Taxicab Geometry

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Taxicab Geometry

• MAEN 504
• George Carbone
• August 18, 2002

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Table Of Contents

• Why Taxicab Geometry?
• History of Taxicab Geometry
• The General Equation for Taxicab Distance
• Taxicab Circles
• An Application - Taxicab Treasure Hunt
• Taxicab Ellipse
• More Applications
• Resources and Bibliography

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Why Taxicab Geometry?
The best way to understand Why Taxicab Geometry
is through a practical application or
problem... John wants to walk from his home to
the library. Based on the diagram at the left
where each square represents one square block,
what is the shortest distance he must walk?
Analysis The library is 4 blocks north and 3
blocks west of John's house. Euclidean geometry
would suggest that the shortest distance is along
the dotted line. Moreover, since we have a right
triangle with legs equaling 3 and 4, then
based on the Pythagorean Theorem, the hypotenuse
or dotted line would equal 5 blocks. However,
unless John is a bird, he can't follow the dotted
path. Instead he must walk along the various
streets or blocks. Therefore, what is the least
number of blocks he must walk? One can see that
least number of blocks is 7. Specifically, he
would need to walk some combination of 4 blocks
north and some combination of 3 blocks
west. Taxicab Geometry provides us with a
Non-Euclidean framework for analyzing problems
based on blocks - much like the grid of an urban
street map - hence the name Taxicab Geometry.
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History Of Taxicab Geometry

• Hermann Minkowski (1864-1909) introduced
  taxicab Geometry over 100 years ago. Minkowski
  was a German mathematician and professor who
  studied at the Universities of Berlin and
  Konigsberg. He taught at several universities in
  Bonn, Konigsberg and Zurich. Interestingly,
  Albert Einstein was among his students in Zurich
  and Minkowski's Non-Euclidean work on a
  four-dimensional space-time continuum provided a
  framework for Einstein's later work on the Theory
  of Relativity.
• Euclidean Geometry measures distance as
  the crow flies. Minkowski recognized that this
  was not necessarily the best model for many real
  world situations, particularly for problems
  involving cities where distances are determined
  along blocks and not as the crow flies.
• Another valuable aspect of Taxicab
  Geometry is its simplicity as a non-Euclidean
  Geometry. It is more easily understood than many
  other non-Euclidean geometries. In fact, given
  its grid or Cartesian based orientation, it can
  be taught with the aid of graph paper as early as
  in the middle school years.
• Since its introduction in the late 1800s,
  Taxicab Geometry has undergone periods of great
  interest and practical application, as well as
  periods of marginalization. It received renewed
  attention in 1975 when Eugene Krause, a
  Mathematics professor at the University of
  Michigan, published a detailed book on the
  subject entitled Taxicab Geometry An Adventure
  in Non-Euclidean Geometry.

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The General Equation for Taxicab Distance
In Euclidean Geometry, the distance between two
points A and B can be derived based on the
Pythagorean Theorem d(A,B) squareroot
((xA-xB)2 (yA-yB)2 ). In the example at left,
this formula yields the crow flies measurement
of 7.81 cm, based on Squareroot(52 62).
However, in Taxicab Geometry, the distance
formula was redefined by Minkowski distance as
d(A,B) xA-xB yA-yB. Thus, the Taxicab
distance in this example is 11 cm, based on 6 5.

• In other words, the distance is defined as
  the sum of the horizontal and vertical distance
  of the two points. This is the minimum distance
  a taxicab would need to travel to reach point B
  from point A, if all streets are only oriented
  horizontally and vertically.
• In the following pages well see how this
  Non-Euclidean measure of distance can be applied
  to practical problems and result in Taxicab
  definition of geometric figures, such as circles
  and ellipses. Not surprisingly, our
  understandings these figures must change.

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Taxicab Circles

• In Euclidean Geometry, a circle represents a
  series of points equidistant from a single point
  or center. If we apply the Taxicab distance to
  the definition of a circle, we get an interesting
  shape of a Taxicab circle.
• For example, the set of points 3 units away from
  point a (1,1) is outlined at left. The dotted
  line provides an example of a distance of 3.
• Note the figure appears to be a square.

