Math course description

Game Mathematics (12 Week Lesson Plan) Lesson 1: Set Theory Te xtbo o k: Chapter One (pgs – 15) Go als : We begin the course by introducing the student to a new vocabulary and set of rules that will be foundational to the m athem atical discussions that follow in subsequent lessons In this first lesson, students are introduced to Set Theory The idea is to learn the basics of this allim portant branch of m athem atics so that students are prepared to tackle and understand the concept of m athem atical functions, which will play a m ajor role in early lessons Students will learn about how entities are grouped into sets and how to conduct various operations on those sets such as unions and intersections (i.e the algebra of sets) We conclude with a brief introduction to the relationship between functions and sets to set the stage for the next lesson Ke y To p ics : Introduction to Set Theory • The Language of Set Theory • Set Mem bership • Subsets, Supersets, and Equality • The Algebra of Set Theory • Set Theory and Functions Pro je cts : Varied Exercises Exam s / Qu izze s : Question Quiz (m ultiple choice) Covers select topics from Chapter/ Lecture One 2% of final grade Re co m m e n d e d Stu d y Tim e ( h o u rs ) : - Lesson 2: Functions Te xtbo o k: Chapter Two (pgs 19 – 47) Go als : In this lesson, students are introduced to m athem atical functions We will begin by talking about the role of functions and look at the concept of m apping values between dom ain and range From there we will spend a good deal of tim e looking at how to visualize various kinds of functions using graphs This will set the stage for discussion of som e of the m ost popular functions that are used in gam e developm ent We will begin with the absolute value function and then m ove on to discuss both exponential and logarithm ic functions Students will get an opportunity to see how these functions can be used to m odel various kinds of phenom ena From the fog used in gam es to calculating how players take weapon dam age, students will see hands-on how such functions play a vital role in creating interesting effects One of the m ore im portant things we will is go step by step through the process of designing a function and com ing up with a m eans for selecting appropriate values that reflect the desired outcomes Ke y To p ics : Mathem atical Functions Graphs o Single-Variable Functions o Two-Variable Functions Fam ilies of Functions o Absolute Value Function o Exponential Functions Fog Density Dam age Calculations o Logarithm ic Functions Using the Log Function for Gam e Developm ent Pro je cts : Varied Exercises Exam s / Qu izze s : Question Quiz (m ultiple choice) Covers select topics from Chapter/ Lecture Two 2% of final grade Re co m m e n d e d Stu d y Tim e ( h o u rs ) : - Lesson 3: Polynomials Te xtbo o k: Chapter Three (pgs 49 – 71) Go als : In this lesson, students will learn about polynom ials We will begin with an exam ination of the algebra of polynom ials, and then m ove on to look at the graphs for various kinds of polynom ial functions Once our theoretical discussions are concluded, we will focus on the application of different kinds of polynom ials in gam e developm ent projects We start with linear interpolation using polynom ials that is com m only used to draw polygons on the display From there we will look at how to take com plex functions that would be too costly to com pute in a real-tim e gam e environm ent and use polynom ials to approxim ate the behavior of the function to produce sim ilar results We will wrap things up by looking at how polynom ials can be used as a m eans for predicting the future values of variables, which can be useful under a num ber of different gam e scenarios (such as m anaging network packet latency for exam ple) Ke y To p ics : Polynom ials Polynom ial Algebra (Single Variable) o Addition/ Subtraction o Scalar Multiplication o Multiplication/ Division Quadratic Equations Graphing Polynom ials Using Polynom ials o Linear Interpolation o Approxim ating Functions o Prediction Pro je cts : Varied Exercises Exam s / Qu izze s : Question Quiz (m ultiple choice) Covers select topics from Chapter/ Lecture Three 2% of final grade Re co m m e n d e d Stu d y Tim e ( h o u rs ) : - Lesson 4: Basic Trigonometry I Te xtbo o k: Chapter Four (pgs 75 – 97) Go als : Triangles are the core prim itive of m ost m odern 3D gam e engines As such it is vital that students have a firm grasp of the properties of triangles, and right triangles in particular In this lesson, students will get a crash course in som e of the core elem ents of trigonometry We will talk about the properties of triangles and look at the relationships that exist between their internal angles and the lengths of their sides This will lead to discussion of the m ost com m only used trigonom etric functions that relate triangle properties to unit circles This includes the sine, cosine and tangent functions We will use these properties and functions to solve a num ber of issues related to graphics program m ing, such as m odeling an anim ated wave function such as m ight be used for water or cloth sim ulation, and also look at how to use these concepts to render circles and ellipses on the display Ke y To p ics : Angles Com m on Angles The Polar Coordinate System Triangles Properties Right Triangles Introduction to Trigonom etry The Trigonom etric Functions Applications of Basic Trigonom etry Solving Triangle Problem s Modeling Phenom ena Modeling Waves Drawing Circles and