3D Math Primer For Graphics And Game Development 1st Edition by Fletcher Dunn, Ian Parberry ISBN 9781284096200 1284096203

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ISBN 10: 1284096203
ISBN 13: 9781284096200
Author: Fletcher Dunn, Ian Parberry

3D Math Primer For Graphics And Game Development 1st Edition Table of contents:

Chapter 1 Introduction

1.1 What is 3D Math?

1.2 Why You Should Read This Book

1.3 What You Should Know Before Reading This Book

1.4 Overview

Chapter 2 The Cartesian Coordinate System

2.1 1D Mathematics

Figure 2.1: One dead sheep

Figure 2.2: A number line for the natural numbers

Figure 2.3: A number line for integers (note the ghost sheep for negative numbers)

2.2 2D Cartesian Mathematics

2.2.1 An Example: The Hypothetical City of Cartesia

Figure 2.4: Map of the hypothetical city of Cartesia

2.2.2 Arbitrary 2D Coordinate Spaces

Figure 2.5: A 2D Cartesian coordinate space

Figure 2.6: Screen coordinate space

Figure 2.7: Possible map axis orientations in 2D

2.2.3 Specifying Locations in 2D Using Cartesian Coordinates

Figure 2.8: How to locate a point using 2D Cartesian coordinates

Figure 2.9: Example points labeled with 2D Cartesian coordinates

2.3 From 2D to 3D

2.3.1 Extra Dimension, Extra Axis

Figure 2.10: A 3D Cartesian coordinate space

2.3.2 Specifying Locations in 3D

Figure 2.11: Locating points in 3D

2.3.3 Left-handed vs. Right-handed Coordinate Spaces

Figure 2.12: Left-handed coordinate space

Figure 2.13: Right-handed coordinate space

Figure 2.14: Comparison of left- and right-handed coordinate systems

Figure 2.15: Viewing a left-handed coordinate space from the positive end of the x-axis

2.3.4 Some Important Conventions Used in This Book

Figure 2.16: The left-handed coordinate system conventions used in this book

2.4 Exercises

Figure 2.17

Chapter 3 Multiple Coordinate Spaces

3.1 Why Multiple Coordinate Spaces?

3.2 Some Useful Coordinate Spaces

3.2.1 World Space

3.2.2 Object Space

3.2.3 Camera Space

Figure 3.1: Camera space

3.2.4 Inertial Space

Figure 3.2: Object, inertial, and world spaces

Figure 3.3: The robot’s object space

Figure 3.4: The robot’s inertial space

Figure 3.5: The world space

3.3 Nested Coordinate Spaces

3.4 Specifying Coordinate Spaces

3.5 Coordinate Space Transformations

Figure 3.6: The robot in object space

Figure 3.7: The robot in inertial space

Figure 3.8: The robot in world space

3.6 Exercises

Chapter 4 Vectors

4.1 Vector — A Mathematical Definition

4.1.1 Vectors vs. Scalars

4.1.2 Vector Dimension

4.1.3 Notation

Equation 4.1: Vector subscript notation

4.2 Vector — A Geometric Definition

4.2.1 What Does a Vector Look Like?

Figure 4.1: A 2D vector

Figure 4.2: A vector has a head and a tail

4.2.2 Position vs. Displacement

4.2.3 Specifying Vectors

Figure 4.3: Vectors are specified by giving the signed displacement in each dimension

Figure 4.4: Examples of 2D vectors and their values

4.2.4 Vectors as a Sequence of Displacements

Figure 4.5: Interpreting a vector as a sequence of displacements

4.3 Vectors vs. Points

Figure 4.6: Locating points vs. specifying vectors

4.3.1 Relative Position

4.3.2 The Relationship Between Points and Vectors

Figure 4.7: The relationship between points and vectors

4.4 Exercises

Chapter 5 Operations on Vectors

5.1 Linear Algebra vs. What We Need

5.2 Typeface Conventions

5.3 The Zero Vector

Figure 5.1: For any positive magnitude, there are an infinite number of vectors with that magnitude.

