1. Representation of Geometry in Model Space
Óbuda University
John von Neumann Faculty of Informatics
Institute of Applied Mathematics
Master in Engineering Informatics
Course Modeling and design
Lecture and laboratory No. 4
Representation of Geometry in Model Space
Dr. László Horváth

2. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/
This presentation is intellectual property. It is available only for students in my courses.
The screen shots in tis presentation was made in the CATIA V5 és V6 PLM systems the Laboratory of Intelligent Engineering systems, in real modeling process.
The CATIA V5 és V6 PLM systems operate in the above laboratory by the help of Dassult Systémes Inc. and CAD-Terv Ltd.
László Horváth UÓ-JNFI-IAM

3. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/
Contents
Lecture
Interpolation and approximation curve
Short history
Parametric curve and its global and local parameters
Parametric equation of surface
B-spline curve
Curve representation using spline base functions
Parameterization of B-spline curve
Control of B-spline curve
Rational B-spline curve
Laboratory tasks
4.1 Building contextual chains
4.2 Definition and analysis of contextual surfaces
László Horváth UÓ-JNFI-IAM

4. Interpolation and approximation curve
Linear
Hermite (applied by: Ferguson and Coons)
Circular
Cubic t2
t2’ t1
Approximation P 1 2 3
Convex hull P 1 2 3
Convex hull
Control polygon P 1 2 3 ,
Control point
László Horváth UÓ-JNFI-IAM

5. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/
Short history
French mathematician Paul Bezier (Renault factory): car body design using the method of approximation
Paul de Casteljau applied the same method at the Citroen factory. However, the method became known under the name of Paul Bezier.
Paul Bezier introduced the control polygon.
Characteristics of the Bezier curve is provided by the Bernstein polynomial basic (blending) functions
The curve passes over the first and last control points.
The first and last segments of control polygon are tangents of curve.
Global control.
Degree of curve = number of control points - 1.
László Horváth UÓ-JNFI-IAM

6. n t b Parametric curve and its local parameters u max P (x,y,z) ( u )
General form of parametric equation of curve
P(u)=[x(u) y(u) z(u)], where umin <= u <= umax
Pu is the position vector to point P. Coordinates of point P in the function of parameter u :
x=x(u), y=y(u) és z=z(u) t n b u
max
P (x,y,z) ( u )
min Z X Y P
Local parameters of curve
Accompanying trihedron (Vector units):
t – tangent,
n- normal
b –binormal
Tangent plane: t and b
Normal plane: b and n Curvature
Cartesian space
László Horváth UÓ-JNFI-IAM

7. Parametric curve and its global parameters
Control polygon consists of a single line.
Control
Local
Global
Degree (D)
Class (N=D+1)
Free or engaged end
Shape of line is not allowed to change
László Horváth UÓ-JNFI-IAM

8. Parametric curve and its global parameters
Control polygon consists of 3 lines.
Degree was changed to 2. Shape of line is allowed to change accordingly.
Degree was changed to 3. Shape of line is allowed to change accordingly
László Horváth UÓ-JNFI-IAM

9. Parametric equation of surface
v=1
u=1
v=0
v=0,8
Isoperimetric curves P v u
(x,y,z) Y X Z (
u, v )
Model coordinate system ,
General form of parametric equation of surface:
P(u,v)=[x(u,v) y(u,v) z(u,v)]
where umin <= u <= umax and vmin <= v <= vmax
x=x(u,v), y=y(u,v) és z=z(u,v)
László Horváth UÓ-JNFI-IAM

10. Parameter line and space
Model space
Parameter line u i u i u
min u
max
Curve
Parameter space v
max v j u i v j u i v
min u
min u
max
Surface
László Horváth UÓ-JNFI-IAM

11. Paraméter egyenes és tér B-spline curve
Flexible steel ribbon in ship building.
It was modeled as B spline.
B-spline curve
Consists of segments.
Continuity at segment borders.
Local control.
Spline base functions.
Degree of the curve is same as degree of the base (blending) function. Different degree of segments is allowed.
Curve goes through of the first and last control points only in case of special parameterization.
t=0
t=1
Sequence of intervals u u u u u 1 2 3 4
Knot vector
László Horváth UÓ-JNFI-IAM

