Math for Programmers: 3D Graphics, Machine Learning, and Simulations with Python

Math for Programmers: 3D graphics, machine learning, and simulations with Python MEAP V08
Copyright
Welcome
Brief contents
1: Learning Math in Code
1.1 Solving lucrative problems with math and software
1.1.1 Predicting financial market movements
1.1.2 Finding a good deal
1.1.3 Building 3D graphics and animations
1.1.4 Modeling the physical world
1.2 How not to learn math
1.2.1 Jane wants to learn some math
1.2.2 Slogging through math textbooks
1.3 Using your well-trained left brain
1.3.1 Using a formal language
1.3.2 Build your own calculator
1.3.3 Building abstractions with functions
1.4 Summary
2: Drawing with 2D Vectors
2.1 Drawing with 2D Vectors
2.1.1 Representing 2D vectors
2.1.2 2D Drawing in Python
2.1.3 Exercises
2.2 Plane vector arithmetic
2.2.1 Vector components and lengths
2.2.2 Multiplying Vectors by Numbers
2.2.3 Subtraction, displacement, and distance
2.2.4 Exercises
2.3 Angles and Trigonometry in the Plane
2.3.1 From angles to components
2.3.2 Radians and trigonometry in Python
2.3.3 From components back to angles
2.3.4 Exercises
2.4 Transforming collections of vectors
2.4.1 Combining vector transformations
2.4.2 Exercises
2.5 Drawing with Matplotlib
2.6 Summary
3: Ascending to the 3D World
3.1 Picturing vectors in three-dimensional space
3.1.1 Representing 3D vectors with coordinates
3.1.2 3D Drawing in Python
3.1.3 Exercises
3.2 Vector arithmetic in 3D
3.2.1 Adding 3D vectors
3.2.2 Scalar Multiplication in 3D
3.2.3 Subtracting 3D vectors
3.2.4 Computing lengths and distances
3.2.5 Computing angles and directions
3.2.6 Exercises
3.3 The dot product: measuring alignment of vectors
3.3.1 Picturing the dot product
3.3.2 Computing the dot product
3.3.3 Dot products by example
3.3.4 Measuring angles with the dot product
3.3.5 Exercises
3.4 The cross product: measuring oriented area
3.4.1 Orienting ourselves in 3D
3.4.2 Finding the direction of the cross product
3.4.3 Finding the length of the cross product
3.4.4 Computing the cross product of 3D vectors
3.4.5 Exercises
3.5 Rendering a 3D object in 2D
3.5.1 Defining a 3D object with vectors
3.5.2 Projecting to 2D
3.5.3 Orienting faces and shading
3.5.4 Exercises
3.6 Summary
4: Transforming Vectors and Graphics
4.1 Transforming 3D objects
4.1.1 Drawing a transformed object
4.1.2 Composing vector transformations
4.1.3 Rotating an object about an axis
4.1.4 Inventing your own geometric transformations
4.1.5 Exercises
4.2 Linear transformations
4.2.1 Preserving vector arithmetic
4.2.2 Picturing linear transformations
4.2.3 Why linear transformations?
4.2.4 Computing linear transformations
4.2.5 Exercises
4.3 Summary
5: Computing Transformations with Matrices
5.1 Representing linear transformations with matrices
5.1.1 Writing vectors and linear transformations as matrices
5.1.2 Multiplying a matrix with a vector
5.1.3 Composing linear transformations by matrix multiplication
5.1.4 Implementing matrix multiplication
5.1.5 3D Animation with matrix transformations
5.1.6 Exercises
5.2 Interpreting matrices of different shapes
5.2.1 Column vectors as matrices
5.2.2 What pairs of matrices can be multiplied?
5.2.3 Viewing square and non-square matrices as vector functions
5.2.4 Projection as a linear map from 3D to 2D
5.2.5 Composing linear maps
5.2.6 Exercises
5.3 Translating vectors with matrices
5.3.1 Making plane translations linear
5.3.2 Finding a 3D matrix for a 2D translation
5.3.3 Combining translation with other linear transformations
5.3.4 Translating 3D objects in a 4D world
5.3.5 Exercises
5.4 Summary
6: Generalizing to Higher Dimensions
6.1 Generalizing our definition of vectors
6.1.1 Creating a class for 2D coordinate vectors
6.1.2 Improving the Vec2 class
6.1.3 Repeating the process with 3D vectors
6.1.4 Building a Vector base class
6.1.5 Defining vector spaces
6.1.6 Unit testing vector space classes
6.1.7 Exercises
6.2 Exploring different vector spaces
6.2.1 Enumerating all coordinate vector spaces
6.2.2 Identifying vector spaces in the wild
6.2.3 Treating functions as vectors
6.2.4 Treating matrices as vectors
6.2.5 Manipulating images with vector operations
6.2.6 Exercises
6.3 Looking for smaller vector spaces
6.3.1 Identifying subspaces
6.3.2 Starting with a single vector
6.3.3 Spanning a bigger space
6.3.4 Defining the word “dimension”
6.3.5 Finding subspaces of the vector space of functions
6.3.6 Subspaces of images
6.3.7 Exercises
6.4 Summary
7: Solving Systems of Linear Equations
7.1 Designing an arcade game
7.1.1 Modeling the game
7.1.2 Rendering the game
7.1.3 Shooting the laser
7.1.4 Exercises
7.2 Finding intersection points of lines
7.2.1 Choosing the right formula for a line
7.2.2 Finding the standard form equation for a line
7.2.3 Linear equations in matrix notation
7.2.4 Solving linear equations with numpy
7.2.5 Deciding whether the laser hits an asteroid
7.2.6 Identifying unsolvable systems
7.2.7 Exercises
7.3 Generalizing linear equations to higher dimensions
