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Shape-centric representation: Unified representation of geometry (NURBS) and topology

This presentation by László Horváth provides an intellectual property that is available only for students in his courses. It covers the boundary representation, parametric representation of curves and surfaces, polynomials, B-spline representation of curves, and non-uniform rational B-spline (NURBS). The presentation also includes a case study on boundary representation.

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Shape-centric representation: Unified representation of geometry (NURBS) and topology

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  1. Óbuda University John von Neumann Faculty of Informatics Institute of Applied Mathematics Master in Mechatronics CourseCAD Systems Laboratory No. 4 Shape centric representation. Unified representation of geometry (NURBS) and topology László Horváth http://users.nik.uni-obuda.hu/lhorvath/

  2. 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 http://users.nik.uni-obuda.hu/lhorvath/

  3. Contents Contents Boundary representation Topology in boundary representation Parametric representation of curves and surfaces Polynomials B-spline representation of curves Non-uniform rational B-spline Boundary representation – Case study CS 4.1 László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  4. Boundary representation G12 F2 F1 What is this? Solid body seems It consists of separated surfaces Solution: Boundary representation= topology (structure) and geometry (shape) Geometry: surfaces and intersection curves László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  5. Topology in boundary representation Él (E - edge) Closed chain of edges E - edge V - vertex F - face Shell Shell+material=body Curve mapped to edge Surface mapped to edge Predecessor edge Successor edge F V 2 2 Winged edge structure V 2 V 1 E Orientations are different. Split edge is applied 1 F E F 1 1 2 Coedge F 1 V 1 Predecessor edge Successor edge László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  6. Parametric representation of curves u max Y n P (x,y,z) u P u ) min ( t X r Z b Parametric equation of a three dimensional curve P(u)=[x(u) y(u) z(u)] x=x(u), y=y(u) és z=z(u) Cartesian space Pu is the position vector to point P. • Cases of connection of two curves: • Non continuous: there is no common point. • O order continuity: there is common point. • 1 order continuity: tangents are the same at the connection point. • 2 order continuity: tangents and curvatures are the same at the connection point. • Local characteristics at a point with parameter value u: • Tangent (t), • Normal (n) • Binormal (b) • Curvature (r) László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  7. Parametric representation of surfaces u=1 u=1 v=0 , v=0 P v u=0,4 v=0,8 u=1 , (x,y,z) P v=1 P u u=0 u=0 v=1 v=0 ) ( P u, v Y u=0 v=1 Isoparametric curves X Z Model coordinate system General form of parametric equation for surface: P(u,v)=[x(u,v) y(u,v) z(u,v)] where umin <= u <= umax and vmin <= v <= vmax The x, y and z coorditates of point P in the model space in the function of parameters u and : x=x(u,v), y=y(u,v) és z=z(u,v) László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  8. Polynomials - = + + + + 1 n n p x a x a x a x a ( ) … = - 1 1 0 n n n ( ) å = i p x a x i = 0 i = The only group of functions in current geometrical modeling For all analytical and free form shapes Differentiation of the function is easy: suitable for determination of tangent, normal, and curvature. General form of a polynomial of degree n is László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  9. B-spline representation of curves -1 segment i u u u u 0 1 2 3 segment i P i -1 P i P +1 i • Spline • Flexible steel ribbon in ship building. • It was modeled as B spline. • B-spline curve characteristics • Consists of segments. • Continuity at segment borders. • Local control. • Spline base functions. • Degree of the curve is same as degree of the base function. Different degree of segments is allowed. • Curve goes through of the first and last control points only in case of special parameterization. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  10. Non-uniform rational B-spline • In the knot vector of non-uniform B-spline the intervals are different in accordance with demand by modeling task: 0,0 0,1 0,33 0,6 0,8 1,0. • The B-spline representation can be considered as generalization of Bezier representation: 00001111. This is a Bezier curve. • Non-uniform rational B-spline (NURBS): • For all shapes. Including exact analytical shape • The rational B-spline curve representation includes weight vector (w) : [1, 4, 1, 1, 1 . Values are mapped to control points. • In case of analytical curves, the segment shape (line, circle, etc.) depends on the relevant w value. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  11. Boundary representation – Case study CS 4.1 Tasks related to this case study Open the model Study definition of form feature sequence step-by-step. Recognize modifications by easch form feature. Study topology and geometry. Study concepts in thematic (See next slide) Do not propagate the model. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  12. Boundary representation – Case study CS 4.1 Thematics Functions for shape definitions in modeling system Topology in the background of shape definition. Solid bodies in model space and lumps in a solid body. Parametric curve and surface definition. Curvature Degree and class of NURBS curve. Isoparametric curves and curve on surface. Connection of curves and surfaces, definition of continuity. Contextual geometry.E. g. contexts of a point. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  13. Boundary representation – Case study CS 4.1 Three reference planes. Curves in reference planes. Two tabulated surfaces in the context of curves. Swept surface in the context of curves. Continued on next slide. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  14. Boundary representation – Case study CS 4.1 Connecting two pairs of surfaces by blend surfaces using continuity definition. Two more reference planes. Close contours in reference planes. NURBS geometry represents straight lines exactly! Flat fill surfaces in the context of close contours. NURBS geometry represents flat surface exactly! Joining five surfaces. Individual surface representations serve as contexts. Join surface can be applied as unit. Definition of solids between surfaces and their ofsets. Definition of tabulated solid. Joining five curve segments and apply join curve as context of rib form feature. Completing solid by its mirror. Shell is a conditioning form feature between boundary and its ofset. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  15. Boundary representation – Case study CS 4.1 Deactivate form feature. Child form features will be also deactivated. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

  16. Boundary representation – Case study CS 4.1 Result of deactivation. Inactive form features still are in the model. However, they do not act as modifiers. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

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