# Geometry for Programmers

- Length: 440 pages
- Edition: 1
- Language: English
- Publisher: Manning
- Publication Date: 2023-06-06
- ISBN-10: 1633439607
- ISBN-13: 9781633439603
- Sales Rank: #2547083 (See Top 100 Books)

**Master the math behind CAD, game engines, GIS, and more! This hands-on book teaches you the geometry used to create simulations, 3D prints, and other models of the physical world.**

In *Geometry for Programmers* you will learn how to:

Speak the language of applied geometry Compose geometric transformations economically Craft custom splines for efficient curves and surface generation Pick and implement the right geometric transformations Confidently use important algorithms that operate on triangle meshes, distance functions, and voxels

*Geometry for Programmers* guides you through the math behind graphics and modeling tools. It’s full of practical examples and clear explanations that make sense even if you don’t have a background in advanced math. You’ll learn how basic geometry can help you avoid code layering and repetition, and even how to drive down cloud hosting costs with more efficient runtimes. Cheerful language, charts, illustrations, equations, and Python code help make geometry instantly relevant to your daily work as a developer.

Purchase of the print book includes a free eBook in PDF, Kindle, and ePub formats from Manning Publications.

**About the Technology **

Geometry is at the heart of game engines, robotics, computer-aided design, GIS, and image processing. This book draws back what is for some a mathematical curtain, giving them insight and control over this central tool. You’ll quickly see how a little geometry can help you design realistic simulations, translate the physical world into code, and even reduce your cloud services bill by improving the efficiency of graphics-intensive applications.

**About the Book **

*Geometry for Programmers *is both practical and entertaining. Fun illustrations and engaging examples show you how to apply geometry to real programming problems, like changing a scan into a CAD model or developing 3D printing contours from a parametric function. And don’t worry if you aren’t a math expert. There’s no heavy theory, and you’ll learn how to offload most equations to the SymPy computer algebra system.

**What’s Inside **

Speak the language of applied geometry Compose geometric transformations economically Craft custom splines for efficient curves and surface generation Confidently use geometry algorithms

**About the Reader**

Examples are in Python, and all you need is high school–level math.

**About the Author **

**Oleksandr Kaleniuk** is the creator of Words and Buttons Online, a collection of interactive tutorials on math and programming.

