve Assgmet Use yor CAD software to draw crves smlar to those Fgre. Idcate the local cotrol of the crves.. The B-sple fcto The B-sple fcto also has the property of recrso, whch s defed as,,, () where,, (), otherwse The are called parametrc ots or ot vales. For a ope crve,, j j,, j j j () where ad the rage of s j () () Relato () shows that (++) ots are eeded to create a (-) degree crve defed by (+) cotrol pots. (7) Ths relato shows that a mmm of two, three, ad for cotrol pots are reqred to defe a lear, qadratc, ad cbc B-sple crve respectvely. A cbc B-sple s sffcet for a large mber of applcatos. Fgre shows the shapes of the B-sple fctos.

5 B-sple crve. (a) Lear fcto ( = ).7.. (b) Qadratc fcto ( = ). (c) Cbc fcto ( = ) Fgre. B-sple fctos.. A Example for Dervg the B-sple Fcto To derstad the recrsve atre of the B-sple fcto, the B-sple fcto for a cbc sple crve defed by cotrol pots s derved. Ths cbc sple has ad. Eght ots are eeded to calclate the B-sple fctos. Eqato () gves the ot vector as 7 The rage of (Eqato ()) s. Eqato () gves, (8),,,, To calclate the above B-sple fctos, se Eqato () ad () together wth the ot vector as follows:

6 B-sple crve,,,,, elsewhere,,, elsewhere,,,,, elsewhere,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 7 7,,,,,,,,,,,,,,,,,,,,,

7 B-sple crve,,,,, 7 7,,,,,,,,,,,,,, 7,, 7 Sbstttg, to Eqato (8) gves,, Sbstttg, to ths eqato gves the crve eqato as, Ths eqato s the same as the oe for the Bezer crve wth the same cotrol pots. Ths the cbc B-sple crve defed by for cotrol pots s detcal to the cbc Bezer crve defed by the same pots. Ths fact ca be geeralzed for a (-)-degree crve defed by cotrol pots. Assgmet Derve the B-sple fctos for =, =, ad =, =. Do t oce yor lfetme. Derve the B-sple fctos for =, =, ad shows ts local cotrol capablty. The gap betwee the eghborg ot vales defe Eqato () s always form, wth the vale of. These ots are called form ots ad a B-sple crve based o form ots s called a form B-sple crve. As we modfy the shape of a crve, we ofte add or delete ot vales ad so prodce o-form gaps betwee the ots, whch reslt o-form B-sple crves. o-form B-sples are 7

8 B-sple crve cosdered the geeral form of B-sple crves. Ths most CAD systems provde the capablty of creatg ad modfyg o-form B-sple.. Ratoal Crves A ratoal crve s defed by the algebrac rato of two polyomals whle a oratoal crve s defed by oe polyomal. I a ratoal crve, the cotrol pots, y, z are gve homogeeos coordates x h, y h, z h, h x. Ths the coordates of a pot o a ratoal crve the homogeeos space, x h, y h, z h, h s obtaed from x h y h z h h h x h y h z, (9), (), () h, () Ths a ratoal B-sple crve ca be expressed as R,, max () where R, are the ratoal B-sple bass fctos ad are gve by h, R, () h, The above eqato shows that R fctos,. If we sbsttte bass fctos R oratoal B-sple coterparts., are a geeralzato of the oratoal bass h the eqato, R. The ratoal,,, have early all the aalytc ad geometrc characterstcs of ther 8

9 B-sple crve Assgmet Show that the ratoal B-sple bass fctos propertes: partto of ty ad postvty. R, have the followg to se ote that The ma dfferece betwee ratoal ad oratoal B-sple crves s the ablty h at each cotrol pot to cotrol the behavor of the ratoal B-sples. h s do ot have to be tegers. The greater the vale of a partclar closer the crve s plled toward the cotrol pot. h, the Ratoal sples provde a exact represetato for qadrc crves (cocs), sch as crcles ad ellpses. oratoal sples, whch are polyomals, ca oly approxmate cocs. Ths allows graphcs pacages to model all crve shapes wth oe represetato, ratoal sples, wthot eedg a lbrary of crve fctos to hadle dfferet desg shapes. For example, sg three cotrol pots,,,,, homogeeos coordates h =h =, h cos ad ther, a crclar arc ca be defed as: () Ad x y () The we ca easly show that x y, that s, Assgmet s a exact crclar arc. Wrte a program Matlab to draw a fll crcle sg Eqatos () ad (). Draw a crcle the tradtoal way ad compare the two crcles. Show yor Matlab program too. 9