Therefore, a single equation that gets both sets of points would be: It seems the another set of points that satisfy the constant difference would be determined by this equation: The definition of a hyperbola is the set of points where the difference in the distance from two fixed points is a constant. I have not figured out a way to trace the segments of slope 1 or -1 from this animation. Thus the ellipse, if it exists and is not degenerate, is a closed figure with 8 or 6 segments.įrom this GSP file you can readily move the focal points A and B. At those times, all the points on the segment that coincides satisfy the equation. Next, there are 4 times at which the sides of the TCcircles coincide on the same segment. Will the region always be a square? Why? Verify that for every point in the square, the sum of the TC distances from A and B is a constant.įOR YOUR GSP CONSTRUCTIONS you want to trace the intersection of the two TC circles you generate.Ĭlick here for a GSP file adapted from Susan Sexton's web pageįirst of all there are 4 intersection points to trace, not 2. If we include it as a TCEllipse it is a degenerate case. VERIFY BY COUNTING THE GRID LINES THAT EVERY POINT ON THE DEPICTED SEGMENTS ARE PART OF THE TCEllipse. These segments are either parallel to the x-axis or y-axis (not shown here) or segments at a slope of 1 or slope of -1. Each TC ellipse in each of the first 5 is made up of six or eight segments. Here are six examples for different choices of A, B, and d. (|x - a| + |y - b|) + (|x - g| + |y - h|) = dĬreate your own GCF file, or use the one in the link above, to explore the shapes and loci that result from this equation. Using the TC distance metric, and the definition of an ellipse as the set points where the sum of the distance from two fixed points is a constant d, we can write an equation for the ellipse with foci at A(a,b) and B(g,h) as Is the ratio of the circumference to the diameter of a TC circle a constant? What is it?

The ratio of the circumference to the diameter of a circle in the Euclidean plane is a constant - pi.

Is there an analogous way to define trigonometric ratios for the Taxicab plane? In the Euclid Plane we use the Unit Circle to define the cosine, sine, and tangent ratios. In particular the equation of the Taxicab Unit Circle is |x| + |y| = 1. If A(a,b) is the origin (0,0), the the equation of the taxicab circle is |x| + |y| = d. Graphing Calculator 3.5 File for center A through B Graphing Calculator 3.5 File for center A and radius d. The claim is made that all of axioms and theorems in Neutral Geometry (Chapter 1) up to the SAS congruence will hold.Ĭonstruct Graphing Calculator and/or GSP files (or GeoGebra files) for Taxi Cab Geometry has the following distance function between points A(x 1,y 1) and B(x 2,y 2):

Elipse equation maker