Hermite Curves. Jim Armstrong Singularity November 2005
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1 TechNoe TN-5- Herie Curves Ji Arsrong Singulariy Noveer 5 This is he second in a series of TechNoes on he sujec of applied curve aheaics in Adoe Flash TM. Each TechNoe provides a aheaical foundaion for a se of Acionscrip eaples. Herie Curves Herie curves are a foundaion of ineracive curve design. All curve design is concerned wih he creaion of sooh curves ased on a sall nuer of user-conrolled paraeers. Applicaions of generaed curves include cross-secional eleens for erusion, par of a syse for creaing D surfaces, inerpolaion eween keyfraes in aniaion, and ploing. I is coon o anchor he curve a wo endpoins, using oher conrols o vary he shape of he curve. Cuic polynoials are very popular in curve design. Two of he four condiions required o specify a cuic are aken y placeen of conrol poins. The oher wo condiions are open o adjus he shape of he curve. Herie curves are designed using wo conrol poins and angen segens a each conrol poin. Tangen handles ay e ineracively dragged o adjus he shape of he curve once conrol poins are placed. While very siple o ipleen, Herie curves have a nuer of pracical drawacks. For eaple, i can e seen fro he Acionscrip deo ha i is difficul o deerine how long o ake a angen handle in order o produce a desired shape. Copensaing for hese drawacks leads naurally ino Bezier curves. Alhough cuic Bezier curves are of priary ineres, in-deph discussion of quadraic Bezier curves is helpful in undersanding he MovieClip.curveTo funcion. In addiion o deriving he equaions o generae Herie curves, his paper provides ackground ha is used in susequen TechNoes. Coninuiy I is rare ha an applicaion requires he generaion of a single curve. Cople curves are ofen creaed segen-y-segen, using sipler curves for each segen. The naure of he coplee curve is deerined y coninuiy condiions a each join poin. If wo curve segens have he sae value a a join poin, he overall curve is said o ehii G or geoeric coninuiy. G coninuiy iplies ha curve segens have aching values a a join poin and firs derivaives ach in direcion, u no in agniude. The overall curve hugs he angen line ore on one side of he join poin han he oher. Anoher ype of coninuiy is called paraeric coninuiy. If wo curve segens have aching agniude and direcion of he k-h derivaive a a join poin, he curve is said o e k C coninuous.
2 Generally speaking, G -coninuous curves appear jus as sooh as C near join poins. The disincion ecoes greaer furher away fro join poins. If a curve is used o inerpolae eween keyfraes, i is coon o consider he aniaed ariue as paraeerized over ie,. As he ariue passes hrough a keyfrae, i is iporan o ainain consisen velociy and acceleraion o avoid jerky oion. C coninuiy is enforced in hese cases even hough a C - coninuous curve ay see o e jus as sooh very near join poins. General For of araeric Equaions Many curves are defined on a paraeer,. Insead of y f, he curve is defined as f and y g, ε [, ]. In order o quickly copare he aheaical aspecs of differen curves, i is helpful o have a coon fraework for deriving he equaions of paraeric curves. One popular ehod is o descrie he curve as a ari equaion involving a asis ari, M, a geoery vecor, g, and a polynoial vecor, p. Define, k k p k [ ] g g g ] [ g k The general for for a paraeric curve in can e wrien as θ [ y z ] p Mg [] The geoery vecor or geoeric consrains are deerined y endpoins, angen vecors, or oher conrol poins used o define and consrain he curve s shape. Equaion [] ay e a i difficul for soe prograers o undersand. Curve equaions generally ake he sae for for, y, and z-coponens, so consider only he equaion for. Le denoe he -coponen of he firs geoeric consrain and consider he equaion of he g -coponen of a cuic curve. Equaion [] siplifies o 4 g g 4 [ ] [] 4g g4 k Soe readers ay find i easier o undersand equaion [] in is fully epanded for, 4 g 4 g g 44 g 4 []
3 Noice ha he geoeric consrains are uliplied coponen-y-coponen y lending funcions, represened y he coponens of p k M. The asis ari, M, deerines he coefficiens of he lending funcions, which give he curve is unique naure or shape. I is coon o differeniae eween differen ypes of curves ased on geoeric consrains and he asis ari. The ne secion discusses how he asis ari is derived for Herie curves. Herie Basis Mari The Herie asis ari is copued y wriing a single equaion for each of he polynoial coefficiens in he lending funcions. The geoeric consrains can e wrien in he for, g R R ] [ Where and are he conrol poins, R represens he angen vecor a he firs conrol poin, and R represens he angen vecor a he second conrol poin, as illusraed in he following diagra. Considering only he -coponen of he Herie curve as he equaions for oher coponens ' ' are he sae, he required equaions are for,,,. The general for for is and o yield a which is copared o equaion [] a a a a [ ] Mg [ ] Mg Since ' [ ] Mg,
4 ] Mg [ ' ] Mg [ ' In ari for, hese equaions can e wrien as * * Mg g Where g ] ' ' [ * The only way his equaion can e saisfied is, A A M The coefficiens of M are easily solved for y seing I AA, fro which M This yields he lending funcions 4 A siple way o epress he -coponen of he Herie curve is R R [4]
5 An alernae epression ha is a i ore copuaionally efficien is [5] 4 R 4 R An even ore efficien approach is o collec polynoial wih nesed uliplicaion.,, and consan ers and hen evaluae he A graph of he lending funcions is shown elow. Noice ha he lending funcions weighing he conrol poins end o overwhel hose weighing he angen vecors. This is ore noiceale in he Acionscrip eaple. iecewise Herie Curve Segens Drawing a single segen of a Herie curve is relaively sraighforward. If desired, a ore cople shape can e creaed wih piecewise Herie curve segens. I is necessary o decide wheher or no o enforce geoeric or ore resricive paraeric coninuiy. If G coninuiy is enforced, hen he geoeric consrains of he wo curve segens a a join poin are R R αr R curve segen lef of join curve segen righ of join
6 where α is a scalar uliplier. If C coninuiy is enforced, hen α. The user inerface ecoes ore coplicaed wih G coninuiy as he direcion of he firs angen handle of he righ curve righ, relaive o join poin us ainain a fied direcion. Is agniude only affecs he righ curve segen. You can iagine how edious i igh e for a user o consruc a cople shape fro a series of Herie curves. In fac, i is no easy o consruc desired shapes for a single segen as he angen handles have o e oved very far away fro he conrol poins o produce significan ends his is eviden fro sudying he graph of he lending funcions. For soe curves, oaining a desired shape requires dragging angen handles ouside he availale display area. An old rick o alleviae his difficuly was o arificially weigh he lending funcions for he angen vecors. Increasing weigh causes he curve o hug each angen segen ore, u producing a desired shape requires even ore work as wo ses of inpus angen handles and weighs us e adjused. I is possile o rela he need for angen handles y convering a Herie curve ino a cardinal spline. A popular for is he TCB Tension-Coninuiy-Bias spline, feaured in os odern D packages. The forulas for auoaically copuing angens were inroduced in []. Anoher ype of curve ha is closely relaed o Herie is Bezier. Bezier curves are copued using nohing u conrol poins and a asis of Bernsein polynoials. This is he sujec of he ne TechNoe in his series. References Ahlerg, J.H., Nielson, E.N., and Walsh, J.L., The Theory of Splines and Their Applicaions, Acadeic ress, NY, 967. Kochaneck, D. and Barels, R., Inerpolaing Splines wih Local Tension, Coninuiy, and Bias conrol, roceedings of he h Inernaional Conference on Copuer Graphics and Ineracive Techniques, ACM ress, pp. -4, 984.
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