Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
Taxicab Geometry
What is Taxicab Geometry?
Taxicab Geometry is used when the shortest distance traveled between two points cannot be a
straight line. This would be the case for a taxicab in a city trying to get from point A to point B.
The taxicab can only move vertically or horizontally in straight lines.
Use the diagram above to answer the following questions.
1. What is the Euclidean distance between A and B?
58
2. The green segments represent possible routes a taxicab might take. What is the distance
of the shortest route? 10 . Is there a unique shortest route? Justify your answer?
No, there is not a unique shortest route. The cab must, somehow, move 3 units up and 7
units to the right. Based on the image, the taxicab has 2 routes displayed that have the
same distance.
3. Write a formula for finding the shortest distance between two points in taxicab geometry.
| x |  | y |
4. What comparison can you make between Euclidean distance and Taxicab distance?
The Taxicab distance is greater than or equal to the Euclidean distance.
Creating a Macro for Taxicab distance. (Using Cabri Software)
1. Select the Show Axes tool. The x and y axes should appear on your screen. Select the
Define grid tool. You will then need to click on one of these axes for the grid to appear.
Use the pointer and drag the origin to the lower lefthand corner of your screen.
Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
2. Use the point tool to place 2 points anywhere on your screen. Label them A and B. Use
the perpendicular line tool to construct through points A and B lines perpendicular to the
appropriated x or y-axis that intersect. Use intersection point tool to place a point at the
intersection of the two perpendicular lines. Label this point C.
3. Use the segment tool to create segments AC and BC . Then use the distance/length
tool to display the lengths of AC and BC .
4. Select the calculate tool. A calculator will appear at the bottom of the screen. Click on
the distance of AC . It will appear in the calculator as “a.” Then, use the addition key.
Click on the distance of BC . It will appear in the calculator as “b.” Select equals and
the calculator will display the taxicab distance. Now, drag the result out into the screen.
5. Use the hide/show tool to hide the perpendicular lines, the intersection point, the
segments, and the distances of AC and BC .
6. Under the Macro toolbox, select initial objects. In this order, click on the y-axis, any grid
point, point A and point B (there are a total of 4 initial objects). Under the Macro toolbox,
select final objects. Click on the number (not the text!) that was given for the taxicab
distance. Next, select Define Macro. Call the new construction and first final object
“Taxicab Distance.” You may wish to write a brief instructions in the “help” box. Click
“ok.”
7. Test your Macro by placing two different points anywhere on the grid. Select Taxicab
Distance from the Macro toolbox. Remember to select your four initial objects in the
same order as above. A dotted box should appear on your screen. Click on this box and
your taxicab distance will appear. Was it correct? Yes.
Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
Investigating Triangles in Taxicab Geometry.
1. Open a new file and use the show axes tool to display the axes. Use the define grid and
click on either axis for the grid to appear. Drag the origin to the lower lefthand corner of
your screen.
2. Use the point tool to plot points at A(1, 5), B(3, 1), and C(1, 1). Use the segment tool to
form AB, BC, and AC.
Find the following taxicab distances:
AC =
m ACB =
4
BC =
90
2
AB =
6
.
3. Write out, in your own words, the Side-Angle-Side Congruence Postulate for Triangles in
Euclidean Geometry.
If two sides and the included angle of one triangle are congruent to another triangle, then
the two triangles are congruent.
4. On the same grid, plot the following points: D(7, 7), E(6, 4), and F(5, 5). Use the segment
tool to form DF, DE, and FE.
Find the following taxicab distances:
Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
DF =
m DFE =
4
FE =
90
2
DE =
4
.
5. What did you notice about your measurements? What does this imply about triangles in
Taxicab Geometry?
The measures of the sides of a triangle are not always the actual length of the sides of the
triangle. In this case, we have a triangle with a side length of 2 and Taxicab geometry
measures it to be 4. Therefore, SAS does not work in taxicab geometry.
2.03 cm
Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
50 cm
Extensions on Taxicab Geometry
Extension 1: What is a circle in Taxicab geometry?
We will be using the Taxicab Macro from the previous instructions.
2.03 cm
O
P
2.38 cm
cm axes tool. The x-axis
1. Select1.50
the show
and y-axis should appear on your screen. Select the
define grid tool. You will then need to click on one of the axes for the grid to appear. Use
the pointer tool and drag the origin to the lower lefthand corner of your screen.
B
2. Draw a circle using the circle tool, be sure to make one of the P
grid points your radius.
3.98itcm
3.98
cm
Using the point tool, draw a point P anywhere on the circle and connect
to the center O
7.19 cm
by using the segment
tool.
7.25 cm
3. Use the Taxicab Macro to measure PO . Move the resultSoff to one side. Now, use the
pointer tool to move point P around the circle. Notice the distance measurement for four
different positions of P.
3.98 cm
A
P1O =
5.31
C P2O =
4.36
3.98 cm
P3O =
4.00
P4O =
5.64
4. 5.37
Whatcm
can you conclude?
2.57 cm
The radius is not constant for a Euclidean circle in Taxicab geometry.
