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Unformatted text preview: 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lecture 1 Nicholas M. Patrikalakis Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA c Copyright 2003 Massachusetts Institute of Technology Contents 1 Introduction and classification of geometric modeling forms 2 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Geometric modeling forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Wireframe modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Surface modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Solid modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.4 Non-two-manifold modeling . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Basic classification of solid modeling methods . . . . . . . . . . . . . . . . . . . 7 1.3.1 Decomposition models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Constructive solid geometry (CSG) . . . . . . . . . . . . . . . . . . . . . 14 1.3.3 Boundary representation (B-Rep) . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Alternate classification of geometric modeling forms . . . . . . . . . . . . . . . 18 1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.2 Unevaluated representation systems . . . . . . . . . . . . . . . . . . . . 18 1.4.3 Evaluated representation systems . . . . . . . . . . . . . . . . . . . . . . 20 Bibliography 21 1 Lecture 1 Introduction and classification of geometric modeling forms 1.1 Motivation Geometric modeling deals with the mathematical representation of curves, surfaces, and solids necessary in the definition of complex physical or engineering objects. The associated field of computational geometry is concerned with the development, analysis, and computer implemen- tation of algorithms encountered in geometric modeling. The objects we are concerned with in engineering range from the simple mechanical parts (machine elements) to complex sculptured objects such as ships, automobiles, airplanes, turbine and propeller blades, etc. Similarly, for the description of the physical environment we need to represent objects such as the ocean bottom as well as three-dimensional scalar or vector physical properties, such as salinity, tem- perature, velocities, chemical concentrations (possibly as a function of time as well). Sculptured objects play a key role in engineering because the shape of such objects (e.g. for aircraft, ships and underwater vehicles) is designed in order to reduce drag or increase the thrust (eg. for propeller blades). At the same time these objects need to satisfy other design constraints to permit them to fulfill certain design requirements (e.g. carry a certain payload, be stable in perturbations, etc). Similarly, there are objects which have significant aesthetic requirements, eg. cars, yachts, consumer products....
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