Creating Exact Bezier Representations of CST Shapes. David D. Marshall. California Polytechnic State University, San Luis Obispo, CA , USA

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1 Creatig Exact Bezier Represetatios of CST Shapes David D. Marshall Califoria Polytechic State Uiversity, Sa Luis Obispo, CA , USA The paper presets a method of expressig CST shapes pioeered by Kulfa ito stadard Bezier curves ad surfaces. Out of the seve stadard shape classes idetified by Kulfa as represetable usig CST curves, four of them, icludig airfoils, ca be represeted as Bezier curves exactly. For the other three, a coverget series expasio proves isufficiet to achieve ay practical accuracy, however these three classes are ot the most commoly used of the CST classes. With the ability to express most commo CST curves ad surfaces as geeric Bezier curves ad surfaces, CST shapes ca be used i CAD, solid modelig, meshig or ay other geometry tool that does ot explicitly support CST. Nomeclature B i, C N1 N N 1 N p i Pp i,j S Covetios CAGD CST i th Berstei polyomial fuctio of degree Class fuctio fuctio First parameter i class fuctio Secod parameter i class fuctio i th Cotrol poit of CST shape fuctio Bezier surface cotrol poit Shape fuctio Computer Aided Graphics Desig Class Shape Trasformatio I. Itroductio There has bee a icreased iterest i usig the class-shape trasformatio method (CST) 1 to represet shapes i a aerodyamics cotext, especially airfoils. A lot of this attetio is due to its use of relatively few umber of parameters eeded to express the shape. Equatio (1) shows the geeral form of the CST method. ζ (ξ) = C N1 N S (ξ) + ξδζ t.e. (1) where N 1 ad N are class parameters that determie the type of geometry to be represeted, S (ξ) is the shape fuctio ad Δζ t.e. is the trailig edge displacemet. The N 1 ad N terms i (1) are collectively kow as the class term ad are sometimes represeted as C N1 (ξ) = ξ N1 (1 ξ) N () N Table 1 shows combiatios for several commo shapes ad the correspodig class parameters. The shape fuctio, S (ξ), is typically show as either a stadard polyomial Associate Professor, Aerospace Egieerig Departmet, Seior Member AIAA. S (ξ) = a i ξ i (3)

2 or as a Bezier curve S (ξ) = B i, (ξ) p i (4) where B i, (ξ) are the Berstei polyomials ad p i are the cotrol poits which represet the covex polygo cotaiig the Bezier curve. Stadard texts o computer aided graphics desig (CAGD), such the oe by Fari or Piegl ad Tiller, 3 will provide a umber of useful properties of Bezier curves ad Berstei polyomials. Table 1. Values of the class fuctio terms for commo geometries from Kulfa. 1 Idex Geometry Class N 1 N 1 Airfoil 1/ 1 Roud Leadig & Trailig Edges 1/ 1/ 3 Bicovex airfoil or Sharp Edges Miimum Wave Drag 3/4 3/4 5 Low Drag Projectile 3/4 1/4 6 Coe or Wedge Rectagular Shape 0 0 II. Exact Trasformatio of CST Airfoils to Bezier Curves This sectio presets the mathematics behid the trasformatio of a CST airfoil, idex 1 i table 1, ito Bezier curves with the use of approximatios. While the stadard CST represetatio is i a explicit form, it is ζ = f (ξ), it is ecessary to covert it to a parameterized form. The stadard CST represetatio, with N 1 ad N values substituted, is the represeted as a pair of parametric equatios for the ξ- ad ζ-coordiates as ξ = t (5a) ζ = t (1 t) B i, (t) p i + tδζ t.e. (5b) To start, the shape fuctio, which is expressed as a order Bezier curve, ca be coverted to a stadard polyomial, i.e. with moomial basis fuctios. The followig relatios ca be foud i ay thorough referece o Bezier curves, such as Piegl ad Tiller. 3 S (t) = B i, (t) p i i = a i t (6) j ( )( ) j j i where a j = ( 1) p i (7) j i Now the ζ-coordiate equatio is i ζ = t (1 t) a i t + tδζ t.e. (8)

