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Taxicab Geometry: Not the Shortest Ride Across Town (Exploring Conics with a Non-Euclidean Metric)

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Taxicab Geometry: Not the Shortest Ride Across Town (Exploring Conics with a Non-Euclidean Metric) Taxicab Geometry: Not the Shortest Ride Across Town (Exploring Conics with a Non-Euclidean Metric) Creative Component Christina Janssen Iowa State University July 2007 1 Introduction Pg 3 Taxicab geometry: What is it and where did come from? What is a metric? Proof Euclidean/Taxicab are metrics Proof Taxicab distanceEuclidean distance Circles Pg13 Circumference Area Equations (two dimensions) Sphere (three dimensions) Formula Summary Ellipses Pg 28 Circumference Area Equations Formula Summary Parabolas Pg 42 Equations Formula Summary Hyperbolas Pg 53 Equations Formula Summary Conclusion Pg 59 Jaunts Pg 61 1 Metrics 2 Triangle Inequality Theorem 3 Right Triangles 4 Congruency 5 Rigid Motion 6 Quadrilaterals 7 Area Limits 8 High School 9 Activities Charts Pg 84 Bibliography Pg 88 2 Introduction This paper journals my investigation through the conic sections with the Taxicab metric. I am a high school teacher and graduate student and attempted to focus not only on my personal learning experience, but also on what I could incorporate into my classroom for the good of my students. As most teachers will attest, learning concepts that you believe will motivate your students is an exciting prospect. My goals in this creative component are multifold: 1. An opportunity to practice and hone my newly acquired problem solving skills in an individual and in-depth investigation. 2. A chance to delve deeper into a topic (Taxicab geometry) that intrigues me. 3. To acquire a deeper understanding of the structure of mathematics. 4. To become more knowledgeable about conics. 5. To discover tools and ideas to bring back to my students. 6. Completion of my Masters in School Mathematics degree. The main text of the paper is broken into four sections, each highlighting a conic section. It begins with Taxicab circles, and continues with Taxicab ellipses, Taxicab parabolas and Taxicab hyperbolas. Each section has various topics and observations. The content, other than definitions and historical information is a strict representation of my own work. Work was performed from scratch, with rudimentary sketches with decisions on future directions being made along the way. The investigation has been empowering. I do not by any stretch know all there is to know, and I am possibly left with more questions then when I started. But, my mathematical growth and increased confidence in my ability to problem solve, has been remarkable. I hope you enjoy the paper. 3 Taxicab geometry: What is it and where did come from? “The usual way to describe a (plane) geometry is to tell what its points are, what its lines are, how distance is measured, and how angle measure is determined.” (Krause 2) Taxicab geometry will use points and lines as defined in Euclidean geometry. “A point is a location.” and “A line is made up of points and has no thickness or width.” (Glencoe 6) Additionally, we shall define an angle to be “the union of two rays that share a common endpoint. The point is called the vertex.” The measure of an angle shall be “the amount of rotation about the vertex needed for one side to overlap the other.”(Geometry to Go 062) The difference in Taxicab geometry shall be that distance measured can be only horizontal and /or vertical. The well-known and loved Euclidian distance formula for points P(x , y ) and Q(x , y ) defined d P, Q = (x ! x )2 + ( y ! y )2 is not P P Q Q E ( ) p Q P Q the metric for your typical cab driver (unless his or her taxi happens to be an air cab). Taxicab geometry is built on the metric where distance is measured d P,Q = x ! x + y ! y and will continue to be measured as the shortest distance T ( ) P Q P Q possible. Taxicab geometry was proposed as a metric long before it was labeled Taxicab. A Russian by the name of Hermann Minkowski wrote and published an entire work of various metrics