5.4 Negating a Vector

5.4.1 Official Linear Algebra Rules

Equation 5.1: Negating a vector

Equation 5.2: Negating 2D, 3D, and 4D vectors

5.4.2 Geometric Interpretation

Figure 5.2: Examples of vectors and their negatives

5.5 Vector Magnitude (Length)

5.5.1 Official Linear Algebra Rules

Equation 5.3: Vector magnitude

Equation 5.4: Vector magnitude for 2D and 3D vectors

5.5.2 Geometric Interpretation

Figure 5.3: Geometric interpretation of the magnitude equation

5.6 Vector Multiplication by a Scalar

5.6.1 Official Linear Algebra Rules

Equation 5.5: Multiplying a vector by a scalar

Equation 5.6: Multiplying a 3D vector by a scalar

Equation 5.7: Dividing a 3D vector by a scalar

5.6.2 Geometric Interpretation

Figure 5.4: A 2D vector multiplied by various scalars

5.7 Normalized Vectors

5.7.1 Official Linear Algebra Rules

Equation 5.8: Normalizing a vector

5.7.2 Geometric Interpretation

Figure 5.5: Normalizing vectors in 2D

5.8 Vector Addition and Subtraction

5.8.1 Official Linear Algebra Rules

Equation 5.9: Adding two vectors

Equation 5.10: Subtracting two vectors

5.8.2 Geometric Interpretation

Figure 5.6: 2D vector addition and subtraction using the triangle rule

Figure 5.7: Extending the triangle rule to more than two vectors

Figure 5.8: Interpreting a vector as a sequence of displacements

5.8.3 Vector from One Point to Another

Figure 5.9: Using 2D vector subtraction to compute the vector from point a to point b

5.9 The Distance Formula

Equation 5.11: The 3D distance formula

Equation 5.12: The 2D distance formula

5.10 Vector Dot Product

5.10.1 Official Linear Algebra Rules

Equation 5.13: Vector dot product

Equation 5.14: Vector dot product expressed using summation notation

Equation 5.15: 2D and 3D dot product

5.10.2 Geometric Interpretation

Figure 5.10: The dot product is related to the angle between two vectors

Equation 5.16: Geometric interpretation of the vector dot product

Equation 5.17: Computing the angle between two vectors using the dot product

Equation 5.18: Computing the angle between two unit vectors

Figure 5.11: The sign of the dot product gives a rough classification of the angle between two vectors

5.10.3 Projecting One Vector onto Another

Figure 5.12: Projecting one vector onto another

Equation 5.19: Projecting one vector onto another

5.11 Vector Cross Product

5.11.1 Official Linear Algebra Rules

Equation 5.20: Cross product

5.11.2 Geometric Interpretation

Figure 5.13: Vector cross product

Equation 5.21: The magnitude of the cross product is related to the sine of the angle between the vectors