Ellipses Projection Pro je cts : Varied Exercises Exam s / Qu izze s : Question Quiz (m ultiple choice) Covers select topics from Chapter/ Lecture Four 2% of final grade Re co m m e n d e d Stu d y Tim e ( h o u rs ) : - Lesson 5: Basic Trigonometry II Te xtbo o k: Chapter Five (pgs 10 – 122) Go als : Picking up where the last lesson left off, students continue their exam ination of trigonom etric functions In this lesson, we will look at the very im portant inverse trig functions such as arcsin, arcos, and arctan, and see how they can be used to determ ine angle values We will also introduce students to the core trig identities such as the reduction and double angle identities and use them as a m eans for deriving proofs Being able to derive proofs are an im portant part of the m athem aticians skill set and we will begin to becom e m ore form al with this concept in this lesson As usual, we will look at applications to gam e technology and see how trig functions can be used to rotate points in two and three dim ensions and also how to construct a proper field of view (FOV) for an in-gam e cam era system Ke y To p ics : Trig Functions Derivative Trigonom etric Functions Inverse Trig Functions Identities o Pythagorean Identities o Reduction Identities o Angle Sum / Difference Identities o Double-Angle Identities o Sum -To-Product Identities o Product-to-Sum Identities o Triangle Laws Applications Point Rotation Field-of-View Pro je cts : Varied Exercises Exam s / Qu izze s : Question Quiz (m ultiple choice) Covers select topics from Chapter/ Lecture Five 2% of final grade Re co m m e n d e d Stu d y Tim e ( h o u rs ) : - Lesson 6: Analytic Geometry I Te xtbo o k: Chapter Six (pgs 125 – 149) Go als : Beyond triangles, students will also need to understand other im portant constructs In this lesson, we will introduce analytic geom etry as the m eans for using functions and polynom ials to m athem atically represent points, lines, planes and ellipses All of these concepts are vital in gam e developm ent since they are used in rendering and optim ization, collision detection and response, gam e physics, and other critical areas We will start with points in space and m ove on to sim ple 2D lines and their various form s (including the all-im portant param etric representation) We will look at intersection form ulas and distance form ulas with respect to lines, points, and planes and also briefly talk about ellipsoidal intersections Ke y To p ics : Points and Lines Two-Dim ensional Lines Param etric Representation Parallel and Perpendicular Lines Intersection of Two Lines Distance from a Point to a Line Angles between Lines Three-Dimensional Lines Ellipses and Ellipsoids Intersecting Lines with Ellipses Intersecting Lines with Spheres Planes Intersecting Lines with Planes Pro je cts : Varied Exercises Exam s / Qu izze s : Question Quiz (m ultiple choice) Covers select topics from Chapter/ Lecture Six 2% of final grade Re co m m e n d e d Stu d y Tim e ( h o u rs ) : – Lesson 7: Vector Mathematics Te xtbo o k: Chapter Seven (pgs 151 – 174) Go als : In this lesson, students are introduced to vector m athem atics – the core of the 3D graphics engine After an introduction to the concept of vectors, we will look at how to perform various im portant m athem atical operations on them This will include addition and subtraction, scalar m ultiplication, and the all-im portant dot and cross products After laying this com putational foundation, we will look at the use of vectors in gam es and talk about their relationship with planes and the plane representation, revisit distance calculations using vectors and see how to rotate and scale geom etry using vector representations of m esh vertices Ke y To p ics : Elem entary Vector Math Linear Com binations Vector Representations Addition/ Subtraction Scalar Multiplication/ Division Vector Magnitude The Dot Product Vector Projection The Cross Product Applications of Vectors Directed Lines Vectors and Planes o Back-face culling o Vector-based Plane Representation Distance Calculations (Points, Planes, Lines) Point Rotation, Scaling, Skewing Pro je cts : Varied Exercises Exam s / Qu izze s : Question Quiz (m ultiple choice) Covers select topics from Chapter/ Lecture Seven 2% of final grade Re co m m e n d e d Stu d y Tim e ( h o u rs ) : - 10 Lesson 8: Matrix Mathematics I Te xtbo o k: Chapter Eight (pgs 177 – 188) Go als : In this lesson, students are introduced to the concept of a m atrix Like vectors, m atrices are one of the core com ponents of every 3D gam e engine and as such are required learning In this first of two lessons, we will look at m atrices from a purely m athem atical perspective We will talk about what m atrices are and what problem s they are intended to solve and then we will look at various operations that can be perform ed using them This will include topics like m atrix addition and subtraction and m ultiplication by scalars or by other m atrices We will conclude the lesson with an overview of the concept of using m atrices to solve system s of linear equations We will this by lightly touching on the notion of Gaussian elim ination Ke y To p ics : Matrices Matrix Relations Matrix Operations o Addition/ Subtraction o Scalar Multiplication o Matrix Multiplication o Transpose o Determ inant o Inverse System s of Linear Equations o Gaussian Elim ination Pro je cts : Varied Exercises Exam s / Qu izze s : Question Quiz (m ultiple choice) Covers select topics from Chapter/ Lecture Eight 2% of final grade Re co m m e n d e