Taxicab Geometry

1
Taxicab Geometry

• Chapter 5

2
Distance

• On a number line
• On a plane with two dimensions
• Coordinate system skew (??) or rectangular

3
Axiom System for Metric Geometry

• Formula for measuring ? metric
• Example seen on previous slide
• Results of Activity 5.4
• Distance ? 0
• PQ QR ? RP(triangle inequality)

4
Axiom System for Metric Geometry

• Axioms for metric space
• d(P, Q) ? 0d(P, Q) 0 iff P Q
• d(P, Q) d(Q, P)
• d(P, Q) d(Q, R) ? d(P, R)

5
Euclidian Distance Formula

• Theorem 5.1Euclidian distance formulasatisfies
  all three metric axiomsHence, the formula is a
  metric in
• Demonstrate satisfaction of all 3 axioms

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Taxicab Distance Formula

• Consider this formula
• Does this distance formula satisfy all three
  axioms?

7
Application of Taxicab Geometry
8
Application of Taxicab Geometry

• A dispatcher for Ideal City Police Department
  receives a report of an accident at X (-1,4).
  There are two police cars located in the area.
  Car C is at (2,1) and car D is at (-1,- 1). Which
  car should be sent?
• Taxicab Dispatch

9
Circles

• Recall circle definitionThe set of all points
  equidistance from a given fixed center
• Or
• Note this definition does not tell us what
  metric to use!

10
Taxi-Circles

• Recall Activity 5.5

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Taxi-Circles

• Place center of taxi-circle at origin
• Determine equationsof lines
• Note how any pointon line has taxi-cabdistance
  r

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Ellipse

• Defined as set off all points, P, sum of whose
  distances from F1 and F2 is a constant

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Ellipse

• Activity 5.2
• Note resultinglocus of points
• Each pointsatisfiesellipse defn.
• What happened with foci closer together?

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Ellipse

• Now use taxicab metric
• First with the two points on a diagonal

15
Ellipse

• End result is an octagon
• Corners are whereboth sidesintersect

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Ellipse

• Now when foci are vertical

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Ellipse

• End result is a hexagon
• Again, four of thesides are wheresides of
  bothcircles intersect

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Distance Point to Line

• In Chapter 4 we used a circle
• Tangent to the line
• Centered at the point
• Distance was radius of circle which intersected
  line in exactlyone point

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Distance Point to Line

• Apply this to taxicab circle
• Activity 5.8, finding radius of smallest circle
  which intersects the line in exactly one point
• Note slopeof line- 1 lt m lt 1
• Rule?

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Distance Point to Line

• When slope, m 1
• What is the rule for the distance?

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Distance Point to Line

• When m gt 1
• What is the rule?

22
Parabolas

• Quadratic equations
• Parabola
• All points equidistant from a fixed point and a
  fixed line
• Fixed linecalleddirectrix

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Taxicab Parabolas

• From the definition
• Consider use of taxicab metric

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Taxicab Parabolas

• Remember
• All distances are taxicab-metric

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Taxicab Parabolas

• When directrix has slope lt 1

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Taxicab Parabolas

• When directrix has slope gt 0

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Taxicab Parabolas

• What does it take to have the parabola open
  downwards?

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Locus of Points Equidistant from Two Points
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Taxicab Hyperbola
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Equilateral Triangle
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Axiom Systems

• Definition of Axiom System
• A formal statement
• Most basic expectations about a concept
• We have seen
• Euclids postulates
• Metric axioms (distance)
• Another axiom system to consider
• What does between mean?

32
Application of Taxicab Geometry
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Application of Taxicab Geometry

• We want to draw school district boundaries such
  that every student is going to the closest
  school. There are three schools Jefferson at
  (-6, -1), Franklin at (-3, -3), and Roosevelt at
  (2,1).
• Find lines equidistant from each set of schools

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Application of Taxicab Geometry

• Solution to school district problem

35
Taxicab Geometry

• Chapter 5