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

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

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 – Exercise CS 4.1 László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/ Program of the course 1. Systems for lifecycle management of product information 2. Functionality of engineering modeling systems. Objects in model space. Structure of engineering configuration 3. Definition of contextual engineering objects for integrated representation of engineering configurations 4. Shape centric representation. Unified representation of geometry (NURBS) and topology. 5. Definition of solid model by modification using form features. Functional definition of shape 6. Analysis on the principle of finite elements (FEM/FEA) 7. Representation of mechanical systems 8. Representation of intents, experiences, and knowledge 9. Connections of virtual space with the physical world 10. Collaboration of engineers in PLM system 11. Representation of engineering configuration in RFLP structure 12. Management of models in PLM systems 13. Model data exchange between different models. ISO 13303. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

Boundary representation What is this? F2 G12 F1 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/

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

n t b Parametric representation of curves r Parametric equation of a three dimensional curve P (x,y,z) ( u ) min max Z X Y P 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. t n b r 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/

Parametric representation of surfaces v=1 u=1 v=0 v=0,8 Isoparametric curves P v u (x,y,z) Y X Z ( u, v ) 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/

László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/ Polynomials 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 ( ) 1 a x p n + = - … ( ) i n x a p å = László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

B-spline representation of curves 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. P i -1 +1 -1 segment segment u 1 2 3 László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/

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/

Boundary representation – Exercise 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/

Boundary representation – Exercise CS 4.1 This laboratory exercise is done in accordance with the thematics in the course of leading and explanation by teacher and without preliminarily developed model. This .ppt will be completed by active viewport snap shots from the model which is developed by the teacher during the laboratory. László Horváth UÓ-JNFI-IAM http://users.nik.uni-obuda.hu/lhorvath/