Curve Modeling (Spline) Dr. S.M. Malaek. Assistant: M. Younesi

Transcription

1 Crve Modeling Sline Dr. S.M. Malae Assistant: M. Yonesi

2 Crve Modeling Sline Crve Control Points Convex Hll Shae of sline crve Interolation Slines Aroximation Slines Continity Conditions Parametric & Geometric Linear Sline Interolation Cbic Sline Interolation. Natral Cbic Slines. Hermit Slines 3. Cardinal Slines

3 Sline Crve

4 Sline Crve In drafting terminology, a sline is a flexible stri sed to rodce a smooth crve throgh a designated set of oints. Flexible Stri

5 Sline Crve Several small weights weights are distribted along the length of the stri to hold it in osition on the drafting table as the crve is drawn. Weights

6 Sline Crve crve originally The term sline crve referred to a crve drawn in this manner.

7 Sline Crve In modeling, the term sline crve refers to a any comosite crve formed with olynomial sections satisfying secified continity conditions at the bondary of the ieces.

8 Control Points A sline crve is secified by given a set of coordinate ositions, called control oints. Control oints indicates the general shae of the crve.

9 Control Points A sline crve is defined, modified, and manilated with oerations on the control oints.

10 Convex Hll The convex olygon bondary that enclose a set control oints is called the convex hll.

11 Convex Hll The convex hll is a rbber band stretched arond the ositions of the control oints so that each control oint is either on the erimeter of the hll or inside it.

12 Remind: Convex A region is convex if the line segment joining any two oints in the region is also within the region.

13 Shae of the Sline Crve

14 Shae of the Crve Control oints are fitted with iecewise continos arametric olynomial fnction in one of two way:. Interolation Slines. Aroximation Slines

15 Interolation Slines When olynomial sections are fitted so that the crve asses throgh each control oint, the reslting crve is said to interolate the set of control oints.

16 Interolation Slines Interolation crves are commonly sed to digitize drawing or to secify animation aths.

17 Aroximation Slines When olynomial sections are fitted to the general control oint ath withot necessarily assing throgh any control oint, the reslting crve is said to aroximate the set if control oints.

18 Aroximation Slines Aroximation crves are ses as design tools to strctre object srfaces.

19 Parametric Continity Conditions

20 Parametric Continity Conditions To reresent a crve as a series of iecewise arametric crves, these crves to fit together reasonably Continity!

21 Parametric Continity Conditions Each section of a sline is described with a set of arametric coordinate fnctions of the form: x x, y y, z z,

22 Parametric Continity Conditions Let C and C, be two arametric Crves. We set arametric continity by matching the arametric derivation of adjoining crve sections at their common bondary.

23 Parametric Continity Conditions Zero order arametric C : Simly the crves meet, C C. First order arametric C : The first arametric derivations for two sccessive crve sections are eqal at their joining oint, C C

24 Parametric Continity Conditions Second order arametric C : Both the first and second arametric derivatives of the two crve sections are the same at the intersection, C C

25 Geometric Continity Conditions

26 Geometric Continity Conditions Geometric Continity Conditions: Only reqire arametric derivatives of the two sections to be roortional to each other at their common bondary instead of eqal to each other.

27 Geometric Continity Conditions Zero order geometric G : Same as C C C First order geometric G : The arametric first derivatives are roortional at the intersection of two sccessive sections. C αc Second order geometric G : Both the first and second arametric derivatives of the two crve sections are roortional at their bondary, C αc

28 Parametric & Geometric Continity A crve generated with geometric continity condition is similar to one generated with arametric continity, bt with slight differences in crve shae. With geometric continity, the crve is lled toward the section with the greater tangent vector.

29 Sline Interolation

30 Sline Interolation Linear Sline Interolation Cbic Sline Interolation

31 Linear Sline Interolation

32 Linear Sline Definition Given a set of control oints, linear interolation sline is obtained by fitting the int oints with a iecewise linear olynomial crve that asses throgh every control oints.

33 Linear Sline Interolation A linear olynomial with nnowns a & b:, b a z b a y b a x z z y y x x, b a

34 Linear Sline Interolation We need to determine the vales of a and b in the linear olynomial for each of the n crve section. Sbstitting endoint vales and for arameter : [ ] b a b a b a

35 Linear Sline Interolation Solving this eqation for the olynomial coefficients: P can be written in terms of the bondary conditions: b a [ ] [ ] b a { { M sline M geometry { U

36 Linear Sline Interolation The matrix reresentation for linear sline reresentation: P U M sline M geometry M geometry is the matrix containing the geometry constraint vales bondary condition on the sline. M sline is the matrix that transform the geometric constraint vales to the olynomial coefficients and rovides a characterization for the sline crve Basis Matrix.

37 Linear Sline Interolation We can exand the matrix reresentation:, P b P { [ ] [ ] Blending Fnctions P P P P P b b

38 Linear Sline Interolation The Linear The Linear sline sline blending fnctions blending fnctions:, P b P P P P P P b b

39 Linear Sline Interolation The crve is linear combination of the blending fnctions. The iecewise linear interolation is C continos. For a linear olynomial, constraints are reqired for each segment. A olynomial of degree has coefficients and ths reqires indeendent constraints to niqely determine it.

