3D VISUALIZATION OF CONIC SECTIONS IN XNA GAME PROGRAMMING FRAMEWORK. A Thesis. Presented to the. Faculty of. San Diego State University

Transcription

1 3D VISUALIZATION OF CONIC SECTIONS IN XNA GAME PROGRAMMING FRAMEWORK A Thesis Presented to the Faculty of San Diego State University In Partial Fulfillment of the Requirements for the Degree Master of Science in Computer Science by Sathyanarayan Chandrashekar Fall 2011

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3 iii Copyright 2011 by Sathyanarayan Chandrashekar All Rights Reserved

4 iv DEDICATION I dedicate this thesis to my beloved mother and sister who have been a constant inspiration for me throughout my life. I also want to dedicate this thesis to my friends and guides at SDSU who have helped and motivated me to perform all the necessary endeavors for the achievement of my goals towards the various development phases. Last but not the least, I would like to convey my sincerest thanks to Prof. Kris Stewart, who have played the role of my mentor, constantly guiding me every step of the way throughout my Master s program along with Prof. Janet Bowers, who suggested this topic and has been very cooperative and helpful throughout my thesis.

5 v ABSTRACT OF THE THESIS 3D Visualization of Conic Sections in XNA Game Programming Framework by Sathyanarayan Chandrashekar Master of Science in Computer Science San Diego State University, 2011 The California State Department of Education (CSDE), who set the academic standards for all the schools in California, decided to incorporate the concepts of conic sections for all the students belonging to the grades from eight through twelve. Students are taught right down from the fundamentals of the conic sections and their relation to quadratic equations. This thesis aims in helping high school students learn and understand the concepts of Conic sections by visually introducing them into learning a concept what is well defined in a 3 Dimensional space. The creation of this 3D environment lead me to make use of the XNA game programming framework that makes the implementation of both 2D and 3D designs very simple and hence, achieving various animations to show off different conic sections views could be obtained much more efficiently and effectively. I have divided the teaching aspect of this project into two parts that tends to give a fine balance between learning, understanding and last but certainly not the least, userinteractivity. As part one, I have designed a set of pre-animated views for each of the conic sections describing the hyperbola, parabola, circle, ellipse, line and finally, a point. As part two, in the free draw mode, the users have full flexibility in moving the 2D plane around and producing any shape desired. Along with this, I also provide an information button for each conic section screen that opens up an image showing necessary information about the same such as, a figure, equations, textual info. Overall, this project is aimed towards achieving a fun-filled, educational based gaming experience and is intended to act as an aid in teaching.

6 vi TABLE OF CONTENTS PAGE ABSTRACT...v LIST OF FIGURES... viii GLOSSARY... ix ACKNOWLEDGEMENTS...x CHAPTER 1 INTRODUCTION History Need for Study California State Department of Education Standards XNA Game Programming Framework Planning and Execution of Thesis LITERATURE SURVEY Detailed Description of Conic Sections Parabola Hyperbola Circle Ellipse Line Point CSDE Standards for Teaching Conic Sections Why XNA? How this Project Would Help Understand Conic Sections? Related Resources that Teach Conic Sections OVERVIEW OF THE IMPLEMENTATION RESULTING SCREEN SHOTS OF CONIC SECTIONS IN XNA Circle Ellipse...21

7 vii 4.3 Hyperbola Arabola Line Point CONCLUSION FUTURE WORKS...28 REFERENCES...29

8 viii LIST OF FIGURES PAGE Figure 1.1. Types of conic sections....1 Figure 1.2. Ball thrown takes a parabolic path....2 Figure 1.3. Parabolic mirrors used in vehicle headlights....3 Figure 1.4. A sundial with sun rays tracing hyperbolic paths....3 Figure 1.5. Elliptical orbital paths of planets around the sun....3 Figure 1.6. An image describing how sounds generated from one focus reaches another in an elliptical auditorium....4 Figure 2.1. List of conic sections with equivalent 2D figures....7 Figure 2.2. Show of various conic sections in an X-Y plane....8 Figure 2.3. Parabola drawn in a 2D space....9 Figure 2.4 (a) Hyperbola with horizontal transverse axis. (b) Hyperbola with vertical transverse axis Figure 2.5. Circle described along with its various properties Figure 2.6. Ellipse drawn on an X-Y plane Figure 3.1. Flow chart of the game play Figure 4.1. Front facing camera view of the circle Figure 4.2. Camera view from the top of the circle Figure 4.3. Front / side facing camera view of the ellipse Figure 4.4. Camera view from the bottom of the ellipse Figure 4.5. Front / side facing camera view of the hyperbola Figure 4.6. Front facing camera view of the hyperbola Figure 4.7. Front / side facing camera view of the parabola Figure 4.8. Camera view from the bottom of the parabola Figure 4.9. Front / side facing camera view of the line Figure Front / side facing camera view of the point...26 Figure Camera view from the top of the point....26

