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Representation of curves using cubic polynomials, Hermite form, Bezier form; Surface modelling representations and Solid modelling using B-Rep and CSG techniques are presented in this slide.

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- 1. IPE-409 CAD/CAM Dr. Nafis Ahmad Professor Department of IPE, BUET Email: nafis@ipe.buet.ac.bd Sept-2020 9/21/2020
- 2. Ch#3. Techniques for Geometric Modelling 2
- 3. 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 9/21/2020
- 4. Techniques for Geometric Modelling • Techniques for surface modelling – Surface patch – The Coons patch – Bicubic patch – Bézier surafces – B-Spline surface • Techniques for volume modelling – Boundary models – Constructive solid geometry – Other modelling techniques 9/21/2020
- 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?? 9/21/2020
- 6. Cont.. y=mx + c .............1 ax+by+c=0 ...........2 ax.x+by.y+2kxy+2fx+2gy+d=0 ..............3 -Value of m (-infinity to +infinity) -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 9/21/2020
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- 8. Cont.. Aero foil and Intersection of two cylinders So, what to do? 9/21/2020
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- 10. 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 9/21/2020
- 11. Cont.. Position of any point on a space curve can be expressed as p = p(u), which is same as x=x(u), y= y(u), z=z(u) Similarly position of any point on a surface or solid can also be expressed by two (u,v) and three (u, v, w) independent parameters respectively. 9/21/2020
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- 13. Parametric cubic polynomial curves • Two points define a line, three points define a circle. So, four points can be used to define a cubic polynomial curve. To find 12 unknown, we need three equations and four points’ coordinates. • Vector form p= p(u) = k0 + k1u1 + k2u2 + k3u3 or 9/21/2020
- 14. Cubic polynomial curves-Hermite form • Hermite cubic form: we can also find 12 unknowns if we know two points and two slopes at the two ends of the curve. – ?? (vector form and matrix form) 9/21/2020
- 15. Cubic polynomial curves-Hermite form Hermite cubic form: two points and two slopes at the two ends of the curve are known. 9/21/2020
- 16. Cubic polynomial curves-Hermite form 9/21/2020
- 17. Cubic polynomial curves-Hermite form 9/21/2020
- 18. Cubic polynomial curves-Hermite form Change value of u from 0 to 1 with an increment of 0.1 and plot the curve in excel, and submit online by tomorrow. 9/21/2020
- 19. Cubic polynomial curves-Bezier form • How Bezier cubic form is different from other forms. • Uses four control points for a 3-degree curve. • Relation between tangent vectors at two end points and other points is shown below. 9/21/2020
- 20. Cubic polynomial curves-Bezier form 9/21/2020
- 21. Cubic polynomial curves-Bezier form 9/21/2020
- 22. Basis functions Hermite vs. Bezier form 9/21/2020
- 23. Parametric cubic polynomial curves 9/21/2020
- 24. Parametric cubic polynomial curves- Bezier form 9/21/2020
- 25. Parametric cubic polynomial curves- Bezier form Curve for different number of control points or degree (no. of points -1) 9/21/2020
- 26. Parametric cubic polynomial curves- Bezier form 9/21/2020
- 27. Parametric cubic polynomial curves- Bezier form P0=(1,1), P1=(3,6), P2=(5,7) and P3=(7,2) P(u=0.5)=? P(u=0.5)=(4, 5.25) 9/21/2020
- 28. Parametric cubic polynomial curves Sample questions: Derive the equation of cubic polynomial of Harmite and Bezier form. Draw and note the differences between blending functions of Hermite and Bezier form. What are the advantages of Bezier curve over Hermit curve Examples of Hermit and Bezier forms of cubic polynomial curves. 9/21/2020
- 29. Parametric cubic polynomial curves- Important considerations • Local modification vs Global modification • Modelling faired shapes found in aircraft and ship 9/21/2020
- 30. Parametric cubic polynomial curves- Important considerations • Degree of continuity • Curve with first degree of continuity C(1) and second degree of continuity C(2) etc. • Parametric continuity (C) and Geometric continuity (G) 9/21/2020
