1. Taxicab Geometry
Chapter 5

2. Distance On a number line On a plane with two dimensions
Coordinate system skew () or rectangular

3. Axiom System for Metric Geometry
Formula for measuring  metric
Example seen on previous slide
Results of Activity 5.4
Distance  0
PQ + QR  RP (triangle inequality)

4. Axiom System for Metric Geometry
Axioms for metric space
d(P, Q)  0 d(P, Q) = 0 iff P = Q
d(P, Q) = d(Q, P)
d(P, Q) + d(Q, R)  d(P, R)

5. Euclidian Distance Formula
Theorem 5.1 Euclidian distance formula satisfies all three metric axioms Hence, the formula is a metric in
Demonstrate satisfaction of all 3 axioms

6. Taxicab Distance Formula
Thus, the taxicab distance formula is a metric in
Consider this formula
Does this distance formula satisfy all three axioms?

7. Application of Taxicab Geometry

8. 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

9. 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!

10. Taxi-Circles
Recall Activity 5.5

11. Taxi-Circles Place center of taxi-circle at origin
Determine equations of lines
Note how any point on line has taxi-cab distance = r

12. Ellipse
Defined as set off all points, P, sum of whose distances from F1 and F2 is a constant

13. Ellipse Activity 5.2 Note resulting locus of points
Each point satisfies ellipse defn.
What happened with foci closer together?

14. Ellipse
Now use taxicab metric
First with the two points on a diagonal

15. Ellipse End result is an octagon
Corners are where both sides intersect

16. Ellipse
Now when foci are vertical

17. Ellipse End result is a hexagon
Again, four of the sides are where sides of both “circles” intersect

18. 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

19. Distance – Point to Line
Apply this to taxicab circle
Activity 5.8, finding radius of smallest circle which intersects the line in exactly one point
Note: slope of line - 1 < m < 1
Rule?

20. Distance – Point to Line
When slope, m = 1
What is the rule for the distance?

21. Distance – Point to Line
When |m| > 1
What is the rule?

22. Parabolas Quadratic equations Parabola
All points equidistant from a fixed point and a fixed line
Fixed line called directrix

23. Taxicab Parabolas
From the definition
Consider use of taxicab metric

24. Taxicab Parabolas
Remember
All distances are taxicab-metric

25. Taxicab Parabolas
When directrix has slope < 1

26. Taxicab Parabolas
When directrix has slope > 0

27. Taxicab Parabolas
What does it take to have the “parabola” open downwards?

28. Locus of Points Equidistant from Two Points

29. Taxicab Hyperbola

30. Equilateral Triangle

31. Axiom Systems Definition of Axiom System: We have seen
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?

32. Application of Taxicab Geometry

33. 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

34. Application of Taxicab Geometry
Solution to school district problem

35. Taxicab Geometry
Chapter 5