Taxicab Geometry

shape

circle

ellipse

hyperbola

parabola

r

B

(4, -6)

(4, -4)

(4, -2)

(4, 0)

(4, 2)

(4, 4)

(4, 6)

L (for parabola only)

y = -3x

y = -x

y = -x/3

The taxicab circle {P :

d

T

(P, B) = 3.}

show Euclidean shape

The traditional (Euclidean) distance between two points in the plane is computed using the Pythagorean theorem and has the familiar formula,

d

E

((

x

1

,

y

1

),(

x

2

,

y

2

))=

2

(

x

1

-

x

2

)

+

2

(

y

1

-

y

2

)

. In taxicab geometry, the distance is instead defined by

d

T

((

x

1

,

y

1

),(

x

2

,

y

2

))=|

x

1

-

x

2

|+|

y

1

-

y

2

|

. This Demonstration allows you to explore the various shapes that circles, ellipses, hyperbolas, and parabolas have when using this distance formula. An option to overlay the corresponding Euclidean shapes is included for purposes of comparison.

The locus is defined:

circle:

{P:

d

T

(P,B)=r}

,

ellipse:

{P:

d

T

(P,A)+

d

T

(P,B)=r}

,

hyperbola:

{P:|

d

T

(P,A)-

d

T

(P,B)|r}

,

parabola:

{P:

d

T

(P,A)=

d

T

(P,

L

)}

.