9 ix GLOSSARY Axis of the cone: The line from the vertex of a cone to the center of its base. Focal distance: Distance from the vertex to the focus. Conjugate axis: The conjugate axis of a hyperbola is the line of symmetry of the hyperbola which cuts the curve twice. Asymptotes: A line which a given curve gets closer and closer to, but never touches, as it gets further from the origin. Major axis: Longer of the two axes. Minor axis: Shorter of the two axes. Apex: The vertex of the cone. Gamescreen: The scenarios/views in the game that show up one at a time while playing the game.

10 x ACKNOWLEDGEMENTS First and foremost, I would like to express my sincere thanks to my thesis advisor Dr Kris Stewart, for helping me complete my thesis. Her guidance and support has been very valuable. I am also very grateful to Prof. Janet Bowers, firstly for suggesting me this thesis when asked if there is any 3D concept which is required in schools that would help benefit the teachers to have an interactive 3D game based environment and secondly, for helping me obtain valuable insights on the topic. Her kindness, co-operation and expertise in the field of mathematics have most certainly aided me in completing this thesis. I also thank Dr Joseph Lewis for his kind co-operation and being part of my thesis committee. Lastly, I want to acknowledge the fact that NSF (NSF DUE ) aided me throughout my research and hence, I would like to convey my sincerest thanks to them as well.

11 1 CHAPTER 1 INTRODUCTION 1.1 HISTORY Conic sections, in its simplest of terms, could be defined as the intersection of a cone with a plane. (see Figure 1.1 [1]) Figure 1.1. Types of conic sections. Source: Conic Sections. Wikipedia website, n.d. accessed May It is an age old topic that was shed light in the third century B.C when Apollonius tried to understand its principles and deduced that all conic section curves could be obtained via a single cone, contrary to its prior beliefs that ellipses could only be achieved via using acute cones, parabolas via right angled cones, etc [2, 3]. He proved that by intersecting the plane and the cone at different angles, different curves could be achieved. This concept was again given light by Kepler in his Laws of Planetary Motion theory [4]. Later on, several others also made use of conic section concepts to come up new theories like the Co-ordinate geometry, Projective geometry, etc., that catapulted conic sections to new heights and thus in very high regard [4]. It was since then that conic sections have become an appealing topic to many and still continue to work with it to bring new and innovative ideas into this world.

12 2 The different types of conic sections are: 1. Hyperbola 2. Parabola 3. Ellipse Along with these, there are 2 other that are termed as degenerate cases: 1. Circle 2. Line 3. Point 1.2 NEED FOR STUDY The practical usage of conic sections seems to be many as they have become an integral part of our day to day life. Whether it being constructing a suspension bridge, that makes use of the concept of a parabola [5] or it being to figure out the orbital paths of planets around the sun that follow an ellipse [6], conics have helped us understand things in a better manner. Some of the other uses include: 1. The trajectory of objects thrown involves a parabolic path (see Figure 1.2 [7]). 2. Parabolic mirrors are used in the head-lights of vehicles (see Figure 1.3 [8]) [4]. 3. Sundials make use of hyperbolas (see Figure 1.4 [9]). 4. Orbital paths of celestial are usually elliptical in shape (see Figure 1.5 [8]) [4]. 5. Properties of ellipse are used in the construction of Whispering Galleries (see Figure 1.6 [8]) [4]. Figure 1.2. Ball thrown takes a parabolic path. Source: Dr. D. Davis. Dr. D s Homepage, n.d. mages/ball.gif, accessed May 2011.

