Geometric Modeling: Exploring Geometric Modeling in Computer Vision

By Fouad Sabry

What is Geometric Modeling

Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes.The shapes studied in geometric modeling are mostly two- or three-dimensional, although many of its tools and principles can be applied to sets of any finite dimension. Today most geometric modeling is done with computers and for computer-based applications. Two-dimensional models are important in computer typography and technical drawing. Three-dimensional models are central to computer-aided design and manufacturing (CAD/CAM), and widely used in many applied technical fields such as civil and mechanical engineering, architecture, geology and medical image processing.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Geometric modeling

Chapter 2: Computer-aided design

Chapter 3: Computational geometry

Chapter 4: Bézier surface

Chapter 5: Constructive solid geometry

Chapter 6: Solid modeling

Chapter 7: Subdivision surface

Chapter 8: Mesh generation

Chapter 9: Procedural modeling

Chapter 10: Geometric constraint solving

(II) Answering the public top questions about geometric modeling.

(III) Real world examples for the usage of geometric modeling in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Geometric Modeling.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: May 4, 2024

Book preview

Chapter 1: Geometric modeling

Geometric modeling is a subfield of applied mathematics and computational geometry that investigates techniques and algorithms for describing shapes mathematically. Although many of its techniques and ideas can be applied to sets of any finite dimension, most of the shapes explored in geometric modeling are two- or three-dimensional (solid figures). Currently, the majority of geometric modeling is performed on computers for computer-based applications. In computer typography and technical drawing, two-dimensional models are essential. Three-dimensional models are fundamental to computer-aided design and manufacturing (CAD/CAM) and are utilized in a variety of applied technical domains, including civil and mechanical engineering, architecture, geology, and medical image processing.

Typically, geometric models are differentiated from procedural and object-oriented models, which specify the shape implicitly through an opaque process that generates the look. They are also compared with digital photographs and volumetric models, which depict the shape as a subset of a fine regular partition of space, and fractal models, which define the shape in an infinitely recursive manner. Nevertheless, these boundaries are sometimes blurred: for example, a digital image might be viewed as a collection of colored squares, while geometric objects such as circles are defined by implicit mathematical equations. In addition, when its recursive description is shortened to a finite depth, a fractal model produces a parametric or implicit model.

The John A. Gregory Memorial Honor is a notable award in the region.

{End Chapter 1}

Chapter 2: Computer-aided design

Using computers (or workstations) to help with the creation, revision, analysis, or optimization of a design is known as computer-aided design (CAD).

Electronic design automation is the practice of using it when creating electronic systems (EDA). Mechanical design automation (MDA), which comprises the process of producing a technical drawing using computer software, is referred to in the field of mechanical design.

The objects of traditional drafting are represented by CAD software for mechanical design using either vector-based visuals or, in some cases, raster graphics that show the overall appearance of planned things. It involves more than simply forms, though. The output of CAD must express information such as materials, procedures, dimensions, and tolerances in accordance with the conventions particular to the application, just like hand drafting of technical and engineering drawings.

Two-dimensional (2D) and three-dimensional (3D) curves, surfaces, and solids can all be designed using computer-aided design (CAD).

One of the various tools used by engineers and designers is computer-aided design, which can be applied in a variety of ways depending on the user's profession and the software in question.

CAD is a component of the entire digital product development (DPD) activity inside the PLM processes, and as such, it is used in conjunction with other tools that are either integrated modules or stand-alone solutions, such as:

Finite element analysis (FEA) and computer-aided engineering (CAE) (FEA, FEM)

Computer numerical control (CNC) machine instructions are part of computer-aided manufacturing (CAM).

Simulating motion and rendering in photorealism

Utilizing product data management for document control and management (PDM)

CAD is also used to accurately create the photo simulations that are frequently needed when preparing environmental impact reports. In these simulations, computer-aided designs of proposed buildings are superimposed onto images of existing environments to represent what that location will look like if the proposed facilities are allowed to be built. CAD is widely used to investigate shadow studies and potential view corridor obstruction.

There are various distinct kinds of CAD, and each one necessitates that the user approach their use and design of virtual components differently.

The makers of cheaper 2D systems are numerous, and many of them offer free and open-source software. Since they may be modified as necessary when creating the final draft, they offer a method of sketching without the complexity that hand drafting entailed about scale and positioning on the drawing page.

A 3D wireframe is essentially a three-dimensional version of 2D drafting, which is no longer commonly utilized. The drawing must be manually updated with each line. The finished product has no mass characteristics and cannot have elements that add directly to it, like holes. Although many 3D systems permit using the wireframe model to create the final engineering drawing views, the operator approaches these in a manner similar to that of the 2D systems.

The creation of 3D dumb solids is similar to manipulating actual objects (not often used today). Solid volumes are added to or deleted from fundamental three-dimensional geometric shapes like prisms, cylinders, spheres, and rectangles to