This definition of a Taxicab Circle provides a
basis for addressing many practical applications
Taxicab Geometry. For example, in the following
section we explore Taxicab Treasure Hunt, a
website (http//www.learner.org/teacherslab/math/g
eometry/shape/taxicab/) for determining the
location of a hidden treasure. The solution is
based on the intersection of respective Taxicab
Circles.
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Taxicab Treasure Hunt
The object of the treasure hunt game is to find a
hidden treasure located somewhere in the mythical
city outlined to the left. You first define a
starting point that I have stated as Third Avenue
and Dogwood Street. The game then tells me my
distance from the treasure - in this example. It
states the distance in 3 blocks. Thus I know the
answer consists of the locus of points a taxicab
circle of 3 blocks from Third and Dogwood. That
circle is outlined at the lower left (note the
entire circle does not appear since it is outside
the bounds of the defined city). I then selected
Fifth and Elm (a point along the prior Taxicab
Circle).
It informs me that I am 4 blocks from the
treasure. Through a second thicker circle now 4
blocks from my new location, I know the solution
must be at the intersection of the 2 circles (see
the middle figure below). Therefore, the
possible solutions are Second and Fir, First and
Elm, or Fourth and Birch. I then selected Fourth
and Fir, and found out I was 2 blocks away. I
drew a third (thin) Taxicab circle 2 blocks from
the new location and found the solution is First
and Elm, the intersection of the 3 circles.
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Taxicab Ellipses

• In Euclidean Geometry, an ellipse is defined as
  the set of all points the sum of whose distance
  from two given points is a fixed distance.
  Again, applying the Taxicab concept of distance
  to this definition of an ellipse yields an
  interesting Taxicab Ellipse.
• For example, let A equal (-2, -1) and B equal
  (2, 2). What is the locus of points where the sum
  of the distance from these points is 9? The
  answer is outlined at below.

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More Applications - A
One problem outlined in Krause's book is the
following...
There are 3 high schools in Ideal City. Fillmore
at (-4, 3), Grant at (2, 1), and Harding at (-1,
-6). Draw in school district boundary lines so
that each student in Ideal City attends the high
school nearest his home. The first thing one must
do is construct boundaries between each pair of
schools. Since Fillmore and Grant are 8 blocks
apart, the locus of equally distant points
between them is the blue line. Since Fillmore and
Harding are 12 blocks apart, the locus of equally
distant points between them is the black line.
And since Grant and Harding are 10 blocks apart,
the locus of equally distant points between them
is the red line. Since there are 3 schools the
actual dividing lines are determined by the
thicker set of lines. Therefore, Grant would
serve those living in the yellow area, Harding
would serve those in the gray area, and Fillmore
would serve those in the teal area.
Follow on question... Burger Baron wants to open
a hamburger stand equally distant from the 3 high
schools. Where should they locate it? Answer
at (-2, -1) as it 6 blocks from each.
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More Applications - B
Another problem outlined by Krause... Ajax
Industrial Corporation wants to build a factory
in Ideal City in a location where the sum of its
distances from the railroad station C (-5, -3)
and the airport D (5, -1) is at most 16 blocks.
For noise control purposes a city ordinance
forbids the
location of any factory within 3 blocks of the
public library L (-4, 2). Where can Ajax
build? The solution to this problem is outlined
by the thicker lined figure outlined at upper
right. This is clearly an ellipse solution with
a wrinkle involving the library. Interestingly,
and suggested by the book, we use a Euclidean
circle not a Taxicab circle for the Library
cut-out. Why? Sound does travel the way the crow
flies. Therefore, a 3 block Euclidean cut-out
which is larger than the potential Taxicab
cut-out (see dotted line) is appropriate.
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Resources and Bibliography

• Biography of Hermann Minkowski
• http//www-gap.dcs.st-and.ac.uk/history/Ma
  thematicians/Minkowski.html
• Hermann Minkowski and Taxicab Geometry
  http//www.mzt.hr/mzt/hrv/informacije/publi/casopi
  si/c-teh/kog/koga5.htm
• Krause, Eugene. Taxicab Geometry An Adventure
  in Non-Euclidean Geometry. Dover Publications,
  Inc. New York. 1975.
• Taxicab Tresure Hunt
• http//www.learner.org/teacherslab/math/geometry
  /shape/taxicab/
• Why Taxicab Geometry?
• http//cgm.cs.mcgill.ca/godfried/teaching/projec
  ts.pr.98/tesson/taxi/why.html