13 3 Figure 1.3. Parabolic mirrors used in vehicle headlights. Source: J. Britton. Welcome to Jill Britton s homepage, n.d. bc.ca/pararay_lg.gif, accessed May Figure 1.4. A sundial with sun rays tracing hyperbolic paths. Source: T. Morawe. The obelisk as a pointer of a sundial, n.d. Accessed May Figure 1.5. Elliptical orbital paths of planets around the sun. Source: J. Britton. Welcome to Jill Britton s homepage, n.d. _lg.gif, accessed May Along with the above practical implementations, the CSDE has set some standards in teaching and requires all the schools in the state of California to educate their students in the field of conic sections and its related topics between the grades of eight through twelve. I try to describe them briefly in the next section.

14 4 Figure 1.6. An image describing how sounds generated from one focus reaches another in an elliptical auditorium. Source: J. Britton. Welcome to Jill Britton s homepage, n.d. a/pararay_lg.gif, accessed May CALIFORNIA STATE DEPARTMENT OF EDUCATION STANDARDS California State Department of Education (CSDE) is the governing body that is responsible for providing a curriculum guideline throughout the schools in California. [10] They decide what information a student should be taught based on which grade he/she belongs to. Their ultimate goal is to make all the students well equipped with the concepts, whether it being mathematical or in physics or English or even literature, that will help them grow further in their field of interest and ultimately help them make their own contribution to it. When it comes to mathematics in school and especially with our topic of conic sections, as per the CSDE standards, high school students belonging to the grades eight through twelve are the ones that are exposed to this. Learning includes understanding of its concepts, deriving quadratic equations to represent the type of conic section and finally, determine type of conic section curve based on the deduced equation. [10] 1.4 XNA GAME PROGRAMMING FRAMEWORK Microsoft's XNA game programming framework is basically a window towards the world of gaming. It provides a programming environment for development and can produce an interactive program to run independent of its development environment. One of the reasons for its popularity is its ability to develop games for both PC and Xbox 360 consoles thus, giving it a wider audience. And the icing on the cake is the fact that it is free to use for all education related purposes. [11]

15 5 I chose to work with the XNA game programming framework due to its capabilities in the 2D and 3D design technology. Whether drawing a 3D model and moving it around the screen at will or working with 2D textures or images, XNA can do it all. Also, since it is programmed primarily in C#, there is no need to learn a new programming language as such (apart from C# and understanding and making use of the XNA libraries). 1.5 PLANNING AND EXECUTION OF THESIS By gaining sufficient knowledge of the conic section principles and also, understanding the advantages of working with XNA game programming framework, I have tried to develop my thesis on the following lines. I plan to draw a 3 dimensional doublynapped cone and a 2 dimensional plane on the screen. Figure 1.1 is typical for how conic sections are defined and I felt that allowing students the ability to control the 2D plane would enhance their learning. Along with the plane, the viewing camera can also be user controlled, giving the user a good amount of flexibility in moving the plane to create various shapes and also, view the whole scene from any angle. The user can also zoom in and out of the screen in order to get a better look and also, there is an information button on the bottom right hand corner which when hovered on, will display the relevant information of the conics, like images and text that will help students understand the concept in a more mathematical sense.

16 6 CHAPTER 2 LITERATURE SURVEY 2.1 DETAILED DESCRIPTION OF CONIC SECTIONS Conic sections are the result of a plane intersecting with a double napped cone. The intersection can occur at any angle and for all of them, there is a type of conic that is formed. The different types of conic sections that can be constructed are depicted on the next page (see Figure 2.1 [12]). Now, let s briefly describe how conics can be produced, beginning with the circle, which is formed with the intersection of a plane perpendicular to the axis of the cone whereas when the plane is not perpendicular and intersects only one of the napes, either an ellipse or a parabola is formed. A hyperbola is obtained when the plane intersects both the napes of the cone and is parallel to its axis. These types are said to be the general conic sections. The 2 other types of conics formed via degenerate cases include: 1. line and, 2. point. A line is formed when the plane is tangential to the outer surface of the 2 napes whereas a point is formed when the plane intersects the cone at the point where the 2 napes touch each other. So far, we have only dealt with the textual description of conic sections so now, it is time we view them in a more algebraic form. Linked with each conic are a couple of variables or terms that define its properties. Some of them are as follows [13]: 1. Locus: a group of points that share some common property as in the case of a circle, the locus of all the points are equidistant to a common point. 2. Focus: A point or a group of points (foci) considered as a reference to creating a shape. In case of a circle, the focus is the center of the circle. 3. Eccentricity: A measure of how deviant a conic is from being circular. For example, if the ecc > 1, we obtain a hyperbola. If 0 < ecc < 1, we obtain an ellipse, if ecc = 0, we obtain a circle and finally when the ecc = 1 then, a parabola is formed. 4. Directrix: a line taken as a reference and does not contain the foci.