40 Linear Sline Alications Constrction of domes with tblar or trss element. Used in some old airlanes bilt b sace trss

41 Cbic Sline Interolation

42 Cbic Sline Interolation Used to: Set aths for object motion A reresentation for an existing object or drawing Design of objects srface Cbic slines are more flexible for modeling arbitrary crve shaes sch as vehicle or aircraft body.

43 Cbic Sline Definition: Given a set of control oints: P x, y, z,,,, K, n Cbic interolation sline is obtained by fitting the int oints with a iecewise cbic olynomial crve that asses throgh every control oints.

44 Cbic Sline Interolation A cbic olynomial:, d c b a z d c b a y d c b a x z z z z y y y y x x x x, 3 d c b a

45 Linear Sline Interolation 3 a b c d, We need to determine the vales of a, b, c and d in the cbic olynomial for each of the n crve section. By setting enogh bondary conditions at the joints between crve sections we can obtain nmerical vales for all the coefficients.

46 Cbic Sline Tyes Methods for setting the bondary conditions for cbic interolation slines:. Natral Cbic Slines. Hermit Slines 3. Cardinal Slines 4. Kochane-Bartels Slines

47 Natral Cbic Slines

48 Natral Cbic Slines Natral cbic slines are develoed for grahics alication. Natral cbic slines have C continity.

49 Natral Cbic Slines If we have n control oints to fit, then: We have n crve sections 4n olynomial coefficients to be determined. 3 control oints n crve sections 4 olynomial coefficients to be determined.

50 Natral Cbic Slines At each n- interior control oints, we have 4 bondary condition: The two crve sections on either side of a control oint mst first and second arametric derivatives at that control oint, and each crve mst ass throgh that control oint. At P interior control:. C C'. C C 3. C P 4. C P

51 Natral Cbic Slines Additional eqation: First control oint P End control oints P n 5. C P 6. C P

52 Natral Cbic Slines We still need two more conditions to be able to determine vales for all coefficients.

53 Natral Cbic Slines Method : Set the second derivatives at and n to 7. C 8. C

54 Natral Cbic Slines Method : Add a control oint P - and a control oint P n. 7. P - 8. P 3 P - P 3

55 Natral Cbic Slines Natral cbic slines are a mathematical model for the drafting sline. Disadvantage: No Local Control : we cannot reconstrct art of the crve withot secifying an entirely new set of control oints.

56 Hermit Slines

57 Given a set of control oints: Hermit Slines Hermit slines is an interolating iecewise cbic olynomial with a secified tangent at each control oint. P x, y, z,,,, K, n

58 Hermit Slines Hermit slines can be adjsted locally becase each crve section is only deendent on its endoint constraints nlie the natral cbic slines.

59 Hermit Slines Examle: Change in Magnitde of T

60 Hermit Slines Examle: Change in Direction of T

61 Hermit Slines P reresents a arametric cbic oint fnction for the section between and. 3 a b c d, The bondary conditions are: D D D and D are the vales for the arametric derivations sloe of the crve at control oints P and P

62 Hermit Slines, 3 d c b a [ ] d c b a 3 [ ] d c b a 3 The derivative of the oint fnction: d c b a D D 3 Constraints in a matrix form: D D Bondary conditions

63 Hermit Slines Solving this eqation for the olynomial coefficients: d c b a D D H D D M D D D D d c b a M H, the Hermit matrix, is the inverse of the bondary constraint matrix.

64 [ ] d c b a 3 [ ] D D [ ] 3 H D D M Blending Fnctions 3 H H H H D D P d c b a D D 3

65 Hermit Slines Hermit Blending Fnctions

66 Hermit Slines Hermit olynomials can be sefl for some digitizing alications where it may not be too difficlt to secify or aroximate the crve sloes. Bt For most roblems in modeling, it is more sefl to generate sline crve withot reqiring int vales for crve sloes.

67 Cardinal Slines

68 Cardinal Slines Cardinal slines are interolating iecewise cbic with secified endoint tangents at the bondary of each crve section. The difference is that we do not have to give the vales for the endoint tangents. For a cardinal Sline, the vale for the sloe at a control oint is aroximately calclated from the coordinates of the two adjacent control oints.

69 Cardinal Slines A cardinal sline section is comletely secified with for consective control oints. The middle two control oints are the section endoints, and the other two oints are sed in the calclation of the endoint sloes.

70 Cardinal Slines The bondary conditions: P P P t P t Ths, the sloes at control oints and are taen to be roortional to the chords and

71 Cardinal Slines The bondary conditions: t t P P P P

72 Cardinal Slines Parameter t is called the tension arameter. Tension arameter controls how loosely or tightly the cardinal sline fits the int control oints. P P P t P t

73 Cardinal Slines When t is called the Catmll- Rom slines, or Overhaser slines.

74 Cardinal Slines t t P P P P [ ] 3 C M P 3 3 s s s s s s s s s s M C t s

75 Cardinal Slines t t P P P P ] 3 [ ] 3 [ CAR CAR CAR CAR s s s s s s s s s s P The olynomials CAR for,,,3 are the cardinal fnctions.

76 Cardinal Slines Cardinal fnctions

77 Homewor

78 Homewor بدنه هواپيمايA-38 را در نظر بگيريد. در مقطعي از بدنه كه مجاور بال و محل تقاطع بال با بدنه است بررسي كنيد شكل مقطع به كدام يك از اسپلاين هاي معرفي شده بيشتر شباهت دارد.