Transcript Taxicab Geometry Chapter 6 Distance • On a number line d ( P, Q)  xP  xQ  x P  xQ  • On a.

Taxicab Geometry
Chapter 6
Distance
• On a number line
d ( P, Q)  xP  xQ 
x
P
 xQ 
2
• On a plane with two dimensions
 Coordinate system skew () or rectangular
x
P
 xQ    yP  yQ   2   xP  xQ    yP  yQ   cos
2
2
x
P
 xQ    yP  yQ 
2
2
Axiom System for Metric
Geometry
• Formula for measuring  metric
 Example seen on previous slide
• Results of Activity 6.4
 Distance  0
 PQ + QR  RP
(triangle inequality)

PR
1
PQ  QR
Axiom System for Metric
Geometry
• Axioms for metric space
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
• Theorem 6.1
Euclidian distance formula
d  P, Q  
x
P
 xQ    yP  yQ 
2
satisfies all three metric axioms
Hence, the formula is a metric in
2

2
• Demonstrate satisfaction of all 3 axioms
Taxicab Distance Formula
• Consider this formula
dT ( P, Q )  xP  xQ  yP  yQ
Thus, the
taxicabformula
distance
• Does this distance
satisfy all three
axioms?
formula is a
 P  Q  dT ( Pmetric
, Q)  0 in  2


dT ( P, Q)  dT (Q, P)
dT ( P, Q)  dT (Q, R)  dT ( R, P)
Application of Taxicab
Geometry
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
Circles
• Recall circle definition:
The set of all points equidistance from a
given fixed center
• Or circle  P : d (P, C)  r, r  0, C is fixed
• Note: this definition does not tell us what
metric to use!
Taxi-Circles
• Recall Activity 6.5
Taxi-Circles
• Place center of taxi-circle at origin
• Determine equations
of lines
• Note how any point
on line has taxi-cab
distance = r
Ellipse
• Defined as set off 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 }
Ellipse
• Activity 6.2
• Note resulting
locus of points
• Each point
satisfies
ellipse defn.
• What happened with foci closer together?
Ellipse
• Now use taxicab metric
• First with the two points on a diagonal
Ellipse
• End result is an octagon
• Corners are where
both sides
intersect
Ellipse
• Now when foci are vertical
Ellipse
• End result is a hexagon
• Again, four of the
sides are where
sides of both
“circles” intersect
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 exactly
one point
Distance – Point to Line
• Apply this to taxicab circle
 Activity 6.8, finding radius of smallest circle
which intersects the line in exactly one point
• Note: slope
of line
-1<m<1
• Rule?
Distance – Point to Line
• When slope, m = 1
• What is the rule for the distance?
Distance – Point to Line
• When |m| > 1
• What is the rule?
Parabolas
• Quadratic equations y  a  x  b  x  c
• Parabola {P : d ( P, F )  d ( P, k )}
2
 All points equidistant from a fixed point and a
fixed line
 Fixed line
called
directrix
Taxicab Parabolas
• From the definition
{P : d ( P, F )  d ( P, k )}
• Consider use of taxicab metric
Taxicab Parabolas
• Remember
 All distances are taxicab-metric
Taxicab Parabolas
• When directrix has slope < 1
Taxicab Parabolas
• When directrix has slope > 0
Taxicab Parabolas
• What does it take to have the “parabola”
open downwards?
Locus of Points Equidistant from
Two Points
Taxicab Hyperbola
Equilateral Triangle
Axiom Systems
• Definition of Axiom System:
 A formal statement
 Most basic expectations about a concept
• We have seen
 Euclid’s postulates
 Metric axioms (distance)
• Another axiom system to consider
 What does between mean?
Application of Taxicab
Geometry
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
Application of Taxicab
Geometry
• Solution to school district problem
Taxicab Geometry
Chapter 6