17 7 # Conic Section Figure 1 Circle 2 Ellipse 3 Parabola 4 Hyperbola 5 Line 6 Point Figure 2.1. List of conic sections with equivalent 2D figures. Source: Glossary for math terms and definition. For My school Stuff website, n.d. math/glossary/c.htm, accessed May 2011.

18 8 The following figure shows some of the conic sections in terms of the above said terminologies (See Figure 2.2 [1]). Figure 2.2. Show of various conic sections in an X-Y plane. Source: Conic Sections. Wikipedia website, n.d. accessed May Now, let us look more deeply into each of the conics as I try to describe each of them individually with the help of some related equations, formulas and 2D diagrams. But, before we venture off into that, I would first want to give out the General Quadratic Equation for representing a Conic Section [14]: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, (2.1) where A,B,C 0 and also, there exists the Discriminant Equation [14]: B 2-4AC, (2.2) that can be used to find the type of conic section produced, as indicated in the sections below Parabola A parabola is constructed when the plane intersects only one of the napes of the cone, at an angle parallel to the outer surface of the cone. In other words, a parabola can be defined as a set of points (loci) that moves in such a way that they are equidistant from a given point called the focus and a line called the directrix (see Figure 2.3 [15]).

19 9 Figure 2.3. Parabola drawn in a 2D space. Source: B. Simmons. Mathwords: Terms and Formulas from Beginning Algebra to Calculus, n.d. accessed May As mentioned in the previous page, if the equation B 2-4AC = 0, then it is said to produce a parabola [14]. Along with this, there are some formulas as well that use to describe the parabola. One of them is said to be the Standard Equation of parabola: y=ax 2 + Bx + C, (2.3) where x is an independent variable and A, B, C are constants. The other equation that is said to describe a parabola is [14]: where P is the focal distance of the parabola. y = x 2 /4P (Vertical Vertex) and, (2.4) x = y 2 /4P (Horizontal Vertex), (2.5) With respect to the above mentioned equation, some the different cases that are observed in the parabola are listed below. With the vertex at the origin, The parabola opens in the positive x direction and has the equation y 2 =4Px, P>0where vertex=(0,0) and focus is the point (P,0). The parabola opens in the negative x direction and has the equation y 2 =4Px, P<0where vertex=(0,0) and focus is the point (P,0). The parabola opens in the positive y direction and has the equation x 2 =4Py, P>0where vertex=(0,0) and focus is the point (0,P).

20 10 The parabola opens in the negative y direction and has the equation x 2 =4Py, P<0 where vertex=(0,0) and focus is the point (0,P). [16] Hyperbola A hyperbola is constructed when the plane intersects both the 2 napes of the cone and is parallel to the axis of the cone. This may not be a strict requirement as some of the nonparallel intersections also result in the formations of hyperbolas. The hyperbola structure consists of 2 curves that are more or less the mirror images of each other and are called as the branches. The branches are exactly identical when the intersecting plane is parallel to the axis of the cone (see Figure 2.4 [17]). Other times, in case of non-parallel intersections, one of the branches is slightly bigger than the other. Figure 2.4 (a) Hyperbola with horizontal transverse axis. (b) Hyperbola with vertical transverse axis. Source: How to graph hyperbola from its equation. Math Warehouse website, n.d. graph-equation-of-a-hyperbola.php, accessed May A hyperbola is a set of all points in a plane, the difference of whose distances from two fixed points (the foci) is a positive constant. If the equation B 2-4AC > 0, it results in a Hyperbola. Another equation that represents it is: where x, y are variables and A, B are constants [14]. x 2 /A 2 y 2 /B 2 = 1, (2.6)