12. { } å ( ) Representation of curve using spline base functions P : i =
Analytical definition of B-spline curve: ( ) P = i 1 n u N k å ,
where:
The normalized B-spline base function: N i, k ( u)
The control points: { } P : i =
0,1, . , n
The B-spline curve includes polynomial represented segments
Order of segment k, its degree k-1.
László Horváth UÓ-JNFI-IAM

13. Parameterization of B-spline curve
The parameter range of curve is divided into intervals for segments:
The knot vector: { } u : i =
0,1, . , n k + u0 u1 u3 u2
where: u i + 1
The curve passes over given number of segment border points. These points are called as knots. Knots carry the parameter values for segment border points.
In case of control points n + 1 , order k, degree k-1, number of knots m:
(m+1) = (n+1) + k
Number of knots: m = n + k
Periodic curve: The parameter intervals are repeated. The uniform B-spline is periodic.
Non-periodic curve: Inside knots are uniform. However, intervals are repeated at beginning and end of the vector. Maximum number of repetition is the order of curve.
The allocation of parameter range can be uniform or non-uniform. Non uniform allocation serves more sophisticated representation of geometry.
László Horváth UÓ-JNFI-IAM

14. Examples for knot vector
"a" curve: k=2, degree=1 b
"b" curve: k=3, degree=2
"c" curve: k=4, degree=3 c
Knot vectors:
"a" curve: 001233
"b" curve:  
"c" curve:  
László Horváth UÓ-JNFI-IAM

15. Control of B-spline curve
A base (blending) function acts only on a segment of the B-spline curve (local control). However, second order continuity (same tangent and curvature) should be ensured at segment borders automatically. This continuity must be saved during any modification of curve. Consequently, the real control is: V 1 2 3 4 5
Uniform, periodic
Control points which act on the highlighted segment
Segments, e.g. 1 V - V 2 2 V - V 1 3 6 V - V 5 1
Effect of B-spline base functions
László Horváth UÓ-JNFI-IAM

16. ( ) ( ) Q = wx , wy , wz , w w ³ x , y , z P w Rational B-spline curve
Representation of rational B-splines uses the concept of homogenous coordinates. This was originally applied at representation of transformation matrices.
Pont defined in the three dimensional Euclidean space: ( ) x , y , z P
Representation of this point in the four dimensional homogenous space w ( ) Q = wx , wy , wz , w
where w is the homogenous coordinate which is often called as weight: w
Values of w are included for the control points in the weight vector.
László Horváth UÓ-JNFI-IAM

17. ( ) å Q u w = , ( ) Q u N V = x y z Rational B-spline curve V V w =
B-spline curve with homogenous coordinates: ( ) Q u N V w i k n = å ,
where:
Qw(u) is the point of curve as expressed using four dimensional homogenous coordinates: ( ) Q u w x y z = , N
i,k ( u )
is the spline base function, and V
is control point in the four dimensional homogenous space: w V w i =
, consequently
László Horváth UÓ-JNFI-IAM

18. Rational B-spline curve
Three dimensional projection of the four dimensional control points by dividing the first three coordinates by the homogenous coordinate:
Similarly, the Q(u) point of curve is represented by as projection of four dimension space to three dimensional space: x w y z = ( ) Q u N w V i k n = , å
The rational B-spline curves are characterized by knot and weight vectors.
At representation of analytical curves value of w determines the shape of curve: straight line, elliptic, parabolic, or hyperbolic the segment.
László Horváth UÓ-JNFI-IAM

19. MD 4.1 laboratory exercise
Building contextual chains
László Horváth UÓ-JNFI-IAM

20. MD 4.1 laboratory exercise
Definition of contexts for point definitions
László Horváth UÓ-JNFI-IAM

21. MD 4.1 laboratory exercise
Definition of point in the context of curve
László Horváth UÓ-JNFI-IAM

22. MD 4.1 laboratory exercise
Definition of point in the context of reference plane
László Horváth UÓ-JNFI-IAM

23. MD 4.1 laboratory exercise
Definition of point in the context of curve
László Horváth UÓ-JNFI-IAM

24. MD 4.1 laboratory exercise
Definition of point in the context of surface in boundary
László Horváth UÓ-JNFI-IAM

25. MD 4.1 laboratory exercise
Definition of curve in the context of points. At one of the points context of curve tangent is defined.
László Horváth UÓ-JNFI-IAM

