IPE-409 CADCAM Geometric Modelling May 2018 (1).pptx
- 1. IPE-409 CAD/CAM Dr. Nafis Ahmad Professor Department of IPE, BUET Copy right: Dr. Nafis Ahmad, Bangladesh University of Engineering & Technology, Nov. 2013
- 4. Techniques for Geometric Modelling • Representation of curves – Parametric representation of geometry, Parametric cubic polynomial curves, Bézier curves – Multi-variable curve fitting, Cubic spline curves, Rational curves • Techniques for surface modelling – Surface patch, The Coons patch, icubic patch – Bézier surafces, B-Spline surface • Techniques for volume modelling – Boundary models, Constructive solid geometry – Other modelling techniques
- 5. Representation of curves • Mathematically straightforward geometries are curves and their representations are most complete. Surfaces are extension of curves • Why we need alternative geometric representation to classical ones? y=mx + c .............1 ax+by+c=0 ...........2 ax.x+by.y+2kxy+2fx+2gy+d=0 ..............3 Problems??
- 6. Cont.. y=mx + c .............1 ax+by+c=0 ...........2 ax.x+by.y+2kxy+2fx+2gy+d=0 ..............3 value of m unbounded geometry, Multi-valued Sequence of points not available Equation changes with coordinate system Other factors: difficulties in faired shapes representation, intersections between solid or surfaces
- 7. Cont.. Aero foil and Intersection of two cylinders So What to do?
- 8. Parametric Representation of geometry • The parametric representation of geometry essentially involves expressing relationships for the x, y and z coordinates of points on a curve or surface or a solid not in terms of each other but of one or more independent variables known as parameters. – For curve a single parameter is used: x, y and z are express in terms of a single variable typically u – For surface two parameters u and v – For solid three parameters u, v and w
- 9. Cont.. Position of any point on a space curve can be expressed as p = p(u) which is same as x=x(u), y= (u), and z=z(u)
- 10. Cont..
- 11. Parametric cubic polynomial curves • Vector form p= p(u) = k0 + k1u1 + k2u2 + k3u3
- 12. Parametric cubic polynomial curves • Hermit cubic form
- 13. Parametric cubic polynomial curves • Hermit cubic form-Blending functions
- 14. Parametric cubic polynomial curves • Bezier cubic form: Control polygon, approximated, degree less one than vertices. In case of cubic bezeir curve:
- 15. Parametric cubic polynomial curves • Bezier cubic form
- 16. Parametric cubic polynomial curves • Bezier cubic form-Blending functions & other examples • Passes through first and last point • Curve lies in the convex hull: minimum convex region enclosing control points
- 17. Parametric cubic polynomial curves
- 18. Parametric cubic polynomial curves • General considerations – Global and Local modification of curves – Degree of continuity: 1st derivative, 2nd derivative, • Cubic Spline Curve, • B-spline curve, • Rational Curves
- 19. Parametric cubic polynomial curves • Cubic Spline Curve: Composite curve with boundary condition of continuity in first and second derivatives at intermediate points. • B-spline curve, • Rational Curves • What is NURBS?
- 20. Parametric polynomial curves • B-spline curve: Pass through start and end points, tangential to the first and last vector. Its generalization of Bezeir curves
- 21. Parametric polynomial curves • Rational polynomials: can represent conic, more general quadric function, various polynomial types
- 22. Parametric polynomial curves • Rational polynomials: can represent conic, more general quadric function, various polynomial types • What is NURBS? Non-uniform rational B- splines(NURBS) can represent non-rational B-spline, Bezeir curves, linear and quadric analytic curves and can also be used for interpolating or approximating mode
- 23. Parametric cubic polynomial curves • Blending functions for Hermit and Bezier curve. (Draw and note the differences) • Advantages of Bezier curve over Hermit curve • Examples of Hermit and Bezier curve • Local modification vs Global modification • Modelling faired shapes found in aircraft and ships
- 24. Surface Modelling • General form of surface modelling • Surface patch • The Coons patch/sculptured surface • Bi-cubic patch • Bezier surface
- 25. Surface Modelling • General form of surface modelling • Surface patch
- 26. Surface Modelling • General form of surface modelling • Surface patch • The Coons patch/sculptured surface • Bicubic patch • Bezier surface
- 27. Volume Modelling • Boundary models/Representation/Graph based model • CSG • Other modelling techniques – Pure primitive instancing (for part family, geometrically, topologically similar not dimensionally similar) – Cell decomposition-use in FEA – Special occupancy enumeration
- 28. B-Rep • Ensures geometric and topological consistency – Faces bounded by single loop – Each edge should adjoin exactly two faces and have a vertex at each end – At least three edges should meet at each vertex – Euler’s rule: V - E + F = 2 – Euler-Poincare formula V – E + F – H + 2P = 2B
- 29. B-Rep
- 30. B-Rep
- 31. CSG
- 32. Other Solid Modeling Techniques • Pure primitive instancing • Cell decomposition • Spatial occupancy enumeration
- 33. Thank You