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Computer Aided Engineering Design
Anupam Saxena Associate Professor Indian Institute of Technology KANPUR
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Recollection! Banana trick, orange trick! Extended Jordon’s curve
Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite Extended Jordon’s curve Theorem A simple, closed Orientable surface bounds a Solid! The bounding surface can be divided into surface patches Each patch has curved boundaries Both GEOMETRIC and CONNECTIVITY INFORMATION SHOULD BE STORED AND RETRIEVED! Each curve has end points! 11/24/2018 Anupam Saxena
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# 6 Representation of Solids
Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite # 6 Representation of Solids
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Half Spaces Half-spaces are unbounded geometric entities
they divide the representation space E3 into two infinite portions one filled with material while the other empty Surfaces are half-space boundaries half-spaces can be treated as directed surfaces the direction vector points towards the material side Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite
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Half spaces: Examples A half-space H can be defined as a regular point set in E3 such that H = {P| P E3 and f(P) < 0} f(P) = 0 defines the surface equation Examples Planar half-space: H = {(x, y, z)| z < 0} Cylindrical half-space: H = {(x, y, z)| x2 + y2 < R2} Spherical half-space: H = {(x, y, z)| x2 + y2 + z2 < R2} Conical half-space : H = {(x, y, z)| x2 + y2 < tan2(/2)[z]2} Toroidal half-space with radii R1 and R2: H = {(x, y, z)| (x2 + y2 + z2 – R22 – R12)2 < 4R22(R12 – z2)} Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite
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Solid with half spaces Complex objects can be modeled as half-spaces combined using set (union, subtraction) operations Half-spaces acting as bounding patches for solids can be free form surfaces Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite WORK ON THE EXAMPLE… WORK OUT THE INTERSECTION, UNION ETC… DISADVANTAGES… BOUNDS ???
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An L Bracket Example H1 H4 H9 Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite H7 H8 H3 H5 H2 H6 Let all be the directed surfaces s.t. the material points into the solid 11/24/2018 Anupam Saxena
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Construction using ½ Spaces
H1 H2 H3 H4 H5 H7 H8 H9 Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite H6 (H1 H2 H3 H4 H6 H7) + (H4 H8 H5 H2 H3 H6) (H9) Unbounded, implicit topological and geometric description 11/24/2018 Anupam Saxena
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Representation scheme
be versatile and capable of modeling a generic solid (b) generate valid solids (c) be complete in the sense that every valid representation (solid) produced is unambiguous (d) generate unique solids in that no two different representations should generate the same object (e) have closure implying that permitted transformation operations on valid solids always yield valid solids (f) be compact and efficient in matters of data storage and retrieval. Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite
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Two representation schemes
Wireframe approach Boundary Representation Approach Constructive Solid Geometry More catered to modeling Work well for Polyhedral solids Can be extended for use with homeomorphic non-polyhedral solids Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite
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Wireframe representation
one of the oldest ways to represent solids representation is essentially through a set of key vertices connected by key edges both topological and (NOT ALL) geometric information is stored The edges may be straight or curved Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite
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Example: Wireframe Tetrahedron
1 2 3 (1) (2) (3) (4) (5) (6) 4 x y z Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite Edge Number Vertex 1 Vertex 2 1 1 2 2 2 3 3 1 3 4 1 4 5 3 4 6 2 4 Vertex Number x y z 3 3 2 0 4 1 2 4
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Example: Wireframe Block
1 2 3 4 5 6 7 8 (1) (2) (3) (4) (9) (10) (11) (12) (5) (6) (7) (8) Edge table 1 2 2 3 3 4 4 1 5 1 6 2 7 3 8 4 5 6 6 7 7 8 8 5 Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite vertex table
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Advantages and Drawbacks
wireframe models are simple, they are non-unique and ambiguous the models do not include the face information. Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite
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Ambiguity with wireframe models
Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite
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Boundary Representation scheme
B-rep for short an extension of wireframe modeling to include the face information The faces can either be analytical or free form surfaces B-rep uses the Jordan’s curve theorem on boundary determinism a closed connected surface determines the interior of a solid both topological and geometric information is stored Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite
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Face orientations with B-rep
Face orientations may be recorded such that a normal to the face points into the solid. This can be ensured by the clockwise ordering of vertices (right-handed rule) associated with the face Using normal vectors, one can distinguish the interior of the solid from its exterior Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite
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Example: Face orientation
1 2 3 (1) (2) (3) (4) (5) (6) 4 x y z Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite Face 1: 1, 4, 2 Face 2: 2, 4, 3 Face 3: 3, 4, 1 Face 4: 2, 3, 1
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Winged Edge data structure
Data storage method with B-Rep Employs only edges to document the connectivity Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite 1 2 3 4 (1) (2) (3) (4) (5) (6) A B C Back face D
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Winged Edge data storage Tetrahedron
1 2 3 4 (1) (2) (3) (4) (5) (6) A B C Back face D Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite Edge Vertices Faces Clockwise Traverse on Face 1 traverse on Face 2 Name Start End Face 1 Face 2 Preceding Suceeding (3) 2 4 A B (5) (1) (2) (6) 1 C (4) 3 D
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Additional tables with Winged Edge Structure
Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite face table vertex table Face Edge A (1) B (2) C (4) D (5) Vertex Edge 1 (1) 2 (2) 3 (3) 4 (6) These are non-unique tables
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Internal loops in Winged Edge Structure
Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite When internal loops present Assign them counterclockwise orientation OR/AND Connect the internal loop with the external one using auxiliary edges Effectively, only one loop will be present An Auxiliary edge will have the same face on the left and right
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Euler-Poincaré formula
How do we know a polyhedral solid is a valid solid? V – E + F – (L – F) – 2(S – G) = 0 * Extended to include pot and through holes V as number of vertices, E as the number of edges, F as the number of faces, G as the number of holes (or genus) penetrating the solid, S as the number of shells and L as the total number of outer and inner loops a shell is an internal void of a solid bounded by a closed connected surface that can have its own genus value Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite * For a valid solid, this is true. Converse, however, is NOT TRUE
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Euler-Poincaré formula
Example 1: V = 8 E = 12 F = 6 L = 6 S = 1 G = 0 Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite V – E + F – (L – F) – 2(S – G) 8 – – (6 – 6) – 2(1 – 0) = 0
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Euler-Poincaré formula
Example 2: V = 16 E = 24 F = 11 L = 12 S = 1 G = 0 Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite V – E + F – (L – F) – 2(S – G) 16 – – (12 – 11) – 2(1 – 0) = 0
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Euler-Poincaré formula
Example 3: V = 16 E = 24 F = 10 L = 12 S = 1 G = 1 Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite V – E + F – (L – F) – 2(S – G) 16 – – (12 – 10) – 2(1 – 1) = 0
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Euler-Poincaré formula
Example 4: V = 24 E = 36 F = 16 L = 18 S = 2 G = 1 Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite V – E + F – (L – F) – 2(S – G) 24 – – (18 – 16) – 2(2 – 1) = 0
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Euler-Poincaré formula
Example 5: Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite V = 10 E = 15 F = 7 L = 7 S = 1 G = 0 This is NOT a valid solid One face is dangling V – E + F – (L – F) – 2(S – G) 10 – – (7 – 7) – 2(1 – 0) = 0
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