1. 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

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. 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

4. 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
László Horváth UÓ-JNFI-IAM

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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:  . 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

12. 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

13. 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