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Taxicab Geometry
Chapter 6
Distance
• On a number line
• On a plane with two dimensions Coordinate system skew () or rectangular
2( , ) P Q P Qd P Q x x x x
2 22 cosP Q P Q P Q P Qx x y y x x y y
2 2
P Q P Qx x y y
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)
1PR
PQ QR
Axiom System for Metric Geometry
• Axioms for metric space
1.d(P, Q) 0d(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 6.1Euclidian distance formula
satisfies all three metric axiomsHence, the formula is a metric in
• Demonstrate satisfaction of all 3 axioms
2 2, P Q P Qd P Q x x y y
2
Taxicab Distance Formula
• Consider this formula
• Does this distance formula satisfy all three axioms?
( , )T P Q P Qd P Q x x y y
( , ) 0TP Q d P Q ( , ) ( , )T Td P Q d Q P( , ) ( , ) ( , )T T Td P Q d Q R d R P
Thus, the taxicab distance formula is a
metric in 2
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
• Note: this definition does not tell us what metric to use!
: ( , ) , 0,circle P d P C r r C is fixed
Taxi-Circles
• Recall Activity 6.5
Taxi-Circles
• Place center of taxi-circle at origin
• Determine equationsof lines
• Note how any pointon line has taxi-cabdistance = r
Ellipse
• Defined as set off all points, P, sum of whose distances from F1 and F2 is a constant
1 2
1 2
{ : ( , ) ( , ) ,
0, , }
ellipse P d P F d P F d
d F F fixed
Ellipse
• Activity 6.2
• Note resultinglocus of points
• Each pointsatisfiesellipse 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 whereboth sidesintersect
Ellipse
• Now when foci are vertical
Ellipse
• End result is a hexagon
• Again, four of thesides are wheresides 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 exactlyone 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: slopeof 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• Parabola
All points equidistant from a fixed point and a fixed line
Fixed linecalleddirectrix
2y a x b x c { : ( , ) ( , )}P d P F d P k
Taxicab Parabolas
• From the definition
• Consider use of taxicab metric
{ : ( , ) ( , )}P d P F d P k
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