Figure 5.14: The cross product and the area of a parallelogram

Figure 5.15: Area of a parallelogram

Figure 5.16: Clockwise turn

Figure 5.17: Counterclockwise turn

5.12 Linear Algebra Identities

Figure 5.18: Table of vector algebra identities

5.13 Exercises

Chapter 6 A Simple 3D Vector Class

6.1 Class Interface

6.2 Class Vector3 Definition

Listing 6.1: Vector3.h

6.3 Design Decisions

6.3.1 Floats vs. Doubles

6.3.2 Operator Overloading

6.3.3 Provide Only the Most Important Operations

6.3.4 Don’t Overload Too Many Operators

6.3.5 Use Const Member Functions

6.3.6 Use Const Reference Arguments

6.3.7 Member vs. Nonmember Functions

Listing 6.2: Member vs. nonmember function semantics

6.3.8 No Default Initialization

6.3.9 Don’t Use Virtual Functions

6.3.10 Don’t Use Information Hiding

6.3.11 Global Zero Vector Constant

6.3.12 No “point3” Class

6.3.13 A Word on Optimization

Chapter 7 Introduction to Matrices

7.1 Matrix — A Mathematical Definition

7.1.1 Matrix Dimensions and Notation

7.1.2 Square Matrices

Equation 7.1: The 3D identity matrix

7.1.3 Vectors as Matrices

7.1.4 Transposition

Equation 7.2: Transposing matrices

Equation 7.3: Transposing converts between row and column vectors

7.1.5 Multiplying a Matrix with a Scalar

Equation 7.4: Multiplying a 4×3 matrix by a scalar

7.1.6 Multiplying Two Matrices

Equation 7.5: 2×2 matrix multiplication

Equation 7.6: 3×3 matrix multiplication

7.1.7 Multiplying a Vector and a Matrix

Equation 7.7: Multiplying 3D row and column vectors with a 3×3 matrix

7.1.8 Row vs. Column Vectors

7.2 Matrix — A Geometric Interpretation

7.2.1 How Does a Matrix Transform Vectors?

Equation 7.8: Expressing a vector as a linear combination of basis vectors

Equation 7.9: Interpreting a matrix as a set of basis vectors

7.2.2 What Does a Matrix Look Like?

Figure 7.1: Visualizing the row vectors of a 2D transform matrix

Figure 7.2: The 2D parallelogram formed by the rows of a matrix

Figure 7.3: Drawing an object inside the box helps visualize the transformation

Figure 7.4: Teapot, unit cube, and basis vectors before transformation

Figure 7.5: Teapot, unit cube, and basis vectors after transformation

7.2.3 Summary

7.3 Exercises

Chapter 8 Matrices and Linear Transformations

8.1 Transforming an Object vs. Transforming the Coordinate Space

Figure 8.1: Rotating an object clockwise 20°

Figure 8.2: Rotating a coordinate space clockwise 20°

Figure 8.3: A useful example of rotating a coordinate space

Figure 8.4: Rotating the coordinate space is the same as rotating the object by the opposite amount

8.2 Rotation

8.2.1 Rotation in 2D

Figure 8.5: Rotation about the origin in 2D

Equation 8.1: 2D rotation matrix

8.2.2 3D Rotation about Cardinal Axes

Figure 8.6: The left-hand rule defines positive rotation in a left-handed coordinate system

Figure 8.7: The right-hand rule defines positive rotation in a right-handed coordinate system

Figure 8.8: Positive and negative rotation about an axis

Figure 8.9: Rotating about the x-axis in 3D

Equation 8.2: 3D matrix to rotate about the x-axis

Figure 8.10: Rotating about the y-axis in 3D

Equation 8.3: 3D matrix to rotate about the y-axis

Figure 8.11: Rotating about the z-axis in 3D

Equation 8.4: 3D matrix to rotate about the z-axis

8.2.3 3D Rotation about an Arbitrary Axis

Figure 8.12: Rotating a vector about an arbitrary axis

Equation 8.5: 3D matrix to rotate about an arbitrary axis

8.3 Scale

8.3.1 Scaling along Cardinal Axes

Figure 8.13: Scaling a 2D object with various factors for kx and ky

Equation 8.6: 2D matrix to scale on cardinal axes

Equation 8.7: 3D matrix to scale on cardinal axes

8.3.2 Scale in an Arbitrary Direction

Figure 8.14: Scaling a vector along an arbitrary direction

Equation 8.8: 2D matrix to scale in an arbitrary direction

Equation 8.9: 3D matrix to scale in an arbitrary direction

8.4 Orthographic Projection

8.4.1 Projecting onto a Cardinal Axis or Plane

Figure 8.15: Projecting a 3D object onto a cardinal plane

Equation 8.10: 2D matrix to project onto the x-axis

Equation 8.11: 2D matrix to project onto the y-axis

Equation 8.12: 3D matrix to project onto the xy-plane

Equation 8.13: 3D matrix to project onto the xz-plane

Equation 8.14: 3D matrix to project onto the yz-plane

8.4.2 Projecting onto an Arbitrary Line or Plane

Equation 8.15: 2D matrix to project onto an arbitrary line

Equation 8.16: 3D matrix to project onto an arbitrary plane

8.5 Reflection

Figure 8.16: Reflecting an object about an axis in 2D

Equation 8.17: 2D matrix to reflect about an arbitrary axis

Equation 8.18: 3D matrix to reflect about an arbitrary plane

8.6 Shearing

Figure 8.17: Shearing in 2D

Equation 8.19: 3D shear matrices

8.7 Combining Transformations

8.8 Classes of Transformations

8.8.1 Linear Transformations

8.8.2 Affine Transformations

8.8.3 Invertible Transformations

8.8.4 Angle-preserving Transformations

8.8.5 Orthogonal Transformations

8.8.6 Rigid Body Transformations

8.8.7 Summary of Types of Transformations

8.9 Exercises

Chapter 9 More on Matrices

9.1 Determinant of a Matrix

9.1.1 Official Linear Algebra Rules

Equation 9.1: Determinant of a 2×2 matrix

Equation 9.2: Determinant of a 3×3 matrix

Equation 9.3: Matrix cofactor

Equation 9.4: Computing an n×n determinant using cofactors

Equation 9.5: Determinant of a 4×4 matrix

9.1.2 Geometric Interpretation

Figure 9.1: The determinant in 2D is the signed area of the skew box formed by the transformed basis vectors

