2/22/2011
Distance
• On a number line
Taxicab Geometry
d ( P, Q ) = xP − xQ =
(x
P
− xQ )
2
• On a plane with two dimensions
Dr. Steve Armstrong
LeTourneau University
(x
2
P
− xQ ) + ( yP − yQ )
2
[email protected]
www.letu.edu/people/stevearmstrong
What Is It ???
Axiom System for Metric
Geometry
• Formula for measuring ⇔ metric
Dist ( P, Q) =
(x
2
P
− xQ ) + ( y P − yQ )
2
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Axiom System for Metric
Geometry
Alternative Distance Formula
• Consider this formula
Axioms for
metric space
dT ( P, Q ) = xP − xQ + yP − yQ
• Does this distance formula satisfy all three
axioms?
P ≠ Q ⇔ dT ( P , Q ) > 0
dT ( P, Q ) = dT (Q, P )
d T ( P , Q ) + d T (Q , R ) ≥ d T ( R , P )
1.d(P, Q) ≥ 0
d(P, Q) = 0 iff P = Q
2.d(P, Q) = d(Q, P)
3.d(P, Q) + d(Q, R) ≥ d(P, R)
Euclidian Distance Formula
• Euclidian distance formula
d ( P, Q ) =
(x
2
P
− xQ ) + ( yP − yQ )
Taxicab Distance
• We call this formula the “taxicab” distance
formula
2
• Satisfies all three metric axioms
Hence, the formula is a metric in
dT ( P, Q) = xP − xQ + yP − yQ
ℜ2
• Distance traveled by city taxicab
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Assumptions
•
•
•
•
Urban geometry
Blocks “nice” squares
No width streets
Buildings “point mass”
Circles
• Circle definition:
circle = { P : d ( P, C ) = r , r > 0, C is fixed }
• But … which metric?
Application of Taxicab
Geometry
• Accident at (-1,4).
• Police Car C
at (2,1) .
• Police Car D
at (-1,- 1).
Distance to Points
• Taxicab distance from C to each point?
P
C
• Which car should
be sent?
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Taxicab “Circle”
“But that’s not
right!”
Nspire TaxiCircle Construction
• Establish
center
• Create
vert. (or
horiz.)
• Use Shapesreg. polygon
" circle " = {all points P : d ( P, C ) = r}
Taxicab Circle?
Ellipse
• Defined as set of all points, P, sum of
whose distances from F1 and F2 is a
constant
ellipse = {P : d ( P, F1 ) + d ( P, F2 ) = d ,
d > 0, F1 , F2 fixed }
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Ellipse
Ellipse
• Centers on horizontal
• Divide line
segment
• Transfer
measurement
of segments
• Note circle
intersection
Taxicab Ellipse
Euclidian Distance
Point to Line
• Taxicab metric
• First with centers on diagonal
• Distance to point
always on a ⊥
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Taxicab Distance – Point to Line
• Apply to taxicab circle
• Note: slope
of line - 1 < m < 1
• Rule?
Distance – Point to Line
• When |m| > 1
• Rule?
Taxicab Distance – Point to Line
• When slope, m = 1
• Rule?
Parabolas
• Quadratic equations y = a ⋅ x 2 + b ⋅ x + c
• Parabola {P : d ( P, F ) = d ( P, k )}
All points equidistant from a fixed point and a
fixed line
Fixed line
called
directrix
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Taxicab Parabolas
• From the definition
Taxicab Parabolas
{P : d ( P, F ) = d ( P, k )}
• What does it take to have the “parabola”
open downwards?
Note: slope of
directrix is
m<1
Taxicab Parabolas
• Horizontal directrix
Locus of Points Equidistant from
Two Points
• Euclidean
(perpendicular
bisector)
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Locus of Points Equidistant from
Two Points
Application of Taxicab
Geometry
Solution
• Taxicab
Chip Reinhardt
The Montana Mathematics Enthusiast, ISSN 1551-3440
Vol2, no.1, 2005 © Montana Council of Teachers of Mathematics
Application of Taxicab
Geometry
• School district boundaries
Hyperbola
• D(A, C) – D(B, C) = Constant
Every student attends closest school.
Schools:
• Jefferson at (-6, -1)
• Franklin at (-3, -3)
• Roosevelt at (2,1)
• Euclidean
• Find “lines” equidistant from each set of
schools
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Hyperbola
• D(A, C) – D(B, C) = Constant
• Taxicab
• Slanted
axis
• What is the taxicab length of the sides of
this triangle?
So how do we
classify this
triangle?
Hyperbola
• D(A, C) – D(B, C) = Constant
• Horizontal
axis
Triangle
Why?
• Personally create math
• Better understand Euclidian
geometry
• Encourage problem solving
• Deeper appreciation of structure
of math/geometry
“Thank you” to Christina Janssen
Taxicab Geometry: Not the Shortest Ride Across Town
Iowa State University
July 2007
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Further Investigations
• Right triangles
Relationships of sides
Taxicab Geometry
• Categories of quadrilaterals
Squares, parallelograms, circles, etc.
• Congruent triangles
SAS, ASA, …
Dr. Steve Armstrong
LeTourneau University
[email protected]
www.letu.edu/people/stevearmstrong
Questions?
10