including what is now known as the taxicab metric. In 1952 an exhibit was displayed at the Museum of Science and Industry of Chicago, which highlighted geometry. A small pamphlet was distributed entitled, “You Will Like Geometry.” It was in the pages of this booklet that the Minkowski’s geometry was coined Taxicab geometry. (Reinhardt 38) 4 What is a metric? A metric is a mathematical function that measures distance. It is important to note that both the Euclidean distance formula and the Taxicab distance formula fulfill the requirements of being a metric. The three axioms for metric space are as follows. Let P, Q, and R be points, and let d(P,Q) denote the distance from P to Q. Metric Axiom 1 d(P,Q) ! 0 and d(P,Q) = 0 if and only if P = Q Metric Axiom 2 d(P,Q) = d(Q, P) Metric Axiom 3 d(P,Q) + d(Q, R) ! d( P, R) (Reynolds 124) In simple terms this means that the distance between two points is always greater than zero and only equal to zero if the two points are actually the same point. The distance between two points is the same despite which point you begin your measure. The distance from a first point to an intermediate point and then from the intermediate point to a final point must be farther than or equal to the distance you would travel if you went directly from the first to a final point. Both distance formulas, Euclidean and Taxicab, satisfy these axioms. Proof is shown below. (See Jaunt 1 for additional metric examples.) Proof Euclidean distance formula is a metric: Let P(x , y ) , Q(x , y ) and R(x , y ) !"2 . P P Q Q R R Metric Axiom 1: d(P,Q) ! 0 and d(P,Q) = 0 if and only if P = Q . If P = Q then d(P,Q) = d(P, P) 2 2 = x ! x + y ! y ( P P ) ( P P ) 5 2 2 = (0) +(0) = 0 . If P ! Q then either xP ! xQ or yP ! yQ . 2 2 And either x ! x > 0 or y ! y > 0 . ( P Q ) ( P Q ) Suppose xP ! xQ 2 2 Then x ! x + y ! y ( P Q ) ( P Q ) 2 ! x " x ( P Q ) = xP ! xQ > 0 2 2 Therefore, x ! x + y ! y > 0 ( P Q ) ( P Q ) Metric Axiom 2: d(P,Q) = d(Q, P) 2 2 d(P,Q) = x ! x + y ! y ( P Q ) ( P Q ) 2 2 = !1 !x + x + !1 !y + y (( )( P Q )) (( )( P Q )) 2 2 = !1 x ! x + !1 y ! y (( )( Q P )) (( )( Q P )) 2 2 2 2 = !1 x ! x + !1 y ! y ( ) ( Q P ) ( ) ( Q P ) 2 2 = x ! x + y ! y ( Q P ) ( Q P ) 6 = d(Q, P) Metric Axiom 3: d(P,Q) + d(Q, R) ! d( P, R) Let P , Q , and R be non-collinear, thus forming a triangle. 2 2 2 2 d(P,Q) + d(Q, R) = x ! x + y ! y + x ! x + y ! y ( P Q ) ( P Q ) ( Q R ) ( Q R ) 2 2 d(P, R) = x ! x + y ! y ( P R ) ( P R ) According to the Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater the length of the third side.” Proof of the theorem is provided in Jaunt 2.(Glencoe 261) 2 2 2 2 2 2 x ! x + y ! y + x ! x + y ! y " x ! x + y ! y ( P Q ) ( P Q ) ( Q R ) ( Q R ) ( P R ) ( P R ) and d(P,Q) + d(Q, R) ! d( P, R) . Let P , Q , and R be collinear. For three points collinear, one point must be between the other two. Betweenness of points requires the points be collinear, and if Q is between P and R then PQ + QR = PR . Proof Taxicab distance formula is a metric: Let P(x , y ) , Q(x , y ) and R(x , y ) !"2 . P P Q Q R R Metric Axiom 1: d(P,Q) ! 0 and d(P,Q) = 0 if and only if P = Q If P = Q then d(P,Q) = d(P, P) = x ! x + y ! y P P P P = 0 + 0 7 = 0 . If P ! Q then either x ! x or y ! y . P Q P Q and x ! x > 0 or y ! y > 0 . P Q P Q Suppose x ! x P Q Then x ! x + y ! y P Q P Q ! x " x P Q > 0 Therefore, x ! x + y ! y " 0 . P Q P Q Metric Axiom 2: d(P,Q) = d(Q, P) d(P,Q) = x ! x + y ! y P Q P Q = !1 !x + x + !1 !y + y ( )( P Q ) ( )( P Q ) = !1 x ! x + !1 y ! y ( )( Q P ) ( )( Q P ) = !1 x ! x + !1 y ! y ( ) ( Q P ) ( ) ( Q P ) = x ! x + y ! y Q P Q P = d(Q, P) Metric Axiom 3: d(P,Q) + d(Q, R) ! d( P, R) Let P , Q , and R be non-collinear, thus forming a triangle. d(P,Q) + d(Q, R) = x ! x + y ! y + x ! x + y ! y P Q P Q Q R Q R 8 = x ! x + x ! x + y ! y + y ! y ( P Q Q R ) ( P Q Q R ) ! x " x + x " x + y " y + y " y ( P Q Q R ) ( P Q Q R ) = x ! x + y ! y P R P R = d( P, R) hence, d(P,Q) + d(Q, R) ! d( P, R) (Reynolds 125) Note that the Triangle Inequality was not used in the proof of axiom 3, as it was in the Euclidean proof.
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