21 Some of the different cases of hyperbolas are: The center is at the origin and the foci are on the x-axis and conjugate axis is the y- axis and has the equation of the form x 2 /A 2 -y 2 /B 2 = 1where the foci and the vertices are on the x-axis, the distance between the foci=2a, the conjugate axis is on the y- axis and the center is at the origin (0,0). The center is at the origin and the foci are on the y-axis and conjugate axis is the x- axis and has the equation of the form y 2 /A 2 -x 2 /B 2 = 1 where the foci and the vertices are on the y-axis, the distance between the foci=2a, the conjugate axis is on the x- axis and the center is at the origin. [16] Asymptotic Equations: The equations of the asymptotes to the hyperbola x 2 /A 2 -y 2 /B 2 = 1 are as follows y = (B/A)x and y = - (B/A)x. [16] Circle A circle is constructed when the plane intersects the one of napes of cone at an angle perpendicular to the axis of the cone. In other words, it can be described as a set of points (loci) that move/revolve around a fixed point known as the focus and are always equidistant from the focus. A circle consists of various components like the radius, diameter, circumference and lastly, the center (see Figure 2.5 [18]). These are helpful in determining the properties of the circle. 11 Figure 2.5. Circle described along with its various properties. Source: Circle. Wikipedia website, n.d. org/wiki/circle, accessed May 2011.

22 the circle is If the discriminant equation B 2-4AC < 0, it results in a Circle [14]. Another equation 12 x 2 + y 2 = r 2, (2.7) where r represents the radius of the Circle [14]. Other properties and equations of circle include, D = 2r, where D is the Diameter and r again acts as the radius of the circle. The Circumference C = 2 πr, Area A = πr 2 and also, Angle covered by the circle is 360 or π. Some of the other properties of the circle include: Tangent: A straight line which touches a given curve exactly once, at a given point. A tangent to a circle is always at right angles to a radius or a diameter drawn from the point of contact. Secant: A straight line joining two points on the perimeter (or edge) of a shape, usually a circle. Chord: Similar to a secant. The diameter is a type of a chord Ellipse An ellipse is constructed when the plane is intersecting one of the napes of the cone at an angle not perpendicular to the axis of the cone. A similar definition could be given for the construction of a parabola but, the difference between the two remains in the fact that for an ellipse, the intersection results in obtaining a closed curve whereas for a parabola, we get an open curve. Another, more mathematical definition of an ellipse can be described as the locus of all points that move around two points known as the foci (see Figure 2.6 [19]). Similar to the circle, if the discriminant equation B 2-4AC < 0, it results in an ellipse [14]. Another equation associated with the ellipse is: x 2 / A 2 + y 2 / B 2 = 1, (2.8) where x, y are independent variables and A, B are constants, represented by the Major radius and the Minor radius respectively [14]. An equation that is used to represent its relation with the focus is: A 2 - B 2 = C 2, (2.9) where C is the distance from the center to the focus [14].

23 13 Figure 2.6. Ellipse drawn on an X-Y plane. Source: Online tutoring. Tutor Next website, n.d. com/system/files/u44/ ellipse_0.jpg, accessed May Different cases of ellipses include: The vertex is at the origin and the foci and the major axis are on the x-axis with the center at the origin and has the equation of the form x 2 /A 2 +y 2 /B 2 = 1where the foci and the major axis are on the x-axis, the length of the major axis is 2A, the minor axis is on the y-axis, the length of minor axis equals to 2B and the center of the origin is at the origin (0,0). The vertex is at the origin and the foci and the major axis are on the y-axis with the center at the origin and has the equation of the form x 2 /B 2 +y 2 /A 2 = 1where the foci and the major axis are on the y-axis, the length of the major axis=2a, the minor axis is on the x-axis, length of the minor axis=2b and the center is at the origin (0,0). [16] Line A line is said to be one of the degenerate cases of the conic sections which is primarily constructed when the plane is at a slant position along the outer surface of the 2 napes and goes through the point of intersection of the 2 napes. In other words, when the plane is at a tangential position to the surface of the cone and passing through the apex, a line is said to be formed. [20]: One of the equations that is closely associated with the line is the Distance formula, (2.10)