9.2 Inverse of a Matrix

9.2.1 Official Linear Algebra Rules

Equation 9.6: Matrix inverse

Equation 9.7: The inverse of a matrix can be computed as the classical adjoint divided by the determinant

9.2.2 Geometric Interpretation

9.3 Orthogonal Matrices

9.3.1 Official Linear Algebra Rules

Equation 9.8: Definition of orthogonal matrix

9.3.2 Geometric Interpretation

9.3.3 Orthogonalizing a Matrix

Equation 9.9: Gram-Schmidt orthogonalization of 3D basis vectors

9.4 4×4 Homogenous Matrices

9.4.1 4D Homogenous Space

Figure 9.2: Projecting homogenous coordinates onto the plane w = 1 in 2D

9.4.2 4×4 Translation Matrices

Equation 9.10: Using a 4×4 matrix to perform translation in 3D

9.4.3 General Affine Transformations

9.4.4 Perspective Projection

Figure 9.3: Orthographic projection uses parallel projectors

Figure 9.4: Perspective projection. The projectors intersect at the center of projection

Figure 9.5: Perspective foreshortening

9.4.5 A Pinhole Camera

Figure 9.6: A pinhole camera

Figure 9.7: A projection plane parallel to the xy plane

Figure 9.8: Viewing the projection plane from the side

Equation 9.11: Projecting onto the plane z = −d

Figure 9.9: Projection plane in front of the center of projection

Equation 9.12: Projecting a point onto the plane z = d

9.4.6 Perspective Projection Using 4×4 Matrices

Equation 9.13: Projecting onto the plane z = d using a 4×4 matrix

9.5 Exercises

Chapter 10 Orientation and Angular Displacement in 3D

10.1 What is Orientation?

Figure 10.1: “Twisting” a vector along its length results in no appreciable change to the vector

Figure 10.2: “Twisting” an object changes its orientation

10.2 Matrix Form

Figure 10.3: Defining an orientation using a matrix

10.2.1 Which Matrix?

10.2.2 Advantages of Matrix Form

10.2.3 Disadvantages of Matrix Form

10.2.4 Summary

10.3 Euler Angles

10.3.1 What are Euler Angles?

Figure 10.4: Heading is the first rotation and rotates about the y-axis

Figure 10.5: Pitch is the second rotation and rotates about the object space x-axis

Figure 10.6: Bank is the third and final rotation; it rotates about the object space z-axis

10.3.2 Other Euler Angle Conventions

10.3.3 Advantages of Euler Angles

10.3.4 Disadvantages of Euler Angles

Figure 10.7: Naïve interpolation can cause excessive rotation

Figure 10.8: Naïve interpolation can rotate “the long way around”

10.3.5 Summary

10.4 Quaternions

10.4.1 Quaternion Notation

10.4.2 Quaternions as Complex Numbers

Equation 10.1: Adding, subtracting, and multiplying complex numbers

Equation 10.2: Complex conjugate

Equation 10.3: Magnitude of a complex number

Figure 10.9: Rotating a complex number in the plane

10.4.3 Quaternions as an Axis-Angle Pair

Equation 10.4: The four values of a quaternion

10.4.4 Quaternion Negation

Equation 10.5: Quaternion negation

10.4.5 Identity Quaternion(s)

10.4.6 Quaternion Magnitude

Equation 10.6: Quaternion magnitude

10.4.7 Quaternion Conjugate and Inverse

Equation 10.7: Quaternion conjugate

Equation 10.8: Quaternion inverse

10.4.8 Quaternion Multiplication (Cross Product)

Equation 10.9: Standard definition of quaternion product

Equation 10.10: Quaternion multiplication is associative, but not commutative

Equation 10.11: Magnitude of quaternion product is equal to the product of the magnitudes

Equation 10.12: Inverse of quaternion product equals the product of inverses, taken in reverse order