Geometry for Programmers brief contents contents preface acknowledgments about this book Who should read this book How this book is organized About the code liveBook discussion forum about the author about the cover illustration 1 Getting started 1.1 Which parts of programming require geometry? 1.2 What is geometry for programmers? 1.3 Why not let my tools take care of geometry? 1.4 Applied geometry has been around forever; why learn it now? 1.5 You don’t have to know much to start 1.6 SymPy will do your math for you Summary 2 Terminology and jargon 2.1 Numbers, points, and vectors 2.1.1 Numbers 2.1.2 Vectors 2.1.3 Section 2.1 summary 2.2 Vertices and triangles 2.2.1 Being pedantic about triangles 2.2.2 Triangle quality 2.2.3 Section 2.2 summary 2.3 Lines, planes, and their equations 2.3.1 Lines and planes 2.3.2 The parametric form for 3D lines 2.3.3 Section 2.3 summary 2.4 Functions and geometric transformations 2.4.1 What is a function? 2.4.2 Function types most commonly used in geometric modeling 2.4.3 Section 2.4 summary 2.5 The shortest possible introduction to matrix algebra 2.5.1 What is algebra? 2.5.2 How does matrix algebra work? 2.5.3 Section 2.5 summary 2.6 Exercises 2.7 Solutions to exercises Summary 3 The geometry of linear equations 3.1 Linear equations as lines and planes 3.1.1 Introducing a hyperplane 3.1.2 A solution is where hyperplanes intersect 3.1.3 Section 3.1 summary 3.2 Overspecified and underspecified systems 3.2.1 Overspecified systems 3.2.2 Underspecified systems 3.2.3 Section 3.2 summary 3.3 A visual example of an interactive linear solver 3.3.1 The basic principle of iteration 3.3.2 Starting point and exit conditions 3.3.3 Convergence and stability 3.3.4 Section 3.3 summary 3.4 Direct solver 3.4.1 Turning an iterative solver into a direct one 3.4.2 Algorithm complexity 3.4.3 Section 3.4 summary 3.5 Linear equations system as matrix multiplication 3.5.1 Matrix equations 3.5.2 What types of matrices we should know about 3.5.3 Things we’re allowed to do with equations 3.5.4 Section 3.5 summary 3.6 Solving linear systems with Gaussian elimination and LU-decomposition 3.6.1 An example of Gaussian elimination 3.6.2 What do “elimination” and “decomposition” mean? 3.6.3 Section 3.6 summary 3.7 Which solver fits my problem best? 3.7.1 When to use an elimination-based one 3.7.2 When to use an iterative one 3.7.3 When to use neither 3.7.4 Section 3.7 summary 3.8 Practical example: Does a ray hit a triangle? 3.8.1 The ray-triangle intersection problem 3.8.2 Forming a system 3.8.3 Making the equations into code 3.8.4 Section 3.8 summary 3.9 Exercises 3.10 Solutions to exercises Summary 4 Projective geometric transformations 4.1 Some special cases of geometric transformations 4.1.1 Translation 4.1.2 Scaling 4.1.3 Rotation 4.1.4 Section 4.1 summary 4.2 Generalizations 4.2.1 Linear transformations in Euclidean space 4.2.2 Bundling rotation, scaling, and translation in a single affine transformation 4.2.3 Generalizing affine transformations to projective transformations 4.2.4 An alternative to projective transformations 4.2.5 Section 4.2 summary 4.3 Projective space and homogeneous coordinates 4.3.1 Expanding the whole space with homogeneous coordinates 4.3.2 Making all the transformations a single matrix multiplication: Why? 4.3.3 Section 4.3 summary 4.4 Practical examples 4.4.1 Scanning with a phone 4.4.2 Does a point belong to a triangle? 4.4.3 Section 4.4 summary 4.5 Exercises 4.6 Solutions to exercises Summary 5 The geometry of calculus 5.1 What is a derivative? 5.1.1 Derivative at a point 5.1.2 Derivative as a function 5.1.3 Rules of differentiation 5.1.4 Using SymPy to do differentiation 5.1.5 Section 5.1 summary 5.2 Smooth piecewise parametric curves 5.2.1 Piecewise functions 5.2.2 Smooth parametric curves 5.2.3 Curvature 5.2.4 Section 5.2 summary 5.3 Practical example: Crafting a curve out of lines and circles 5.3.1 The biarc building block 5.3.2 The line segment and arc building block 5.3.3 The combination of both 5.3.4 Section 5.3 summary 5.4 Exercises 5.5 Solutions to exercises Summary 6 Polynomial approximation and interpolation 6.1 What are polynomials? 6.1.1 Axis intersections and roots of polynomial equations 6.1.2 Polynomial derivatives 6.1.3 Section 6.1 summary 6.2 Polynomial approximation 6.2.1 Maclaurin and Taylor series 6.2.2 The method of least squares 6.2.3 Practical example: Showing a trend with approximation 6.2.4 Section 6.2 summary 6.3 Polynomial interpolation 6.3.1 Using Vandermonde matrix to get the interpolating polynomial 6.3.2 What limits polynomial interpolation application to small data only? 6.3.3 How to lessen unwanted oscillations 6.3.4 Lagrange interpolation: Simple, genius, unnecessary 6.3.5 Practical example: Showing the trend with interpolation 6.3.6 Section 6.3 summary 6.4 Practical example: Showing a trend with both approximation and interpolation 6.4.1 The problem 6.4.2 The solution 6.4.3 Section 6.4 summary 6.5 Exercises 6.6 Solutions to exercises Summary 7 Splines 7.1 Going beyond the interpolation 7.1.1 Making polynomial graphs with tangent constraints 