TAXICABGEOMETRYBy: Cathy Dowdhttp://www.animationfactory.com/en/search/close-up.mc?&oid=4965062&s=1&sc=1&st=12&q=taxicab&spage=1&hoid=1c79551bc87749a83c31ead9f429a42b

Introduction to Taxicab GeometryHistoryComponents of a mathematical systemReview 13 Euclidean AssumptionsCompare Taxicab geometry to Euclidean geometryDistance functionExamples: circle, ellipse, and parabolaAssumptions that Taxicab does meetAssumption twelve: SASWhy Taxicab geometry is considered a non-Euclidean geometry.

Taxicab HistoryTaxicab Geometry is a form of non-Euclidean geometries introduced by Hermann Minkowski (Gesammelte Abhandlungen) over 100 years ago.

Although interest has varied, today there is a whole spectrum of applications for this kind of geometry.

http://en.wikipedia.org/wiki/Image:Hermann_Minkowski.jpg#file

For ExampleUrban Planning

Emergency RoutesAmbulancesPoliceFire

Components of a Mathematical SystemObjectsMetricAssumptionsDefinitionsTheorems

GEOMETRYGeometric ObjectsA set P of points,A collection L of subsets of P called lines,MetricsAn angle measure function m,A distance function dE for ordinary geometry, dT for taxicab geometry.Notation[P,L,dE,m] Euclidean geometry[P,L,dT,m] Taxicab geometry

Euclidean GeometryObjects Normal points & lines in the Cartesian PlaneMetricsDistance function measures distances between two points as the crow flies.Angle Measure function measures angles in the usual way.Thirteen fundamental assumptions for modern Euclidean geometry.Euclidean Geometry [P,L,dE,m] What are these assumptions?Glad you asked

Euclidean Assumptions

Euclidean Assumptions

Euclidean GeometryIn Euclidean coordinate geometry, the distance is measured as the way the crow flies.

Taxicab Geometry

Taxicab GeometryThe taxicab distance is defined as the sum of the lengths of the vertical and horizontal distances from A to B.

Taxicab Geometry Objects Normal points & lines in the Cartesian PlaneMetricsDistance function measures distances between two points using only horizontal and vertical distances.Angle Measure function measures angles in the usual way.

Explore CirclesEuclideanRadius 3Center (0,0)Definition: The set of all points equidistant from a given point.

Explore Taxicab CircleNow you try..Radius of 3Center (0,0)

Explore Taxicab CircleRadius of 3Center (0,0)This circle is also defined as the set of all points that are a distance of 3 from A.

Explore EllipsesDefinition:

The set of all points where the sum of the distances from two fixed points to a point in the set is constant.

EuclideanTaxicab

Explore Ellipseshttp://jwilson.coe.uga.edu/EMAT6680Fa06/Sexton/GeoFinalProject/Taxicab/ellipse.htmlThe Euclidean ellipse is defined as the set of all points where the AC + CB = 6

Taxicab Ellipsehttp://jwilson.coe.uga.edu/EMAT6680Fa06/Sexton/GeoFinalProject/Taxicab/ellipse.htmlThe ellipse is defined as the set of all points wheretaxicab distance AC + taxicab distance CB=9

Explore Parabolashttp://jwilson.coe.uga.edu/EMAT6680Fa06/Sexton/GeoFinalProject/Taxicab/Parabola.htmlHere the parabola is defined as the set of all points where AC is the same distance as A to the line and BC is the same distance as B to the line.The Taxicab parabola is defined as the set of all points where taxicab distance AC = taxicab distance of A to the line and taxicab distance CB = taxicab distance of B to the line

Comparing Taxicab and Euclidean GeometriesHow many of the Euclidean assumptions are also true for Taxicab geometry?

Assumptions: One, Two, Three, FourAll of these hold for Euclidean and Taxicab Geometry

Assumption Five Two CasesCase One: A,B,C are non-collinearCase Two: A,B,C are collinear

Assumption Six(Distances are preserved)

Assumption SixTo prove this assumption we must check for the following:

One to OneOntoDistances are preserved

Assumption Six: Looking for the functionTwo CasesL is vertical

L contains a defined slope

Assumption Six ContinuedL is Vertical

Assumption Six ContinuedL is Vertical

Assumption Six ContinuedL is Vertical

Assumption Six Conclusion for Vertical LinesTRUE!!

Assumption Six ContinuedL Contains a Slope

Assumption Six ContinuedL Contains a Slope

Assumption Six ContinuedL Contains a Slope

Assumption Six ContinuedL Contains a Slope

Assumption Six ContinuedL Contains a Slope

Assumption Six ContinuedL Contains a Slope

Assumption Six ContinuedL Contains a Slope

Assumption Six ContinuedL Contains a Slope

Assumption Six ConclusionL Contains a SlopeTRUE!!!

Assumptions: Seven, Eight, Nine, Ten, Eleven, and ThirteenAll of these hold for Euclidean and Taxicab Geometry

Assumption Twelve: SASThis is the one basic property of [P,L,dE,m] that [P,L,dT,m] does not have!!

Better known as Side-Angle-Side Axiom.

Assumption Twelve Continued:SASExampleClearly you can tell that these two triangles are not the same.In the Taxicab plane, triangles that meet the SAS criteria are not congruent. The problem is due to how distance is measured!

Conclusion: Taxicab GeometryThis is why [P,L,dT,m] is a non-Euclidean geometry!!!

TAXICABGEOMETRYBy: Cathy Dowd