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