© 2020

Fundamentals of Geometry Construction

The Math Behind the CAD


Part of the Springer Tracts in Mechanical Engineering book series (STME)

Table of contents

  1. Front Matter
    Pages i-xx
  2. Jorge Angeles, Damiano Pasini
    Pages 1-43
  3. Jorge Angeles, Damiano Pasini
    Pages 45-75
  4. Jorge Angeles, Damiano Pasini
    Pages 77-101
  5. Jorge Angeles, Damiano Pasini
    Pages 103-163
  6. Back Matter
    Pages 165-168

About this book


The textbook provides both beginner and experienced CAD users with the math behind the CAD. The geometry tools introduced here help the reader exploit commercial CAD software to its fullest extent. In fact, the book enables the reader to go beyond what CAD software packages offer in their menus.

Chapter 1 summarizes the basic Linear and Vector Algebra pertinent to vectors in 3D, with some novelties: the 2D form of the vector product and the manipulation of “larger" matrices and vectors by means of block-partitioning of larger arrays. In chapter 2 the relations among points, lines and curves in the plane are revised accordingly; the difference between curves representing functions and their geometric counterparts is emphasized. Geometric objects in 3D, namely, points, planes, lines and surfaces are the subject of chapter 3; of the latter, only quadrics are studied, to keep the discussion at an elementary level, but the interested reader is guided to the literature on splines. The concept of affine transformations, at the core of CAD software, is introduced in chapter 4, which includes applications of these transformations to the synthesis of curves and surfaces that would be extremely cumbersome to produce otherwise.

The book, catering to various disciplines such as engineering, graphic design, animation and architecture, is kept discipline-independent, while including examples of interest to the various disciplines. Furthermore, the book can be an invaluable complement to undergraduate lectures on CAD.


3D Objects CAD software geometric objects affine transformations Vector Algebra Linear Algebra engineering design

Authors and affiliations

  1. 1.Department of Mechanical EngineeringMcGill UniversityMontrealCanada
  2. 2.Department of Mechanical EngineeringMcGill UniversityMontrealCanada

About the authors

Jorge Angeles is professor at the Department of Mechanical Engineering and at the Centre for Intelligent Machines at McGill University, Montreal, Canada.

Bibliographic information

  • Book Title Fundamentals of Geometry Construction
  • Book Subtitle The Math Behind the CAD
  • Authors Jorge Angeles
    Damiano Pasini
  • Series Title Springer Tracts in Mechanical Engineering
  • Series Abbreviated Title Springer Tracts in Mechanical Engineering
  • DOI
  • Copyright Information The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
  • Publisher Name Springer, Cham
  • eBook Packages Engineering Engineering (R0)
  • Hardcover ISBN 978-3-030-43130-3
  • Softcover ISBN 978-3-030-43133-4
  • eBook ISBN 978-3-030-43131-0
  • Series ISSN 2195-9862
  • Series E-ISSN 2195-9870
  • Edition Number 1
  • Number of Pages XX, 168
  • Number of Illustrations 60 b/w illustrations, 49 illustrations in colour
  • Topics Engineering Design
    Computer-Aided Engineering (CAD, CAE) and Design
    Algebraic Geometry
  • Buy this book on publisher's site
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“The book serves as a timely and relevant body of concepts and mathematical fundamentals behind the computer aided design (CAD) technology. … The presentation of the material is formal, precise and convincing. Overall, the book is a useful reference and textbook material to a broad community users of CAD tools who wish to gain underlying mathematical knowledge.” (Witold Pedrycz, zbMATH 1458.51001, 2021)