3 Next, the N term of the class fuctio ca be combied with the shape fuctio to obtai a polyomial of order ζ = t a 0 + (a i a i 1 ) t i a t + tδζ t.e. (9) i=1 What is left is a polyomial i t multiplied by a o-polyomial term, t. If equatio (9) ca be expressed as a polyomial, the it ca be coverted to a Bezier curve usig the iverse of equatio (9). Itroducig the parameter substitutio of t s does covert the ξ- ad ζ-coordiate expressios to polyomials ξ = s (+1) (10a) ζ = s a 0 + (a i a i 1 ) s i a s + s Δζ t.e. (10b) i=1 Multiplyig through by s ad simplifyig yields ξ = s (11a) i+1 ζ = a a s +3 0 s + s Δζ t.e. + (a i a i 1 ) s (11b) +3 = b i s i i=1 Δζ t.e. if i = 0 if i eve ad i = where b i = a 0 if i = 1 a if i = + 3 a i a i 1 otherwise The fial step i the process is covertig the polyomials back to a Bezier curve usig the followig relatio that coverts a degree m moomial basis polyomial with coefficiets d j ito a Bezier represetatio with cotrol poits q i i ( ) 1 m j q i = m dj (1) i j i j=0 I usig equatio (1), both the ξ- ad ζ-coordiate expressios must be trasformed. Thus istead of the oe-dimesioal cotrol poits that the CST formulatio uses, the trasformed bezier curve represetatio has two-dimesioal cotrol poits ad ca be expressed as (with m = + 3) Pξ = ξ = m ζ where dp i = c i 1 if i = ad c i = b i 0 otherwise m dp i t i = B i,m (t) Pq i (13) Equatio (13) represets a exact, stadard Bezier curve represetatio of a CST airfoil. This curve ca the be used i CAD, solid modelig, meshig or ay other geometry tool that does ot explicitly support CST airfoils. Figure 1 shows a sample calculatio of the correspodig Bezier curve for a 8 th order CST represetatio of a upper surface of a airfoil usig the above relatios. III. Applicatio to Other CST Shapes While most of the uses of CST i the literature are for airfoils, the other classes i CST are also of iterest. For the typical geometry classes from table 1, each oe ca either be represeted exactly as a Bezier curve or approximated usig a covergece series expasio for the class fuctio.

4 (a) Curves (b) ζ-coordiates (c) ξ-coordiates (d) Absolute Error Figure 1. A example of a CST airfoil ad the resultig Bezier curve with the correspodig error o the trasformed Bezier curve. Note that a miimum threshold of was used i the absolute error calculatio. A. Exact Represetatios If the N term i the CST class fuctio is a iteger, i.e. either 0 or 1 sice N 1 ad N are bouded betwee those values iclusively, the the CST curve ca be represeted exactly as a Bezier curve. This meas that geometry classes 1, 3, 6 ad 7 ca be represeted exactly usig this techique. Geometry class 1 is the airfoil case from above. 1. Geometry Class 3 Geometry class 3 has the followig form ζ = ξ (1 ξ) S (ξ) + ξδζ t.e. (14) The combied class ad shape fuctios will result i a polyomial of degree + for a degree shape fuctio. The trailig edge offset i the just a liear term that is added to the polyomial. Clearly this expressio is a polyomial ad thus ca be coverted to a explicit Bezier curve of degree +.

