TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Geometry(A Story of Alex and Blake)
Don Hickethier
Flathead Valley Community College
MCTM Conference, MissoulaOctober 19, 2017
[email protected] / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Distance Between Alex and Blake
Alex and Blake met at a local coffee shop, The DailyDose.After some small talk they realize they live in the same area.
Find the distance they each walk from their apartments to TheDailyDose.
Find the distance between their apartments.
2 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Distance
The Taxicab distance, dT , between two points is the number ofvertical and horizontal units between the two points.
dT (A,D) =
dT (B,D) =
dT (A,B) =
Are A, B and D collinear?
Does the route matter?
How many paths are there from Blake’s apartment to the DailyDose? (Another talk on Pascal’s Triangle.)
3 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Distance
The Taxicab distance, dT , between two points is the number ofvertical and horizontal units between the two points.
dT (A,D) =
dT (B,D) =
dT (A,B) =
Are A, B and D collinear?
Does the route matter?
How many paths are there from Blake’s apartment to the DailyDose? (Another talk on Pascal’s Triangle.)
3 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Distance
The Taxicab distance, dT , between two points is the number ofvertical and horizontal units between the two points.
dT (A,D) =
dT (B,D) =
dT (A,B) =
Are A, B and D collinear?
Does the route matter?
How many paths are there from Blake’s apartment to the DailyDose? (Another talk on Pascal’s Triangle.)
3 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Distance
The Taxicab distance, dT , between two points is the number ofvertical and horizontal units between the two points.
dT (A,D) =
dT (B,D) =
dT (A,B) =
Are A, B and D collinear?
Does the route matter?
How many paths are there from Blake’s apartment to the DailyDose? (Another talk on Pascal’s Triangle.)
3 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Distance
The Taxicab distance, dT , between two points is the number ofvertical and horizontal units between the two points.
dT (A,D) =
dT (B,D) =
dT (A,B) =
Are A, B and D collinear?
Does the route matter?
How many paths are there from Blake’s apartment to the DailyDose?
(Another talk on Pascal’s Triangle.)
3 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Distance
The Taxicab distance, dT , between two points is the number ofvertical and horizontal units between the two points.
dT (A,D) =
dT (B,D) =
dT (A,B) =
Are A, B and D collinear?
Does the route matter?
How many paths are there from Blake’s apartment to the DailyDose? (Another talk on Pascal’s Triangle.)
3 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Midpoint Between Alex and Blake
After many conversations over coffees Alex and Blake realizethey are quite compatible and both very broke. They decide toget an apartment together. To be totally fair to each otherthey decide they need to find an apartment that is equaldistance from each apartment.
Find all of the possible locations to look for an apartment.
4 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Midline
The midline is the set of points equidistant from two givenpoints.
d(M,A) = d(M,B)
5 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Equal Distance from the The Comfy Complex
After much searching they settle on a place with a much bettername than facilities, The Comfy Complex. Again in an effort tobe totally fair to one another they decide to look for new jobsthat are equal distance from the Complex.
Find all locations that are 4, 8, 12 and 16 units from C.
6 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Circle
A circle is a set of points that are equidistance from a fixedpoint.
d(P,C) = r
7 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Total Distance to Work
Ironically Alex and Blake found jobs at Almost Antiques andthe Busy Bakery, just below their old apartments. Now thatthey both have jobs they decide they can look for morefavorable housing than the Comfy Complex. Still on a bit of abudget they decide that the total combined distance they travelto work must be no more than 30 units.
Find the boundary of the search region for a new apartment.
8 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Ellipse
An ellipse is a set of points such that the sum of the distancesfrom two fixed points is a constant.
d(P,A) + d(P,B) = s
9 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Difference Traveled
Not having much luck in the house search Alex and Blakedecided that as long as one person did not have to travel 6more units to work than the other, that would be okay.
Find the new boundary for their search.
10 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Hyperbola
A hyperbola is a set of points such that the difference of thedistances from two fixed points is a constant.
|d(P,A)− d(P,B)| = k
11 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Between Busy Bakery and the By-Pass
Still not having much luck in the house search they realize thatthe new by-pass has just been completed making it much easierto get to Almost Antiques. This time Alex and Blake decidethat as long as the distance from the Busy Bakery and theby-pass is the same that would be pretty much fair.
How far is it from the Busy Bakery to the by-pass?
Find the new possible search region.
12 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Between Busy Bakery and the By-Pass
Still not having much luck in the house search they realize thatthe new by-pass has just been completed making it much easierto get to Almost Antiques. This time Alex and Blake decidethat as long as the distance from the Busy Bakery and theby-pass is the same that would be pretty much fair.
How far is it from the Busy Bakery to the by-pass?
Find the new possible search region.
12 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab Parabola
A parabola is a set of points equal distances from a fixed pointand a line.
d(P,A) = d(P,L)
13 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Divided SpaceThree Ways
After weeks of fruitless searches disagreements on the “fairest”way to locate a house begin to surface. Alex and Blake realizethey really had nothing in common other than coffee. Theydecide to split up. Not wanting to accidentally run into eachother at the Daily Dose they decide they will each look forhousing no more than midway between their work, the DailyDose and the others work.
Find the new areas they may isolate themselves for their finalhousing searches.
14 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Thanks for Playing
Taxicab Geometry, An Adventure in Non-Euclidean GeometryEugene F. Krause, Dover
15 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Taxicab is Non-EuclideanSAS fails
Given a 1-1 correspondence between the vertex set of twotriangles. If two sides and the included angle of the firsttriangle are congruent to the corresponding parts of the secondtriangle, then the correspondence is a congruence.
16 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Applications and Further Study
City and Emergency Planning
Amazon robots
3 Dimensional Taxicab
Trigonometry in Taxicab
17 / 18
TaxicabGeometry(A Story ofAlex andBlake)
DonHickethier
NewAcquaintances
Meeting HalfWay
Finding Jobs
NewApartment(on a budget)
NewApartment(almost fair)
The By-Pass
The Break-Up
The End
Euclidean vs Taxicab
dE(A,D) =√
(a1 − b1)2 + (a2 − b2)2
dT (A,D) = |a1 − b1|+ |a2 − b2|
18 / 18
