Conics in Taxicab Geometry

The notion of distance is different in Euclidean and taxicab geometry. This affects what the circle, parabola, hyperbola, and ellipse looks like in each geometry. In fact, some results may prove to be surprising, but all are based on the fundamental assumptions of taxicab geometry.

Equation of a Circle

In both geometries the circle is defined the same: the set of all points that are equidistant from a single point. The set of all points that are the same Euclidean distance from a single point look like the figure on the left:


Here the circle is defined as the set of all points that are a distance of 3 from A. The set of all points that are the same taxicab distance from a single point look like the figure on the right. This circle is also defined as the set of all points that are a distance of 3 from A.

Because of the definition of distance, it is easy to deduce the formula for the unit circle. The formula for the unit circle in taxicab geometry is |x| + |y| = 1, as opposed to x2 + y2 = 1, in Euclidian geometry.

Expanding on this knowledge it is possible to construct a general equation of a circle.

Taxicab representations of other conics

With a bit of envisioning, one can construct sketches and derive formulas for other types of conic sections in taxicab geometry.

A parabola is the set of all points that are equidistant from a fixed point and a fixed line. For example, the parabola shown has directrix y = x - 4. Its equation is in the form of three lines, y = -2, y = x, and x = 2.


The hyperbola in both Euclidian and Taxicab geometry is defined as the set of all points where the difference of the distances from two fixed points to a point in the set is constant. As can be seen, the taxicab hyperbola is also a combination of lines. Verify that the sketch indeed fits the definition.

In both geometries the ellipse is defined the same: the set of all points where the sum of the distances from two fixed points to a point in the set is constant. a sketch of the taxicab ellipse is also possible.

Interesting Remarks


As can be verified, the above image is also a correct representation of a hyperbola! Therefore, equations in taxicab form are not unique and there are many cases to be considered.