Equation 10.13: Definition of quaternion multiplication used in this book

10.4.9 Quaternion “Difference”

10.4.10 Quaternion Dot Product

Equation 10.14: Quaternion dot product

10.4.11 Quaternion Log, Exp, and Multiplication by a Scalar

Equation 10.15: The logarithm of a quaternion

Equation 10.16: The exponential function of a quaternion

Equation 10.17: Multiplying a quaternion by a scalar

10.4.12 Quaternion Exponentiation

Equation 10.18: Raising a quaternion to a power

Listing 10.1: Code to raise a quaternion to a power

10.4.13 Quaternion Interpolation — aka “Slerp”

Equation 10.19: Quaternion slerp in theory

Figure 10.10: Interpolating a rotation

Figure 10.11: Interpolating a vector about an arc

Listing 10.2: How slerp is computed in practice

10.4.14 Quaternion Splines — aka “Squad”

Equation 10.20: Squad

10.4.15 Advantages/Disadvantages of Quaternions

10.5 Comparison of Methods

Figure 10.12: Comparison of matrices, Euler angles, and quaternions

10.6 Converting between Representations

10.6.1 Converting Euler Angles to a Matrix

Equation 10.21: Computing the inertial-to-object rotation matrix from a set of Euler angles

Equation 10.22: Computing the object-to-inertia l rotation matrix from a set of Euler angles

10.6.2 Converting a Matrix to Euler Angles

Listing 10.3: Extracting Euler angles from an inertial-to-object rotation matrix

10.6.3 Converting a Quaternion to a Matrix

Equation 10.23: Converting a quaternion to matrix form

10.6.4 Converting a Matrix to a Quaternion

Listing 10.4: Converting a rotation matrix to a quaternion

10.6.5 Converting Euler Angles to a Quaternion

Equation 10.24: Computing the inertial-to-object quaternion from a set of Euler angles

Equation 10.25: Computing the object-to-inertial quaternion from a set of Euler angles

10.6.6 Converting a Quaternion to Euler Angles

Listing 10.5: Converting an inertial-to-object quaternion to Euler angles

Listing 10.6: Converting an object-to-inertial quaternion to Euler angles

10.7 Exercises

Chapter 11 Transformations in C++

11.1 Overview

Listing 11.1: MathUtil.h

11.2 Class EulerAngles

Listing 11.2: EulerAngles.h

Listing 11.3: EulerAngles.cpp

11.3 Class Quaternion

Listing 11.4: Quaternion.h

Listing 11.5: Quaternion.cpp

11.4 Class RotationMatrix

Listing 11.6: RotationMatrix.h

Listing 11.7: RotationMatrix.cpp

11.5 Class Matrix4×3

Listing 11.8: Matrix4×3.h

Listing 11.9: Matrix4×3.cpp

Chapter 12 Geometric Primitives

12.1 Representation Techniques

12.1.1 Implicit Form

12.1.2 Parametric Form

Figure 12.1: Parametric circle

12.1.3 “Straightforward” Forms

12.1.4 Degrees of Freedom

12.2 Lines and Rays

Figure 12.2: Line vs. line segment vs. ray

12.2.1 Two Points Representation

Figure 12.3: Defining a ray using the starting and ending points

12.2.2 Parametric Representation of Rays

Equation 12.1: Parametric definition of a 2D ray

Equation 12.2: Parametric definition of a ray using vector notation

Figure 12.4: Defining a ray parametrically

12.2.3 Special 2D Representations of Lines

Equation 12.3: Implicit definition of infinite line in 2D

Equation 12.4: Implicit definition of infinite 2D line using vector notation

Equation 12.5: Slope-intercept form

Figure 12.5: The slope and y-intercept of a line

Figure 12.6: Defining a line using a perpendicular vector and distance to the origin

Figure 12.7: Defining a line using a perpendicular vector and a point on the line