7 cm
5. Draw a square using the regular polygon tool. Label S the center of the square. Use the
point tool to draw a point on the square (point P) and connect it to the center S by using
the segment tool.
2.00 cm
P
S
6. Use the Taxicab Macro to measure PS Move the result off to one side. Now, use the
pointer tool to move point P around the circle. Notice the distance measurement for four
different positions of P.
Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
P1S =
4
P2S =
4
P3S =
4
P4S =
4
7. What do your results from steps 3-6 tell you about squares in Taxicab geometry?
Euclidean squares or square diamonds are circles in Taxicab geometry.
Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
Extension 2: Exploring regular polygons in Taxicab geometry.
1. Draw an equilateral triangle using the regular polygon tool. Name the vertices A, B, and
C.
2. Using the Taxicab Macro, measure the distances between the vertices.
AB =
6.63
BC =
8.37
CA =
8.66
3. What do your results indicate?
Taxicab geometry does not measure the lengths of the sides of a triangle correctly.
4. Using the regular polygon tool, discover which, if any, regular polygons you can have in
Taxicab Geometry.
After doing a bit of research, I have come to the conclusion that only one regular
polygon, a square, works in Taxicab geometry. You can manipulate a octagon (to have
alternating angles) to also work, but it is not a regular polygon.
Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
Extension 3: What is an ellipse in Taxicab geometry?
An ellipse is a curve which is the locus of all points in a plane, the sum of whose distances r1 and
r2 from two fixed points F1 and F2 (the foci) separated by a distance of 2c is given positive
constant 2a (Hilbert and Cohn-Vossen 1999, p. 2). This result in the two-center bipolar
coordinate equation r1 + r2 = 2a.
1. Draw an ellipse using the conic tool. Make sure that you select the five points on the grid,
centroid around the origin, and equidistant from the axis. Use the point tool to draw a
point P on the ellipse and connect it to the center S by using the segment tool.
2. Use the Taxicab Macro to measure the distance PS. Move the result off to one side. Now
use the pointer tool to move point P around the square, notice the distance measurement
for four different positions of P.
P1S =
1.15
P2S =
2.21
P3S =
2.0
P4S =
1.67
3. What do you conclude?
The properties for an ellipse from Euclidean geometry do not hold in taxicab geometry.
The major axis is not the longest axis. In fact, the point where PS is the longest is when
you make about a 24.6 degree angle with the x-axis.
4. Use Cabri to draw a hexagon in the coordinate
plane with two opposite sides lying on vertical lines
with the top and bottom consisting of isosceles
right triangles (as shown).
Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
5. Use the Taxicab macro, find the distance from the vertex V to the vertex T along the
vertical line. Call this distance 2a. VT = 2a =
8.00
.
6. Using the point tool, define point A on a side of the hexagon. Use the segment tool to
connect point A to each of the focus points, F1 and F2.
7. Use the taxicab macro to find the distances from A to F1 and to F2. When the distances
are added together, what distance do you get?
AF1 =
2.68
AF2 =
8. The distance obtained is the same as?
5.32
AF1 + AF2 =
8.00
2a
9. Use the pointer tool to grab and move the point A around the hexagon. How do the three
distances AF1, AF2, and AF1 + AF2 change?
AF1 + AF2 stays constant the entire way around the
hexagon. AF1 and AF2 change on the vertical
sides and stay constant on the angled sides of the
hexagon.
V
AF1 + AF2= 8.00 cm
AF1=2.68 cm
F1
A
10. What do you conclude?
VT=2a=8.00 cm
AF2=5.32 cm
This is the shape of an ellipse in Taxicab geometry.
F2
T
2.03 cm
Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
Extension 4: Locus of Points Equidistant from two given points
Question: Verify to your own satisfaction that all points in the shaded region are equidistant
from A and B in taxicab geometry.
1. Define point A at the origin and point B any place in the first quadrant. Connect the two
points using the segment tool. Make a vertical line through point B.
2. Make a point P in the region below the x-axis and to the right of the vertical line through
B. Connect the point P to A and B.
3. Using the Taxicab Macro, find the distance measures of AP and BP.
AP =
10.00
BP =
10.00
4. Using the pointer tool, move the point P anywhere in the shaded region in the picture
above. What do you notice about the distances AP and BP?
They remain the same
Kuzas, Richmond, Tocchi – Team 6 – Taxicab Geometry
5. How is it call in Euclidean geometry the locus of points equidistant from to points A and
B? How will that locus be redefined in taxicab geometry?
In Euclidean geometry, all points that are equidistant from two points A and B are on the
perpendicular bisector of segment AB. In Taxicab geometry, all points that are
equidistant from two points A and B (where A, B are on grid points) are located in a
square area rather than just on a line. From the image below, we can see where these
points are located.
B
10.00 cm
1
A
1
10.00 cm
P
However, this idea seems to only hold true when AO = BO, or when the segment AB is
the diagonal to a square. Figure 10 illustrates how the taxicab distances turn out when
AO  BO. With this type of segment, the taxicab distances in the same corner area are
always one unit apart, which is consistent with AO being one unit shorter than BO.