- 31. Parametric cubic polynomial curves • Degree of continuity 9/21/2020
- 32. Parametric cubic polynomial curves • Cubic spline curve/Composite curve: a series of curve(cubic) segments are jointed end to end. • Use knot point and boundary conditions continuity at first(tangent) and second(curvature) derivatives at intermediate points. • Global modification is a problem, though damped at the remote points. • For n control point or knot points, (n-1) spans, 4(n-1) co-efficient vectors • Number of point boundary condition is 2(n-1), plus (n-2) slope conditions and (n-2) curvature conditions • Remaining conditions- slope/tangent vectors at two ends. • Accumulated cord length is used. 9/21/2020
- 33. Parametric cubic polynomial curves • B-Spline curve: Local modification and degree of the curve are the reasons • Use blending function to combine influence of a series of control points. • In Bezier curve degree depends on control points, but here degree is independent of the number of control points. • In Bezier curve blending function is non-zero in entire range of u. In B- Spline curve they may be non-zero for a limited range. • For a series of n+1 points Pi is: • Blending functions of order 2(linear), 3( quadratic) and 4(cubic) • How these functions define the curve. Curve do not pass through two end points 9/21/2020
- 34. Parametric cubic polynomial curves • Cubic blending functions to approximate eight track points and localized effect. • Localized effect can be changed by changing polynomial order (order reduced localized effect increase) and repeating points. 9/21/2020
- 35. Parametric cubic polynomial curves • Non-uniform Rational: Though integer knots are commonly used, knots can be of arbitrary ascending numerical value as shown below. 9/21/2020
- 36. Parametric cubic polynomial curves • Besides representation of free form curves and data, most of the engineering design uses standard analytic shapes like arcs, cylinders, cones, lines and planes. • To generalize further or include all kind of geometric entities, Non-uniform Rational B-Spline (NURBS) representation is used. 9/21/2020
- 37. Surface Modelling • General form of surface modelling • Surface patch • The Coons patch/sculptured surface • Bicubic patch- extension on cubic spline curve. Use points and tangent(tensor) • Bezier surface-extension of Bezier curve 9/21/2020
- 38. Surface Modelling • General form of surface modelling: Modelling of free form surface is the extensions into the second dimension of polynomial curve techniques. • Position of any point on a surface or solid can also be expressed by two (u,v) 9/21/2020
- 39. Surface Modelling • Linearly blended patch. • A Coons patch, is a type of manifold parametrization to smoothly join other surfaces together, in finite element method and boundary element method, to mesh problem domains into elements. 9/21/2020
- 40. Surface Modelling General form of surface modelling • Bezier Surfaces 9/21/2020
- 41. Surface Modelling General form of surface modelling • B-Spline surface 9/21/2020
- 42. 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 9/21/2020
- 43. Volume Modelling • Boundary models/Representation/Graph based model – Face should be bounded by a single ring or 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 should apply V – E + F = 2 – Euler-Poincare formula should apply V – E + F – H + 2P = 2B 9/21/2020
- 45. Volume Modelling – Euler-Poincare formula should apply V – E + F – H + 2P = 2B 9/21/2020
- 47. Volume Modelling • Other modelling techniques – Pure primitive instancing (for part family, geometrically, topologically similar not dimensionally similar) – Cell decomposition-use in FEA – Special occupancy enumeration 9/21/2020
- 48. Volume Modelling Other modelling techniques • Pure primitive instancing (for part family, geometrically, topologically similar not dimensionally similar) 9/21/2020
- 50. Important Links 9/21/2020 Ref: Approximation and Modeling with B-Splines https://archive.siam.org/books/ot132/ot132-program-collection-index.php