Figure 12.8: Defining a line as the perpendicular bisector of a line segment

12.2.4 Converting between Representations

12.3 Spheres and Circles

Figure 12.9: A sphere is defined by its center and radius

Equation 12.6: Implicit definition of a sphere using vector notation

Equation 12.7: Implicit definition of a sphere

Equation 12.8: Diameter and circumference of circles and spheres

Equation 12.9: Area of a circle

Equation 12.10: Surface area and volume of a sphere

12.4 Bounding Boxes

Figure 12.10: 3D objects and their AABBs

12.4.1 Representing AABBs

12.4.2 Computing AABBs

Listing 12.1: Computing the AABB for a set of points

12.4.3 AABBs vs. Bounding Spheres

Figure 12.11: The AABB and bounding sphere for various objects

12.4.4 Transforming AABBs

Figure 12.12: The AABB of a transformed box

12.5 Planes

12.5.1 Implicit Definition — The Plane Equation

Equation 12.11: The plane equation

Figure 12.13: The front and back sides of a plane

12.5.2 Definition Using Three Points

Figure 12.14: Computing a plane normal from three points in the plane

Equation 12.12: The normal of a plane containing three points

12.5.3 “Best-fit” Plane for More Than Three Points

Equation 12.13: Computing the best-fit plane normal from n points

Listing 12.2: Computing the best-fit plane normal for a set of points

12.5.4 Distance from Point to Plane

Figure 12.15: Computing the distance between a point and a plane

Equation 12.14: Computing the signed distance from a plane to an arbitrary 3D point

12.6 Triangles

12.6.1 Basic Properties of a Triangle

Figure 12.16: Labeling triangles

Equation 12.15: Law of sines

Equation 12.16: Law of cosines

Equation 12.17: Perimeter of a triangle

12.6.2 Area of a Triangle

Figure 12.17: A triangle enclosed in a parallelogram

Equation 12.18: The area of a triangle is half the area of the parallelogram that shares two sides with the triangle

Equation 12.19: Heron’s formula for the area of a triangle

Figure 12.18: The area “beneath” an edge vector

Equation 12.20: Computing the area of a 2D triangle from the coordinates of the vertices

12.6.3 Barycentric Space

Equation 12.21: Computing a 3D point from barycentric coordinates

Figure 12.19: Examples of barycentric coordinates

Figure 12.20: Barycentric coordinates tessellate the plane

Figure 12.21: Computing the barycentric coordinates for an arbitrary point p

Equation 12.22: Computing barycentric coordinates for a 2D point

Equation 12.23: Interpreting barycentric coordinates as ratios of areas

Listing 12.3: Computing barycentric coordinates in 3D

Figure 12.22: Computing barycentric coordinates in 3D

Equation 12.24: Computing barycentric coordinates in 3D

12.6.4 Special Points

Figure 12.23: The center of gravity of a triangle

Figure 12.24: The incenter of a triangle

Figure 12.25: The circumcenter of a triangle

12.7 Polygons

12.7.1 Simple vs. Complex Polygons

Figure 12.26: Simple vs. complex polygons

Figure 12.27: Turning complex polygons into simple ones by adding pairs of seam edges

12.7.2 Self-intersecting Polygons

Figure 12.28: A self-intersecting polygon

12.7.3 Convex vs. Concave Polygons

Figure 12.29: Convex vs. concave polygons

Listing 12.4: 3D polygon convexity test using angle sum

12.7.4 Triangulation and Fanning

Figure 12.30: Triangulating a convex polygon by fanning

12.8 Exercises

Chapter 13 Geometric Tests

13.1 Closest Point on 2D Implicit Line

Figure 13.1: Finding the closest point on a 2D implicit line

Equation 13.1: Computing the closest point on a 2D implicit line

13.2 Closest Point on Parametric Ray

Figure 13.2: Finding the closest point on a ray

Equation 13.2: Computing the closest point on a parametric ray

13.3 Closest Point on Plane

Equation 13.3: Computing the closest point on a plane

13.4 Closest Point on Circle/Sphere

Figure 13.3: Finding the closest point on a circle

Equation 13.4: Computing the closest point on a circle or sphere

13.5 Closest Point in AABB

Listing 13.1: Computing the closest point in an AABB to a point

13.6 Intersection Tests

13.7 Intersection of Two Implicit Lines in 2D

Equation 13.5: Computing the intersection of two lines in 2D

Figure 13.4: Intersection of two lines in 2D — the three cases

13.8 Intersection of Two Rays in 3D

Figure 13.5: Skew lines in 3D do not share a common plane or intersect

13.9 Intersection of Ray and Plane

Figure 13.6: Intersection of a ray and plane in 3D

Equation 13.6: Parametric intersection of a ray and a plane

13.10 Intersection of AABB and Plane

13.11 Intersection of Three Planes

Figure 13.7: Three planes intersect at a point in 3D

Equation 13.7: Three planes intersect at a point

13.12 Intersection of Ray and Circle/Sphere

Figure 13.8: Intersection of a ray and sphere

Equation 13.8: Parametric intersection of a ray and circle/sphere

13.13 Intersection of Two Circles/Spheres

Figure 13.9: Intersection of two spheres

Figure 13.10: Two moving spheres

Figure 13.11: Combining displacement vectors so that one sphere is considered stationary