7.3.1 Representing planes in 3D
7.3.2 Solving linear equations in 3D
7.3.3 Studying hyperplanes algebraically
7.3.4 Counting dimensions, equations, and solutions
7.3.5 Exercises
7.4 Changing basis by solving linear equations
7.4.1 Solving a 3D example
7.4.2 Exercises
7.5 Summary
8: Understanding Rates of Change
8.1 Calculating average flow rates from volumes
8.1.1 Implementing an average_flow_rate function
8.1.2 Picturing the average flow rate with a secant line
8.1.3 Negative rates of change
8.1.4 Exercises
8.2 Plotting the average flow rate over time
8.2.1 Finding the average flow rate in different time intervals
8.2.2 Plotting the interval flow rates alongside the flow rate function
8.2.3 Exercises
8.3 Approximating instantaneous flow rates
8.3.1 Finding the slope of very small secant lines
8.3.2 Building the instantaneous flow rate function
8.3.3 Currying and plotting the instantaneous flow rate function
8.3.4 Exercises
8.4 Approximating the change in volume
8.4.1 Finding the change in volume on a short time interval
8.4.2 Breaking up time into small intervals
8.4.3 Picturing the volume change on the flow rate graph
8.4.4 Exercises
8.5 Plotting the volume over time
8.5.1 Finding the volume over time
8.5.2 Picturing Riemann sums for the volume function
8.5.3 Improving the approximation
8.5.4 Definite and indefinite integrals
8.6 Summary
9: Simulating Moving Objects
9.1 Simulating constant-speed motion
9.1.1 Intuiting speed
9.1.2 Thinking of velocity as a vector
9.1.3 Animating the asteroids
9.1.4 Exercises
9.2 Simulating acceleration
9.2.1 Picturing acceleration in various directions
9.2.2 Quantifying Acceleration
9.2.3 Accelerating the spaceship
9.2.4 Exercises
9.3 Digging deeper into Euler’s method
9.3.1 Stepping through Euler’s method
9.3.2 Implementing the algorithm in Python
9.3.3 Picturing the approximation
9.3.4 Applying Euler’s method to other problems
9.3.5 Exercises
9.4 Calculating exact trajectories
9.4.1 Writing position, velocity, and acceleration as functions of time
9.4.2 Using the terminology of integration
9.4.3 Calculating integrals
9.4.4 Exercises
9.5 Summary
10: Working with Symbolic Expressions
10.1 Modeling algebraic expressions
10.1.1 Breaking an expression into pieces
10.1.2 Building an expression tree
10.1.3 Translating the expression tree to Python
10.1.4 Exercises
10.2 Putting a symbolic expression to work
10.2.1 Finding all the variables in an expression
10.2.2 Evaluating an expression
10.2.3 Expanding an expression
10.2.4 Exercises
10.3 Finding the derivative of a function
10.3.1 Derivatives of powers
10.3.2 Derivatives of transformed functions
10.3.3 Derivatives of some special functions
10.3.4 Derivatives of products and compositions
10.3.5 Exercises
10.4 Taking derivatives automatically
10.4.1 Implementing a derivative method for expressions
10.4.2 Implementing the product rule and chain rule
10.4.3 Implementing the power rule
10.4.4 Exercises
10.5 Integrating functions symbolically
10.5.1 Integrals as antiderivatives
10.5.2 Introducing the SymPy library
10.5.3 Exercises
10.6 Summary
11: Simulating Force Fields
11.1 1 Modeling gravitational fields
11.1.1 Defining a vector field
11.1.2 Defining a simple force field
11.2 Adding gravity to the asteroid game
11.2.1 Making game objects feel gravity
11.2.2 Exercises
11.3 Introducing potential energy
11.3.1 Defining a potential energy scalar field
11.3.2 Plotting a scalar field as a heatmap
11.3.3 Plotting a scalar field as a contour map
11.4 4 Connecting energy and forces with the gradient
11.4.1 Measuring steepness with cross sections
11.4.2 Calculating partial derivatives
11.4.3 Finding the steepness of a graph with the gradient
11.4.4 Calculating force fields from potential energy with the gradient
11.4.5 Exercises
11.5 5 Summary
12: Optimizing a Physical System
12.1 Testing a projectile simulation
12.1.1 Building a simulation with Euler’s method
12.1.2 Measuring properties of the trajectory
12.1.3 Exploring different launch angles
12.1.4 Exercises
12.2 Calculating the optimal range
12.2.1 Finding the projectile range as a function of the launch angle
12.2.2 Solving for the maximum range
12.2.3 Identifying maxima and minima
12.2.4 Exercises
12.3 Enhancing our simulation
12.3.1 Adding another dimension
12.3.2 Modeling terrain around the cannon
12.3.3 Solving for the range of the projectile in 3D
12.3.4 Exercises
12.4 Optimizing range using gradient ascent
12.4.1 Plotting range versus launch parameters
12.4.2 The gradient of the range function
12.4.3 Finding the uphill direction with the gradient
12.4.4 Implementing gradient ascent
12.4.5 Exercises
12.5 Summary
A: Loading and Rendering 3D Models with OpenGL and PyGame
A.1 Recreating the octahedron from Chapter 3
A.2 Changing our perspective
A.3 Loading and rendering the Utah teapot
A.4 Exercises
B: Getting set up with Python
B.1 Checking for an existing Python installation
B.2 Downloading and installing Anaconda
B.3 Using Python in interactive mode
B.4 Creating and running a Python script file
B.5 Using Jupyter notebooks