24 14 where x 1, x 2, y 1, y 2 represent the two end points of the line on the X and Y axis respectively. Another equation that is also associated with the line is: y = Mx + C, (2.11) where M is the slope of the line. This equation is famously known as the Slope-Intercept equation of the line [21] Point Another degenerate case of the conic section is the formation of a point where the plane passes through the apex of the cone without intersecting anywhere else. In other words, when the plane passes through the intersection of the two napes, it results in a point being formed. 2.2 CSDE STANDARDS FOR TEACHING CONIC SECTIONS California State Department of Education (CSDE) is the governing body responsible for setting up the standards and protocols for learning and teaching for all the schools located in the state of California. In other words, all schools adopt the procedures and guidelines set by CSDE. [10] CSDE aims to bring the students closer to the world of mathematics and develop their skills in the same. CSDE realizes the fact that mathematics is just not restricted to learning within the schools but, have a much more of a practical application as well as they are constantly used in our day to day lives. CSDE tries to set the standards for mathematics for all the grades from one through twelve, bringing the students step by step into deeper and complex levels of understanding and problem solving. The standards are set with the notion that students are capable of learning and understanding the concepts and use these concepts while climbing up the ladder and building on them. Students attaining proficiency in learning is not only limited to what is taught by the teachers but, requires rigorous practice and effort on the part of the students as well. Thus, effective teaching and regular practice by the students allows them to get comfortable with the taught concepts and builds the foundation for newer and more complex contexts in mathematics. [10] Hence, the overall goal of CSDE can be summed up by the following points: [10] 1. Improve the fluency and understanding of mathematical concepts.

25 15 2. Help the students become better at solving problems. 3. Make the students be more analytical and have a better sense of reasoning. 4. Help connect mathematics to other related or non-related areas. In the Algebra-ΙΙ item# 17 of their guideline handout, CSDE has set standards of learning conic sections primarily for the students belonging to the grades eight through twelve. Students are taught the fundamentals of the conic sections and their relation to quadratic equations. At the end of it all, the students are expected to demonstrate their knowledge and understanding by determining the type of conic based on the quadratic equation and the reverse, given the conic provide the equation that determines it. 2.3 WHY XNA? Microsoft's XNA game programming environment facilitates video game development and management. [11] Essentially a framework, it comes equipped all the necessary libraries to write world class games and for multiple platforms like the Xbox 360 and Windows PC. [22] This flexibility not only helps the game programmers reduce their programming worries of having to switch between different programming environments while coding games for multiple platforms but also, increases the spectrum of game play. For example, if someone likes to play games on their PC, they would love if it were available on the game consoles. With the XNA game programming framework, this feat can be achieved. As a game programming environment, it comes loaded with all the essential technology that is required to build a working and a good looking game. XNA is based on the.net framework so, people who are familiar with the Visual Studio environment and know the coding styles of C# would find it really easy to adapt to this framework. [22] One of the great features of XNA is the ease with which things can be accomplished. Whether it being drawing an image or a text or rendering a 3D model in the screen, XNA can do it all. Not only this but, changing from one game screen to another or even, creating various animations is a task not too difficult with XNA. In my project, I render a 3D model in the shape of a doubly napped cone and also a 2D plane, which is user controlled via the arrow keys. With the movement of the plane and its intersection to cone, various conics are produced and thus could be seen quite easily in this project.