d Stu d y Tim e ( h o u rs ) : - 10 Lesson 9: Matrix Mathematics II Te xtbo o k: Chapter Nine (pgs 191 – 210 ) Go als : In this lesson, we continue our discussion of m atrix m athem atics and introduce the student to the problem that m atrices are generally used to solve in 3D gam es: transform ations After introducing the idea of linear transform ations, we will take a brief detour to exam ine how an im portant non-linear operation like translation (used to reposition points in 3D gam e worlds) can be m ade com pliant with our m atrix operations by introducing 4D hom ogenous coordinates Once done, we will exam ine a num ber of comm on m atrices used to effect transform ations in 3D gam es This will include projection, translation, scaling and skewing, as well as rotations around all three coordinate axes We will wrap up with the actual vector/ m atrix transform ation operation (multiplication) which represents the foundation of the 3D graphics rendering pipeline Ke y To p ics : Linear Transform ations Com puting Linear Transform ation Matrices Translation and Hom ogeneous Coordinates Transform ation Matrices o The Scaling Matrix o The Skewing Matrix o The Translation Matrix o The Rotation Matrices o The Projection Matrix Linear Transform ations in 3D Gam es Pro je cts : Varied Exercises Exam s / Qu izze s : Question Quiz (m ultiple choice) Covers select topics from Chapter/ Lecture Nine 2% of final grade Re co m m e n d e d Stu d y Tim e ( h o u rs ) : 10 - 12 Lesson 10: Quaternion Mathematics Te xtbo o k: Chapter Ten (pgs 211 – 227) Go als : In this lesson, students are introduced to quaternion m athem atics To set the stage for quaternions, which are hyper-com plex num bers, we will first exam ine the concept of im aginary num bers and look at the various arithm etical operations that can be perform ed on them We will look at the sim ilarities and differences with respect to the real num bers Once done, we will introduce com plex num bers and again look at the algebra involved Finally we will exam ine the quaternion and its associated algebra With the form alities out of the way we will look at applications of the quaternion in gam e developm ent Prim arily the focus will be on how to accom plish rotations about arbitrary axes and how to solve the gim bal lock problem encountered with Euler angles We put this concept to use to create an updated world to view space transform ation m atrix that is derived from a quaternion after rotation has taken place Ke y To p ics : Im aginary Num bers Powers Multiplication/ Division Addition/ Subtraction Com plex Num bers Addition/ Subtraction Multiplication/ Division Powers Com plex Conjugates Magnitude Quaternions Addition/ Subtraction Multiplication Com plex Conjugates Magnitude Inverse Rotations World-to-View Transform ation Pro je cts : Varied Exercises Exam s / Qu izze s : Question Quiz (m ultiple choice) Covers select topics from Chapter/ Lecture Ten 2% of final grade Re co m m e n d e d Stu d y Tim e ( h o u rs ) : 10 - 12 Lesson 11: Analytic Geometry II Te xtbo o k: Chapter Eleven (pgs 229 – 251) Go als : In this lesson, we will focus on som e of the practical applications of m athem atics In this particular case we will look at how analytic geom etry plays an im portant role in a num ber of different areas of gam e developm ent We will start by looking at how to design a sim ple collision/ response system in 2D using lines and planes as a m eans for m odeling a sim ple billiards sim ulation We will continue our intersection discussion by looking at a way to detect collision between two convex polygons of arbitrary shape From there we will see how to use vectors and planes to create reflections such as m ight be seen in a m irror Then we will talk about the use of a convex volum e to create shadows in the gam e world Finally we will wrap things up with a look at the Lam bertian diffuse lighting m odel to see how vector dot products can be used to determine the lighting and shading of points across a surface Ke y To p ics : 2D Collisions Reflections Polygon/ Polygon Intersection Shadow Casting Lighting Pro je cts : Varied Exercises Exam s / Qu izze s : NONE Re co m m e n d e d Stu d y Tim e ( h o u rs ) : - 10 Lesson 12: Exam Preparation and Course Review Te xtbo o k: NONE Go als : In this final lesson we will leave the student free to prepare for and take their final exam ination Multiple office hours will be held for student questions and answers Ke y To p ics : NONE Pro je cts : NONE Exam s / Qu izze s : NONE Re co m m e n d e d Stu d y Tim e ( h o u rs ) : 15 - 20 Final Examination The final exam ination in this course will consist of 25 m ultiple-choice and true/ false questions pulled from the first 10 textbook chapters Students are encouraged to use the lecture presentation slides as a m eans for reviewing the key m aterial prior to the exam ination The exam should take no m ore than three hours to com plete It is worth 80 % of student final grade  ...Lesson 1: Set Theory Te xtbo o k: Chapter One (pgs – 15) Go als : We begin the course by introducing the student to a new vocabulary and set of rules that will be foundational... with a m eans for selecting appropriate values that reflect the desired outcomes Ke y To p ics : Mathem atical Functions Graphs o Single-Variable Functions o Two-Variable Functions Fam ilies of... properties of triangles, and right triangles in particular In this lesson, students will get a crash course in som e of the core elem ents of trigonometry We will talk about the properties of triangles