7.1.2 Practical example: Approximating the sine function for a space simulator game 7.1.3 Unexpected fact: Explicit polynomial modeling generalizes power series 7.1.4 Section 7.1 summary 7.2 Understanding polynomial splines and Bézier curves 7.2.1 Explicit polynomial parametric curves 7.2.2 Practical example: Crafting a spline for points that aren’t there yet 7.2.3 Bézier curves 7.2.4 Section 7.2 summary 7.3 Understanding NURBS 7.3.1 BS stands for “basis spline” 7.3.2 NU stands for “nonuniform” 7.3.3 R stands for “rational” 7.3.4 Not-so-practical example: Building a circle with NURBS 7.3.5 Section 7.3 summary 7.4 Exercises 7.5 Solutions to exercises Summary 8 Nonlinear transformations and surfaces 8.1 Polynomial transformations in multidimensional space 8.1.1 The straightforward approach to polynomial transformations 8.1.2 The fast and simple approach to polynomial transformations 8.1.3 Practical example: Adding the “unbending” feature to the scanning app 8.1.4 Section 8.1 summary 8.2 3D surface modeling 8.2.1 A surface is just a 3D transformation of a plane 8.2.2 Practical example: Planting polynomial mushrooms 8.2.3 Section 8.2 summary 8.3 Using nonpolynomial spatial interpolation in geometry 8.3.1 Inverse distance interpolation 8.3.2 Inverse distance interpolation in space 8.3.3 Practical example: Using localized inverse distance method to make a bitmap continuous and smooth 8.3.4 Practical example: Building a deformation field with multivariable interpolation to generate better mushrooms 8.3.5 Section 8.3 summary 8.4 Exercises 8.5 Solutions to exercises Summary 9 The geometry of vector algebra 9.1 Vector addition and scalar multiplication as transformations 9.2 Dot product: Projection and angle 9.2.1 Other names and motivations behind them 9.2.2 Vectors and their dot product in SymPy 9.2.3 Practical example: Using a dot product to light a scene 9.2.4 Section 9.2 summary 9.3 Cross product: Normal vector and the parallelogram area 9.3.1 Other names and motivations behind them 9.3.2 The cross product in SymPy 9.3.3 Practical example: Check whether a triangle contains a point in 2D 9.3.4 Section 9.3 summary 9.4 Triple product: The parallelepiped volume 9.4.1 The geometric sense of the triple product 9.4.2 The triple product in SymPy 9.4.3 Practical example: Distance to a plane 9.4.4 Section 9.4 summary 9.5 Generalization for parallelotopes 9.5.1 The meaning 9.5.2 The math 9.5.3 The code 9.5.4 Section 9.5 summary 9.6 Exercises 9.7 Solutions to exercises Summary 10 Modeling shapes with signed distance functions and surrogates 10.1 What’s an SDF? 10.1.1 Does it have to be an SDF? 10.1.2 How to program SDFs? 10.1.3 Theoretical example: Making an SDF of a triangle mesh 10.1.4 Practical example: Making an SDF of a rectangle in 2D 10.1.5 Section 10.1 summary 10.2 How to work with SDFs 10.2.1 How to translate, rotate, or scale an SDF 10.2.2 How to unite, intersect, and subtract SDFs 10.2.3 How to dilate and erode an SDF 10.2.4 How to hollow an SDF 10.2.5 Section 10.2 summary 10.3 Some techniques of not-really-SDF implicit modeling 10.3.1 Tri-periodic minimal surfaces 10.3.2 Practical example: A gyroid with variable thickness 10.3.3 Metaballs 10.3.4 Practical example: Localizing the metaballs for speed and better governance 10.3.5 Multifocal lemniscates 10.3.6 Practical example: A play button made of a multifocal lemniscate 10.3.7 Section 10.3 summary 10.4 Exercises 10.5 Solutions to exercises Summary 11 Modeling surfaces with boundary representations and triangle meshes 11.1 Smooth curves and surfaces 11.1.1 Data representation 11.1.2 Operations are also data 11.1.3 Section 11.1 summary 11.2 Segments and triangles 11.2.1 Vertices and triangles vs. the half-edge representation 11.2.2 Pros and cons of triangle meshes 11.2.3 Section 11.2 summary 11.3 Practical example: Contouring with marching cubes and dual contouring algorithms 11.3.1 Marching cubes 11.3.2 Dual contouring 11.3.3 Is it that simple in practice, too? 11.3.4 Section 11.3 summary 11.4 Practical example: Smooth contouring 11.4.1 The algorithm 11.4.2 The implementation 11.4.3 Section 11.4 summary 11.5 Exercises 11.6 Solutions to exercises Summary 12 Modeling bodies with images and voxels 12.1 How does computed tomography work? 12.2 Segmentation by a threshold 12.3 Typical operations on 3D images: Dilation, erosion, cavity fill, and Boolean 12.3.1 Dilation 12.3.2 Erosion 12.3.3 Practical example: Denoising 12.3.4 Boolean operations on voxel models 12.3.5 A few other uses of dilation and erosion 12.3.6 Section 12.3 summary 12.4 Practical example: Image vectorization 12.4.1 Input image 12.4.2 Step 1: Obtain a contour 12.4.3 Step 2: Fit the contour 12.4.4 Step 3: Make the contour smooth 12.4.5 Implementation 12.4.6 Section 12.4 summary 12.5 How voxels, triangles, parametric surfaces, and SDFs work together 12.6 Exercises 12.7 Solutions to exercises Summary Appendix—Sources, references, and further reading Books Papers Websites Interactive web pages index Symbols Numerics A B C D E F G H I L M N O P Q R S T U V W

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