9769

Taxicab Geometry

The traditional (Euclidean) distance between two points in the plane is computed using the Pythagorean theorem and has the familiar formula, . In taxicab geometry, the distance is instead defined by . 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: ,
ellipse: ,
hyperbola: ,
parabola: .
  • Contributed by: Marc Brodie (Wheeling Jesuit University)

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

The seven possible points B serve as the center of the circle, the focus of the parabola, and one of the foci for the ellipse and hyperbola. (The other focus A is fixed at the origin.) The three lines are the directrix when plotting a parabola. These choices for B and are inclusive enough to show all the interesting variations of shapes that can occur. The value serves to vary the fixed distance in the locus of points definitions of the four figures. For those familiar with Krause's book, the "midset" can be displayed using when plotting hyperbolas.
Snapshot 2 shows a degenerate ellipse. In Euclidean geometry, is the line segment from A to B, in other words all points on the shortest path from A to B. In taxicab geometry, there are many shortest paths from A to B, and is the rectangle with A and B at diametrically opposed corners.
Snapshot 4 shows a taxicab hyperbola in which two entire quarter-planes of points satisfy the relationship .
Note: Due to the different distance measures, for certain included values of , the locus of points satisfying the definition of a Euclidean hyperbola is empty, causing the Euclidean hyperbola to "disappear".
Based on: Eugene Krause, Taxicab Geometry: An Adventure in Non-Euclidean Geometry, Mineola, NY: Dover Publications, 1987.

PERMANENT CITATION

    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+