Figure 13.12: Dynamic intersection of circles/spheres

13.14 Intersection of Sphere and AABB

13.15 Intersection of Sphere and Plane

Listing 13.2: Determining which side of a plane a sphere is on

Figure 13.13: A sphere moving toward a plane

Figure 13.14: Point of contact between a sphere and a plane

Equation 13.9: Dynamic intersection of a sphere and plane

13.16 Intersection of Ray and Triangle

13.17 Intersection of Ray and AABB

13.18 Intersection of Two AABBs

Figure 13.15: Projecting the dynamic AABB intersection problem onto one axis

Figure 13.16: Intersecting two intervals of time

13.19 Other Tests

13.20 Class AABB3

Listing 13.3: AABB3.h

Listing 13.4: AABB3.cpp

13.21 Exercises

Chapter 14 Triangle Meshes

Figure 14.1: Examples of triangle meshes

14.1 Representing Meshes

Listing 14.1: A trivial representation of a triangle mesh

14.1.1 Indexed Triangle Mesh

Listing 14.2: Indexed triangle mesh

14.1.2 Advanced Techniques

14.1.3 Specialized Representations for Rendering

14.1.4 Vertex Caching

14.1.5 Triangle Strips

Figure 14.2: A triangle strip

Figure 14.3: The triangles in a triangle strip alternate between clockwise and counterclockwise vertex ordering

Figure 14.4: Stitching together two triangle strips using degenerate triangles

Figure 14.5: The triangles obtained by stitching together two triangle strips

Figure 14.6: Joining two triangle strips by skipping vertices

14.1.6 Triangle Fans

Figure 14.7: A triangle fan

14.2 Additional Mesh Information

14.2.1 Texture Mapping Coordinates

14.2.2 Surface Normals

Figure 14.8: On the right, the box edges are not visible because there is only one normal at each corner

14.2.3 Lighting Values

14.3 Topology and Consistency

14.4 Triangle Mesh Operations

14.4.1 Piecewise Operations

14.4.2 Welding Vertices

Figure 14.9: Welding two nearly coincident vertices

Figure 14.10: A vertex is welded to an isolated vertex, causing unnecessary distortion to the mesh

Figure 14.11: Welding two vertices of a sliver triangle causes the triangle to become degenerate

Figure 14.12: A domino effect created by averaging vertex positions during welding

