AML710 CAD LECTURE 12 CUBIC SPLINE CURVES. Cubic Splines Matrix formulation Normalised cubic splines Alternate end conditions Parabolic blending
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1 CUIC SLINE CURVES Cubc Sples Marx formulao Normalsed cubc sples Alerae ed codos arabolc bledg AML7 CAD LECTURE CUIC SLINE The ame sple comes from he physcal srume sple drafsme use o produce curves A geeral cubc polyomal s represeed by: Mahemacally sple s a pecewse polyomal of degree wh couy of dervaves of order - a he commo jos bewee he segmes Thus, he cubc sple has secod order or C couy aramerc represeao of cubc sple: A sgle segme of cubc sple s represeed as D Cx x Ax y z y x z y x z y x
2 CUIC SLINE cod Here ad are he values of he parameer a he wo eds of he curve segme We ca wre he expaded form of he cubc sple as follows The age vecors are obaed by dffereag he above expresso z y x aramerc orm of Cubc Sple Whou loss of geeraly we ca assume ad wre he ew rage as Solvg for, gves
3 aramerc orm of Cubc Sple Mulple Segme Sple curve Curvaure of paramerc Cubc Sple Curvaure a s couous
4 Mul-segme Cubc Sple Mulplyg boh sdes by 6 6 rs-segme Secod-segme Wh curvaure couy esured by
5 5 Marx form for calculag eror age vecors roperes? 5 5 Marx form for calculag eror age vecors 5 5 R M R M
6 6 eeralzg Coeffces Coeffce Marx ad ledg fucos
7 7 ledg fucos Subsug from prevous eq for [] where / ledg fucos Usg he defo of bledg fucos where
8 ledg fucos 8 6 f f f f ,, Sple roperes The curve s produced wh ed pos age vecors ad values of he parameer The curvaure machg a he eral jos self s o suffce o esure he smoohess To ge a smooh curve he rgh choce of s o mmze he ad values s recommeded As creases he compuaoal effor, for may praccal applcaos we ca ge smooh sples by chord legh approxmao 8
9 9 Algebrac form o eomerc form Le us cosder he algebrac form of a cubc Herme curve Whou loss of geeraly we ca assume ad wre he ew rage as Solvg for algebrac coeffces, gves Subsug he algebrac form ad rearragg erms: Ths ca be smplfed o he followg geomerc form: Where are called geomerc coeffces,,, Normalzed Cubc Sples If daa pos are uformly dsrbued he accepably smooh curves ca be obaed by ormalsg e, ag for each segme The Herme bass fucos or bledg fucos are ow gve as follows: N T N T
10 Normalzed Cubc Sples N T Here [T] ad [N] are cosa for all segmes We eed o deerme [] for each of hem Normalzed Cubc Sples - Example N T The poso vecors of a ormalzed cubc sple are gve as,, - ad The age vecors a he eds are boh The eral age vecors are calculaed ad boh are equal o 8 Calculae he pos o he curve a / ad / for a leas oe segmes a / -> [/ 6/5][ 67] ad a / -> [/ 6/5][667 9]
11 Cubc Sple Ed Codos Clamped Ed codos Example Ths llusraes he use of clamped or complee sple erpolao where ed slopes are prescrbed Zero slopes a he eds of a erpola o he values of a cera dsrbuo are eforced x -:; y [ ]; cs splex,[ y ]; xx lspace-,,; plox,y,'o',xx,ppvalcs,xx,'-'; Alerae Ed Codos Relaxed Ed codos If he curvaure a he eds are assumed zero relaxed or aural ed codo resuls 6 The frs ad las rows of marces [M] ad [R] are modfed respecvely as 6 [ / L ][ ] [ L ][ ] ad Clamped ad Relaxed ed codos ca be mxed 6
12 Alerae Ed Codos Cyclc ed codos Ths requres boh he age vecor ad he curvaures a he eds be equal These codos are useful o produce closed or repeag cubc curves A-cyclc ed codos Effec of Ed Tage Vecor
13 Effec of Tage Vecor Magude Effec of Tage Vecor Magude
14 Effec of Tage Vecor Magude Lmaos of Cubc Sples If he umber of daa pos are large compuao me requred o ver age vecor marx ca be excessve aramerc cubc curves cao represe coc secos exacly They poorly approxmae asympoc curves If o corolled properly, exhb spurous oscllaos Dscoues hrd dervave may gve rse o uwaed fleco pos Soluos: Addg physcal aalog of logudal eso Modfyg he bledg fucos arabolc
15 arabolc ledg A parabolcally bleded curve ca be represeed as: where r, s ad are parameers r, qs are paramerc parabola hrough,, ad,, respecvely as show C p r q s 5
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