26. MD 4.1 laboratory exercise
Activation of close parameter of curve.
László Horváth UÓ-JNFI-IAM

27. MD 4.1 laboratory exercise
Definition of surface by tabulation of the closed curve.
László Horváth UÓ-JNFI-IAM

28. MD 4.1 laboratory exercise
Definition of reference plane in the context of three points.
László Horváth UÓ-JNFI-IAM

29. MD 4.1 laboratory exercise
Definition of point in the context of the reference surface Plane2.
László Horváth UÓ-JNFI-IAM

30. MD 4.1 laboratory exercise
One point of the earlier defined curve was replaced by the Point4. One of the ends of the modified tabulated surface was closed by the first order contextual surface Fill1.
László Horváth UÓ-JNFI-IAM

31. MD 4.1 laboratory exercise
The other end of modified tabulated surface was similarly closed by the surface Fill2.
László Horváth UÓ-JNFI-IAM

32. MD 4.1 laboratory exercise
The three highlighted surfaces are joined in the feature Join1 in order to make them eligible as common context.
László Horváth UÓ-JNFI-IAM

33. MD 4.1 laboratory exercise
Solid feature was defined between the Join1 feature and its offset. For further development of solid body new reference plane will be required.
László Horváth UÓ-JNFI-IAM

34. MD 4.1 laboratory exercise
The tabulated solid form feature was regenerated into shell shaped by activation of the relevant parameter.
László Horváth UÓ-JNFI-IAM

35. MD 4.1 laboratory exercise
Additional line definition is needed.
László Horváth UÓ-JNFI-IAM

36. MD 4.1 laboratory exercise
Rib form feature is defined.
László Horváth UÓ-JNFI-IAM

37. MD 4.1 laboratory exercise
Pipe stub is required in order to provide connection to the interior. The solution to the problem begins by defining a hole (topologically breakthrough!)
László Horváth UÓ-JNFI-IAM

38. MD 4.1 laboratory exercise
The hole feature extended the boundary by a cylindrical surface which is extrapolated.
László Horváth UÓ-JNFI-IAM

39. MD 4.1 laboratory exercise
Solid form feature is defined between the extrapolate surface and its offset.
László Horváth UÓ-JNFI-IAM

40. MD 4.1 laboratory exercise
Fillet features are defined.
László Horváth UÓ-JNFI-IAM

41. MD 4.1 laboratory exercise
Model is ready. Survey the contextual chains!
László Horváth UÓ-JNFI-IAM

42. MD 4.1 laboratory exercise
Analysis of curvature in an important context.
László Horváth UÓ-JNFI-IAM

43. MD 4.1 laboratory exercise
Visualization of control polygon mesh of one of the surfaces.
László Horváth UÓ-JNFI-IAM

44. MD 4.1 laboratory exercise
Curve parallel with the closed Spline1 is defined for later use.
László Horváth UÓ-JNFI-IAM

45. MD 4.2 laboratory exercise
Definition and analysis of contextual surfaces
No comments included. Please understand model using screen shots.
László Horváth UÓ-JNFI-IAM

46. MD 4.2 laboratory exercise
László Horváth UÓ-JNFI-IAM

47. MD 4.2 laboratory exercise
László Horváth UÓ-JNFI-IAM

48. MD 4.2 laboratory exercise
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49. MD 4.2 laboratory exercise
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50. MD 4.2 laboratory exercise
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51. MD 4.2 laboratory exercise
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52. MD 4.2 laboratory exercise
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53. MD 4.2 laboratory exercise
László Horváth UÓ-JNFI-IAM

54. MD 4.2 laboratory exercise
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55. MD 4.2 laboratory exercise
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56. MD 4.2 laboratory exercise
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57. MD 4.2 laboratory exercise
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58. MD 4.2 laboratory exercise
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59. MD 4.2 laboratory exercise
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60. MD 4.2 laboratory exercise
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61. MD 4.2 laboratory exercise
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62. MD 4.2 laboratory exercise
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63. MD 4.2 laboratory exercise
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64. MD 4.2 laboratory exercise
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65. MD 4.2 laboratory exercise
László Horváth UÓ-JNFI-IAM