5 . Geometry Class 6 Geometry class 6 has the followig form ζ = ξs (ξ) + ξδζ t.e. (15) which agai results i a polyomial. This time the combiatio of the class ad shape fuctios yields a degree + 1 polyomial, ad the trailig edge offset agai is a liear term added to the polyomial. The resultig polyomial ca the easily be trasformed ito a explicit Bezier curve. 3. Geometry Class 7 Geometry class 7 has the followig form ζ = S (ξ) + ξδζ t.e. (16) which, similarly to the other cases, ca easily be coverted ito a degree explicit Bezier curve. B. Approximate Represetatios The three remaiig geometry classes:, 4 ad 5, have a o-iteger N term which makes the precedig process of trasformatio ot possible. Istead, a Taylor series approximatio for the class fuctio is used to develop a approximate polyomial represetatio. For Geometry class, the CST curve represetatio is ζ = ξ 1 ξ S (ξ) + ξδζ t.e. (17) Covertig these to a pair of parametric equatios, covertig the shape fuctio to stadard polyomial form ad performig the t = s trasformatio as doe above yields ξ = s The leadig 1 s term is ot a polyomial, however it ca be expressed as a biomial series, which is coverget over the etire iterval of s [0, 1], ad has the followig form, see ay stadard calculus book such as4 ( ) i 1/ i i 1 s = ( 1) s α i s (19) i+1 (18a) ζ = 1 s a i s + s Δζ t.e. (18b) i where the geeralized biomial coefficiet is used. Substitutig back ito (18b) yields i i+1 ζ = α i s a i s + s Δζ t.e. (0) Trucatig the ifiite series to the first m terms yields ito (0) yields m i i+1 ζ = α i s a i s + s Δζ t.e. (1) Ufortuately, eve a Biomial series expasio of degree 5, m = 5, yields a uacceptably large error over the majority of the parameter space with the maximum error occurrig at s = 1 of 0.5. Figure shows the results up to m = 5. While the series is absolutely coverget, the rate of covergece is quite slow. Also, recall that for m = 5 the resultig polyomial will be m = 10. Thus the resultig polyomial from this process will be of degree m( + 1), which quickly becomes too large to be practical.

6 (a) Curves (b) Absolute Error Figure. Biomial series represetatio of the class type equatio ad the correspodig errors for polyomials of icreasig degree usig equatio (19). IV. Coclusio A exact coversio betwee the most commo CST shapes to Bezier curves has bee preseted ad show to work to covert airfoils with oly egligible error. For the cases where N is ot a iteger, a biomial series approximatio was preseted that has theoretical covergece properties but has uacceptably large errors for practical uses. A alterative approach, such as usig ratioal bezier curves, will be eeded to develop a practical approximate coversio for these geometry classes. Extedig this work to three-dimesios is a trivial effort oce the CST curve trasformatio has bee accomplished. Kulfa 1 preseted two methods to create surfaces from CST shapes: (1) CST curves are used as cross-sectios of a geometry ad () the shape fuctio i the curve represetatio is replaced with a surface shape fuctio via Bezier surfaces i a local chord ad spa parametric coordiate system. Both approaches rely o usig Bezier surfaces i the spa-wise directio ad CST curves i the chord-wise directio. The stadard represetatio of a m Bezier surface is m Bi j p i,j j=0 ξ P(u, v) = (u) B m (v) P () For ay fixed i or j the resultig curve is a Bezier curve, so the coversio process meas replacig the surface cotrol poits, Pp i,j, with the correspodig trasformed cotrol poits, d P i, from (13). Refereces 1 Kulfa, B. M., A Uiversal Parametric Geometry Represetatio Method CST, 45 th AIAA Aerospace Scieces Meetig ad Exhibit, AIAA, Reo, NV, Jauary 007, AIAA Fari, G., Curves ad Surfaces for CAGD: A Practical Guide, The Morga Kaufma Series i Computer Graphics ad Geometric Modelig, Morga Kaufma Publishers, Ic., Sa Fracisco, CA, 5th ed., Piegl, L. ad Tiller, W., The NURBS Book, Moographs i Visual Commuicatio, Spriger, New York, d ed., Hurley, J. F., Calculus, Wadsworth Publishig, Belmot, CA, 1987.