Geometric Modeling: A First Course

Introduction

Preamble

• Geometric models are key in various applications including special effects, simulations, and robotics.
• They capture spatial aspects of objects for both real and virtual applications.
• Course focuses on 3D objects and associated computer algorithms.
• Relevant fields include VLSI layout, GIS, electronic packaging, graphics, and vision.
• Emphasis on 3D modeling, balancing graphics/multimedia with robotics/automation.
• Course complements existing Computer Graphics textbooks.
• Audience: Upper-level undergraduates in CS or Engineering, and practicing engineers.
• Mathematical prerequisites: Calculus, analytic geometry, basic linear algebra.

Course Structure

• Sections cover applications of geometric modeling in CAD/CAM and 3D graphics.
• Systematic approach to study geometric modeling proposed.
• Historical background provided.
• Chapters cover Euclidean and projective geometry, representations for curves, surfaces, solids, and geometric algorithms.

The Role of Geometry in CAD/CAM

• 3-D geometry is crucial in the traditional product life-cycle stages from design to manufacturing.
• Design involves geometric and non-geometric specifications.
• Concurrent engineering principles improve quality and time to market by integrating parallel processes.
• Engineering Environments support concurrent processes and rely on geometric modeling tools among others.

The Role of Geometry in 3-D Graphics

• Early graphics were 2-D, evolving to distinguish between model and display.
• 3D graphics involves geometric models augmented with color, texture, and other data.
• Image-based rendering and Computer Vision techniques are employed.
• Graphics models for CAD/CAM require accuracy; for graphics, realism in appearance may suffice.
• Increasing overlap between graphics and CAD/CAM methodologies anticipated due to shared needs like collision detection.

Models, Representations, Algorithms, and Systems

• Distinction between mathematical and computational models.
• Validity of models ascertained through experimental predictions.
• System architecture includes geometric models, algorithmic process, input/output facilities.
• Fundamental and application-specific algorithms discussed.

Historical Summary

• Traces origins of geometric modeling to the 1950s with the advent of computer graphics and NC machining.
• Four main streams evolved: computer graphics, wireframe, free-form curves, and solid modeling.
• Computational Geometry and spatial reasoning in robotics as additional streams.
• Convergence of these subfields is becoming evident with shared ideas and methodologies.
• Geometric modeling adoption slow due to high computational resource requirements.
• 3-D graphics gained popularity in the 1990s, with solid modeling steadily gaining acceptance.