26 2.4 HOW THIS PROJECT WOULD HELP UNDERSTAND CONIC SECTIONS? As the days have gone by and technology is becoming more and more an integral part of our day to day lives, I believe that children now are more interested looking into different methods of studying that allowing learning faster and more efficiently rather than the old fashion way of reading big and bulky books since it takes more time and also, getting the exact book you want isn t always possible. Thus, it makes good sense to teach them something that doesn t just involve much reading but also makes them more interactive. Something a student enjoys doing is one of those things that they tend to learn the fastest. Conic sections, in its natural form, consists of a 3-Dimensional doubly napped cone and a 2-Dimensional plane and by giving the students the ability to move the object in 3D representation would certainly help them better understand the concept, components and learn the most. Thus, in my project, I aim to combine both the above said features and try to deliver what I call an Interactive learning experience. Students involve themselves by playing around the 2D plane, moving it around here and there and, trying to understand how different intersections with the cone results in different conics being generated RELATED RESOURCES THAT TEACH CONIC SECTIONS Conic sections are a traditional topic and we find a lot of resources either via books or the internet that can help us understand the concept better. Some of these which I came across were mainly in the form of textual description, basically describing each of the conic section, its properties, equations, usage and of course, a picture depicting the same. At the end of it all, the students might be asked to take some quizzes that act as a confirmation on their part in learning. Some of the other learning methodologies included Animations. [23] By showing an animation via either a video or an image, the gist of concept is learned. How the plane while intersecting the cone at different angles makes different conics is certainly well understood with the help of these animations, and the one that I found online was by Prof. Louis A. Talman. [23] He prepared a quicktime movie showing the movement of the plane and the equivalent conic sections that are produced. Even though it provides adequate details to the formations of conic sections, it has certain limitations. For example, the

27 17 animation does not show the two degenerate cases: Line and Point. Another thing to notice is that, the view is fixed. In other words, the viewing camera is stationary and cannot be moved in order to see the conic from a different angle. In my thesis, I have taken into consideration the above limitations and have incorporated both of them for the user to have better experience. Another fascinating resource out on the internet is the Java-based Applets. [24, 25] These are basically learning tools that involve some amount of user interactivity by asking them to enter a value or move things around with their mouse etc. All these learning tools and methodologies are good in their manner and are able to help the students understand various aspects of conic section. My thesis topic allows me to make my own contribution towards this step of teaching by not only giving students the basic information in terms of definitions and equations of each of the conic but also, allowing the students to interact with the tool and having a little bit of fun while learning.

28 18 CHAPTER 3 OVERVIEW OF THE IMPLEMENTATION This project can be seen as a 3D game based simulated learning environment that would help the high school students understand the concepts of conic sections. The game essentially has a splash screen displaying the name of the project (or game) along with user selectable option like Press, Instructions and Exit. By selecting Instructions, the user can learn the basic gameplay like which keys help them to move the plane and camera. On our splash screen, each of the conic animation is represented by a number. By choosing the desired number, the user is taken to an appropriate screen having a preanimated view of conic section. In this view, the user has the ability to move the camera in any direction and also, zoom in and out to see details of the conic being produced. Along with this, there is also an Information button, which when hovered on, displays relevant information about that conic section, something that the user might be interested in learning. The game is conceptually divided into 2 parts, the first contains the above mentioned preanimated views of the conic sections and the second is a free-draw mode where userinteractivity is available. When the user selects this mode, the user has full control over the 2D plane and moves it around the screen as they wish. Apart from normal linear translation, they can also perform rotation in X-Y-Z axes. These feature help the user create any conic he/she wishes. Below I have created a flow chart that diagrammatically describes the above said points (see Figure 3.1).

29 Figure 3.1. Flow chart of the game play. 19

30 20 CHAPTER 4 RESULTING SCREEN SHOTS OF CONIC SECTIONS IN XNA In this section, I present the resulting screen shots of my work, by categorized them based on the type of conic it represents, there are two images per conic produced, which gives us a decent idea of how it looks like from the front view and also, a top or a bottom view. 4.1 CIRCLE Below is the snapshot of the 2D plane (shown in red) intersecting the 3D cone to produce a circle (see Figure 4.1). The image represents the front-facing view of the conic and hence, we see that the plane is perpendicular to the axis of the plane, resulting in a circle. Figure 4.1. Front facing camera view of the circle.

31 21 The second image below is the result of viewing the intersection from a different camera angle, from the top (see Figure 4.2). As we can see, this view gives us a good idea in the formation of the circle. Figure 4.2. Camera view from the top of the circle. 4.2 ELLIPSE Similar to the circle, the snap shot of the ellipse below is taken via a front facing camera angle (see Figure 4.3). This camera angle helps us understand the relation of the plane with the cone. As we can see, the intersection of the plane is not perpendicular to the axis of the cone and results in a closed curve. The following image was produced when the camera is placed below the two models in 3D space (see Figure 4.4). In other words, it represents a bottom up view and thus, helping us understand the nature of intersection from a different perspective. With this image, the formation of an ellipse can be clearly seen.

