The Elements of Taxicab Geometry A NonEuclidean Geometry

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The Elements ofTaxicab GeometryA Non-Euclidean
Geometry
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Taxicab geometry

• Taxicab geometry was developed by Hermann
  Minkowski in the 19th century and is a form of
  geometry in which the usual metric of Euclidean
  geometry is replaced by a new metric in which the
  distance between two points is the sum of the
  (absolute) differences of their coordinates.
• Taxicab Geometry, An Adventure in Non-Euclidean
  Geometry by Eugene F. Krause published by Dover
  Publications.

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Taxicab Geometry undefined terms and axions

• Undefined Elements point, block
• Undefined Terms adjacent, congruent, incident
• Axiom 1 Each point is adjacent to exactly four
  distinct points.
• Axiom 2 Each block contains (is incident with)
  exactly two points.
• Axiom 3 Two points on (incident with) the same
  block are adjacent.
• Axiom 4 Each pair of adjacent points are on
  (incident with) exactly one block.
• Axiom 5 Two distinct blocks intersect (are
  incident) in at most one point.
• Axiom 6 Each block is congruent to every other
  block.

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Some Definitions

• Def direct trip
• A trip such that
• Def distance between two points
• A(a,b) and B(c,d)
• ABca db

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Theorem 1 Each point is on exactly four blocks

• Proof
• By Axiom 1 , each point is adjacent to exactly 4
  points,
• by Axiom 4, each pair of points is on exactly 1
  block,
• so a point lies on at least 4 blocks.
• Suppose a point were on 5 blocks.
• Then by Axiom 3, it would be adjacent to 5
  points.
• This is a contradiction to Axiom 1.
• Thus each point is on exactly 4 blocks.

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Theorem 2 If T(P,R) is a direct trip from P to
R, passing through Q and T(P,R) is the
union of T1(P,Q) and T2(P,Q) ,
then T1 and T2 are direct trips.

• Proof
• Suppose that T1 is not a direct trip.
• Then there is a direct trip S1(P,Q) such that
• Further, the union of S1 and T2 is a trip S(P,R)
  that is shorter than T. But T is a direct trip
  and is at least as short as any other trip from P
  to R.
• Thus, T1 is a direct trip. Similarly, T2 is a
  direct trip.

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Theorem 3 If P, Q, and R are distinct points,
then Q lies on a direct trip
from P to R if and only if PQ QR PR.

• Proof
• If Q lies on a direct trip, say T(P,R),
• then by Theorem 2 T is the union of two direct
  trips, so PQ QR PR.
• Conversely, if PQ QR PR and
• T1(P,Q) and T2(Q,R) are direct trips,
• Then the union of T(P,R) of T1 and T2
• has the length PR. Then by definition, T1 is a
  direct trip. Since Q lies on T, the theorem is
  proved.

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Theorem 4 If P, Q, and R are distinct points,
T1(P,Q) and T2(P,Q) are
direct trips, and PQ QR PR, then the union
is a direct trip from P to R.

• Proof
• The theorem follows immediately from the fact
  that the length of the union of two is equal to
  the sum of the lengths.

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Theorem 5 If P(x,y), Q(x,y) and R(r,s) are
three distinct points, then
R lies on a direct trip
between P and Q if and only if
2rxx xx and 2syy yy

• Proof

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The triangle whose vertices are A(2,2), B(-1,1),
C(1,-1) in the xy plane is equilateral under the
taxicab metric. Find the following lengths.

• AB
• BC
• AC

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The analogue of the Euclidean parabola y(1/4)x2
in the taxicab plane (having focus at F(0,1)
directrix y1). Find the general location of a
variable point P(x,y) on the parabola using the
defining property PFPQy1. Q is on the
directrix.
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Is the Pythagorean Theorem valid in Taxicab
geometry? If not, give a counter-example
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Taxicab Geometry Rules

• Same rules as the normal Euclidean geometry
• Distance is measured differently

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Distance
Euclidean dE (A, B) normal Taxicab dT (A, B)B
CC A
B
A
C
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Distance
Euclidean dE(A, B)5 Taxicab dT(A, B)7
B
A
C
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What Changes?

• Same
• Points, lines, angles
• triangles
• Perpendicular bisectors
• Different
• Equilateral, isosceles triangles
• Circles
• Points equidistant from two given points

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Taxi-Circles dT (A, B) 5
A
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What is closer to the Post Office?
Museum
CityHall
PostOffice
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Equidistant Set of Points
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Equidistant Set of Points
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Simplifying Assumptions

• A better model for urban geometry.
• In making models assumptions are always made.
• All streets are nice squares.
• Streets have no width.
• Buildings are point masses.

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Ellipse dT (A, P) dT (B, P) k
Red 5 Blue 9
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Ellipse dT (A, P) dT (B, P) k
Red 5 Blue 9
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Hyperbola dT (A, P)dT (B, P)3
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Hyperbolas dT (A, P)dT (B, P)2
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Hyperbolas dT (A, P)dT (B, P)2
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