Part 1

Geometric Modeling Computer Aided Design history

D. CHABLAT / S. CARO [email protected]

Geometric Modeling • Definition – A geometric data model is a set of data characterizing the geometric shape of a part. • Properties – Rich semantics, – Exact or approximate modeling, – Easy creation and modification, – Gateways to the visualization, kinematics, machining, FEA, dynamic, …

D. Chablat / S. Caro -- Institut de Recherche en Communications et Cybernétique de Nantes

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Geometric Modeling • There are 5 main families – 2D Modeling – Wireframe 3D modeling – Surface modeling – Volume Modeling – Parametric modeling ¾order of increasing richness of semantic and historical appearance.


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2D Modeling • Interests – Easy calculations – Uses the basic principles of industrial design – The correspondence between the views is obtained manually. • Main problems – No verification of consistency between the views, – Accuracy of the model.


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2D Modeling • Entities manipulated – The point, segment, arc, ellipse. As sketch entities, the point, the line and the circle point. – Planar curves, in the form of Bézier or B-Splines curves – High redundancy of data. • Calculations of intersection, surfaces and hatch. • Dimensioning and Classifications.


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2D Modeling • Definition of the features – By numerical values (with the keyboard or mouse), – By "hanging" or construction constraints, • • • • •

tangent to the line or to the circle, intersection, through an existing point, middle point, perpendicular to a segment ....

– Creating sketches – Features for direct creation D. Chablat / S. Caro -- Institut de Recherche en Communications et Cybernétique de Nantes

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2D Modeling • Geometric transformations – change of scale, symmetry, rotations, translations, mixed transformations... • Insertion of a sub-model – fixed shape or defined by parameters – possibility of pointers to avoid copies (library of shape) • Modification of templates – changing contours created – parametric contours sometimes ...


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Wireframe 3D modeling • Extension of the 2D model by adding a third component to the coordinates of the elements. • There is talk of wireframe representation. • Adding surface elements not limited defined only by their analytical expression, – The plan, infinite cylinder, infinite cone, the sphere. – The model is obtained by intersection of these elements. • Main problems – No concept of volume, the gravity center...


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Surface modeling • Objectives of the surface modeling in the 60 – Making the simulation of machining, – Calculations of aerodynamics – Evaluate from the aesthetics of images. • In France, we note the work of Bezier (Renault) and Casteljau (Citroën / Peugeot).

– Creation of Euclid Software from Unisurf of Renault – Created of CATIA by Dassault Systems subsidiary of Dassault Aviation. • CATIA= Computer Aided Three Dimensional Interactive Application


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Surface modeling • Constraints for the design of free forms, – The model should belong to a class of functions continuous and sufficiently differentiable, – The model should be easy to manipulate (computer) and stored on a computer, – The evaluation of a current point must be made with a minimum of operations, – The model should allow to develop algorithms intersection low cost, – The model should be able to cover a wide variety of forms. D. Chablat / S. Caro -- Institut de Recherche en Communications et Cybernétique de Nantes

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Coons’s Models • Introduced by Steve Coons in 1966, the method is based on the definition of a tile of surface defined by its edges P(u,0), P(u,1), P(0,v) et P(1,v) and the intersection of two interpolation functions interpolation ⎧ f1 (t ) = 2t 3 − 3t 2 + 1 ⎨ 3 2 = − + f ( t ) 2 t 3 t ⎩ 2 • A point on the tile surface is defined as P(u, v) = P(u,0) f1(v) + P(u,1) f 2 (v) + P(0, v) f1(u ) + P(1, v) f 2 (u ) − P (0,0) f1 (u ) f1 (v) − P (0,1) f1 (u ) f 2 (v) − P (1,0) f 2 (u ) f1 (v) − P(1,1) f 2 (u ) f 2 (v) D. Chablat / S. Caro -- Institut de Recherche en Communications et Cybernétique de Nantes

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Bézier’s model • Using the concept of Bernstein’s polynomials. • Principle – The standard monomials form a polynomial curve of degree less than n is given by x(u)= a0 + a1u+…+anun – where u is the parameter of the local curve segment • Benefits – Polynomials are continuous functions, – Storage, handling and evaluation of polynomials are particularly adapted to computer – The derivative of a polynomial are polynomials whose coefficients can be found algebraically.


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Bézier’s model • Disadvantages – The coefficients of polynomials give no idea about the nature of the curve, – A small variation on a factor may cause a large deformation of the curve, – Polynomials are unstable for the interpolation of a large number of data.


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Bézier’s model • Bézier’s curves n

C (u ) = ∑ Bin (u ) Pi avec u ∈ [0,1] i =0

• Bernstein’s polynomials n i n −1⎛ n ⎞ Bi (u ) = u (1 − u ) ⎜ ⎟

⎝i⎠

• for n=6


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Bézier’s model • Properties – The manipulation of control points and easy and intuitive – Its calculation can be done recursively.


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B-Spline model • Principes – generalization of Bézier’s curves – solve the problem of oscillation of interpolation polynomials. • Basic function: – De Boor’s algorithm u − ti ti + k − u k −1 k k −1 N i (u ) = N i (u ) + N i −1 (u ) ti + k −1 − ti + k ti + k − ti +1 ⎧1 si ti ≤ u ≤ ti +1 1 N i (u ) = ⎨

⎩ 0 ailleurs


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B-Spline model • Current point

k

C (u ) = ∑ N lk− k (u ) Pl − k +i i =1

• Note – The Bezier curve is farthest from the polygon that the BSpline curve.


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