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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.