14.4.3 Detaching Faces

Figure 14.13: Detaching two triangles

14.4.4 Edge Collapse

Figure 14.14: Edge collapse and vertex split

14.4.5 Mesh Decimation

14.5 A C++ Triangle Mesh Class

Listing 14.3: EditTriMesh.h

Chapter 15 3D Math for Graphics

15.1 Graphics Pipeline Overview

Listing 15.1: Pseudocode for the graphics pipeline

15.2 Setting the View Parameters

15.2.1 Specifying the Output Window

Figure 15.1: Specifying the output window

15.2.2 Pixel Aspect Ratio

Equation 15.1: Computing the pixel aspect

15.2.3 The View Frustum

Figure 15.2: The 3D view frustum

15.2.4 Field of View and Zoom

Figure 15.3: Horizontal field of view

Figure 15.4: Geometric interpretation of zoom

Equation 15.2: Converting between zoom and field of view

15.3 Coordinate Spaces

15.3.1 Modeling and World Space

15.3.2 Camera Space

Figure 15.5: Camera space

15.3.3 Clip Space

Equation 15.3: The six planes of the view frustum in clip space

Equation 15.4: OpenGL-style clip matrix

Equation 15.5: DirectX-style clip matrix

15.3.4 Screen Space

Figure 15.6: The output window in screen space

Equation 15.6: Projecting and mapping to screen space

15.4 Lighting and Fog

15.4.1 Math on Colors

Figure 15.7: The color space cube

15.4.2 Light Sources

Figure 15.8: A point light

Figure 15.9: A conical spotlight

15.4.3 The Standard Lighting Equation — Overview

Equation 15.7: Overview of the standard lighting equation

15.4.4 The Specular Component

Figure 15.10: Phong model for specular reflection

Figure 15.11: Constructing the reflection vector r

Equation 15.8: Phong model for specular reflection

Figure 15.12: Different values for mgls and mspec

Figure 15.13: Blinn model for specular reflection

Equation 15.9: The halfway vector h, used in the Blinn model

Equation 15.10: The Blinn model for specular reflection

15.4.5 The Diffuse Component

Figure 15.14: Diffuse lighting models scattered reflections

Figure 15.15: Surfaces more perpendicular to the light rays receive more light per unit area

Equation 15.11: Calculating the diffuse component using Lambert’s law

15.4.6 The Ambient Component

15.4.7 Light Attenuation

Equation 15.12: Real-world light attenuation is inversely proportional to the square of the distance

15.4.8 The Lighting Equation — Putting It All Together

Equation 15.13: The standard lighting equation for one light source

Figure 15.16: The visual contribution of each of the components of the lighting equation

Equation 15.14: The standard lighting equation for multiple lights

15.4.9 Fog

Equation 15.15: Fog value computation using minimum and maximum fog distances

15.4.10 Flat Shading and Gouraud Shading

Figure 15.17: A flat shaded teapot

Figure 15.18: A Gouraud shaded teapot

Figure 15.19: A Phong shaded teapot

15.5 Buffers

Figure 15.20: Double buffering

15.6 Texture Mapping

Figure 15.21: A 3D model before and after texture mapping

Figure 15.22: An example texture map

15.7 Geometry Generation/Delivery

15.7.1 LOD Selection and Procedural Modeling

15.7.2 Delivery of Geometry to the API

15.8 Transformation and Lighting

15.8.1 Transformation to Clip Space

15.8.2 Vertex Lighting

Equation 15.16: Rearranging the standard lighting equation to make it more suitable for vertex-level lighting computations

Equation 15.17: Vertex-level diffuse and specular lighting values

Equation 15.18: Shading pixels using interpolated lighting values

15.9 Backface Culling and Clipping

15.9.1 Backface Culling

Figure 15.23: Backface culling in 3D

Figure 15.24: Backface culling of triangles with vertices enumerated counterclockwise in screen space

15.9.2 Clipping

Figure 15.25: Clipping a single edge – the four cases

Figure 15.26: Clipping a polygon against the right clip plane

15.10 Rasterization

Chapter 16 Visibility Determination

16.1 Bounding Volume Tests

16.1.1 Testing Against the View Frustum

Figure 16.1: Testing a bounding box against the view frustum

Figure 16.2: Outcodes use a simple bit encoding scheme to store the results of clip plane testing

Listing 16.1: Computing an outcode for a point in clip space

16.1.2 Testing for Occlusion

16.2 Space Partitioning Techniques

16.3 Grid Systems

Figure 16.3: Using 2D grid system for visibility determination

16.4 Quadtrees and Octrees

Figure 16.4: A quadtree

Figure 16.5: Assigning objects to a quadtree

Figure 16.6: Adaptive subdivision using a quadtree

Listing 16.2: Raytracing a quadtree

16.5 BSP Trees

Figure 16.7: A 2D BSP tree

Figure 16.8: The hierarchy of a BSP tree

Figure 16.9: A BSP node represents a volume of space, not just a dividing plane

16.5.1 “Old School” BSPs

16.5.2 Arbitrary Dividing Planes

16.6 Occlusion Culling Techniques

16.6.1 Potentially Visible Sets

16.6.2 Portal Techniques

Figure 16.10: Apartment floor plan

Figure 16.11: The apartment viewed from the inside

Figure 16.12: The apartment floor plan as a graph

Figure 16.13: The bounding box of a portal polygon

Figure 16.14: The second portal is visible through the first portal

Chapter 17 Afterword

Back Matter

Appendix A Math Review

Summation Notation

Angles, Degrees, and Radians

Trig Functions

Trig Identities

Appendix B References

Index

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