32 22 Figure 4.3. Front / side facing camera view of the ellipse. Figure 4.4. Camera view from the bottom of the ellipse. 4.3 HYPERBOLA The image below depicts to us that intersection of the plane with the cone at an angle parallel to its central axis results in the formation of a hyperbola (see Figure 4.5). The plane must also need to pass both the napes to result in a hyperbola as shown in the image below.

33 23 Figure 4.5. Front / side facing camera view of the hyperbola. As compared to the above image, that acts more like a side view, the following image is the result of a front facing camera which clearly shows us the 2 open curves of the hyperbola being formed (see Figure 4.6). Figure 4.6. Front facing camera view of the hyperbola.

34 ARABOLA As seen in the image below, the formation of the parabola requires the plane to intersect the cone at an angle not perpendicular to the axis of the cone and also, the intersection must pass through only one of the napes, along with the fact that it should result in an open curve. These are the necessary conditions that results in a parabola and are depicted in the image below (see Figure 4.7). Figure 4.7. Front / side facing camera view of the parabola. The image below is produced when the camera was viewing the intersection from the bottom, more like a bottom up view, giving a picture of the parabola (see Figure 4.8). 4.5 LINE Now we head into viewing our degenerate cases and the first one that we will have a look at is the line. As we can from the image below, the plane passes through the point of intersection of the two napes, while being tangential their outer surfaces (see Figure 4.9). 4.6 POINT The last of the conic section that we will be discussing is the point, which also is another one of the degenerate cases of the conic sections. A point formed when the plane passes through the point of intersection of the two napes and also, does not intersect either of the 2 napes of the cone, which is depicted in the image below (see Figure 4.10).

35 25 Figure 4.8. Camera view from the bottom of the parabola. Figure 4.9. Front / side facing camera view of the line.

36 26 Figure Front / side facing camera view of the point The image below is the result of the camera looking at the intersection from a top (see Figure 4.11). A point can be seen in the middle of the image, which is the result of the intersection mentioned above. Figure Camera view from the top of the point.

37 27 CHAPTER 5 CONCLUSION After deciding to take this project as my thesis and working on all the tasks that came along with it such as gather valuable information in order to start, development environment to use, implementation design, UI design, learning experience, interactivity, I believe that it maintains a balance between learning and enjoying while you learn, as it is a game to begin with. The animations tend to provide a generic picture on how to on how to produce the conic, on the other hand, the free draw mode allows the students to try things for themselves, putting their knowledge into practice. Last but not the least, by providing the students an option to use their Xbox game consoles to play this game, adds that little bit of charm to the whole project.

38 28 CHAPTER 6 FUTURE WORKS Since this project was developed using the XNA Game Programming Framework (3.1) and is the first of its kind, I would like to call this version 1.0. There are good amount of enhancements that could be made in the future releases to make this game more interactive and effective. Some of these include: Since the current game play is more on the simplistic side, improvements could be made by introducing a more traditional game-like flow by adding better UI, textures and 3D models. The user interactivity could also be enhanced by incorporating quizzes at the end of each animation and collecting the results for statistical purposes. Narrative audio assistance could also be added that helps the user to smoothly progress through the game. Some of the online resources that I have come across [24] provides the user the ability to enter numerical inputs such as the focal distance into an algebraic equations, that generates a conic section. The shape of the conic generated would vary based on the input provided. This feature could also be incorporated in the future releases to help the students better understand the algebraic part of the conic sections. The ability to capture the points of intersection between the cone and the plane and, highlighting them to show the conic generated in a more visually appealing manner. Ability to export the game into Xbox consoles and Kinect so that user has a choice to play it either on a PC or a console. Ability to update the values of variables in the equation and the image being displayed via the information button while the animation is being played. Ability to add more data into the models such the asymptotes, focus, directrix to better explain the concept and update the same while the animation is being played.

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