Parabolas in Taxicab Geometry Everyone Knows What a Circle Looks
Everyone knows what a circle looks like, and geometry students can recite the fact that a circle is the set of points equidistant to a given center point. Equations for parabolas have been memorized, and students might remember that the definition involves a focus point and a directrix. But do students really pay attention to these definitions? Changing the way the distance between two points is measured is an excellent way to challenge geometry student’s understanding. The easiest, visually intuitive, metric to work with is the taxicab metric.
Definition: Let A and B be points with coordinates (푥1, 푦1) and (푥2, 푦2), respectively. The taxicab distance from A to B is defined as 푑𝑖푠푡(퐴, 퐵) = |푥1 − 푥2| + |푦1 − 푦2|.
I do not use the words “taxicab distance” or “taxicab geometry” in my class because I do not want students to google the terms. The fact that this seemingly strange distance measurement occurs naturally when maneuvering on a city grid is left for students to discover themselves. My first assignment is to plot several points on graph paper and compute the distance between them by following the definition, i.e. adding their horizontal and vertical distances. In practice, this means to count the horizontal and vertical steps needed to get from A to B on the grid. Our first observation is that the shortest path between two points is not always unique because, as long as there is no backtracking, you can alternate horizontal and vertical steps. Students should also observe that distance is invariant under translation, but not rotation. A circle is defined as the set of points equidistant from a given center point. The next assignment is to mark two points C and A on graph paper and construct the circle with center C, containing point A. Some students plot a few points and then try to connect them with a curve. The point of working with taxicab geometry is to help students understand that they need to pay careful attention to definitions and not have preconceived notions of what these figures should look like. The fact that a circle with center (ℎ, 푘) consists only of linear parts can be seen from the defining equation: |푥 − ℎ| + |푦 − 푘| = 푟.
Observation: In taxicab geometry a circle consists of four congruent segments of slope ±1.
Once students have accepted that a circle is a diamond shape, we move on to the next assignment: Look up the definition of parabola and draw a parabola using our new definition of distance.
Definition: Given a line l (directrix), and a point F (focus) not on the line, a parabola is the set of all points P that have the same distance to l as to F. {푃 ∶ 푑𝑖푠푡(푃, 푙) = 푑𝑖푠푡(푃, 퐹)}
Almost all students will initially use a horizontal line as directrix, see figure 1, so the fact that they will probably assume (from their Euclidean experience) that the shortest distance from a point to a line is along the perpendicular from P to l, will accidentally be correct. This is a great opportunity for a class discussion on “What do we mean by the distance of a point to a line?”
Definition: The distance of point P to line l is the smallest of the distances from P to any point L on l. 푑𝑖푠푡(푃, 푙) = 푚𝑖푛{푑𝑖푠푡(푃, 퐿): 퐿 표푛 푙}
To find the minimal distance, consider expanding circles around P until a circle touches the line. The radius of that circle is the shortest distance from P to l, see figure 2. Circles have sides of slope ±1, so if the line also has slope 푚 = ±1 there will be a quarter-circle segment of points on the line that all have the same minimal distance to P.
Observation: In taxicab geometry the shortest path from P to l depends on the slope m of the line as follows:
If |푚| < 1: go vertically from P to l. (this includes horizontal lines) If |푚| > 1: go horizontally from P to l. (this also applies to vertical lines) If |푚| = 1: go vertically or horizontally or anything in between!
Now we can investigate the general shape of a parabola in taxicab geometry.
Figure 1 shows a parabola with horizontal directrix y=0, and focus F(0,a). All points 푃(±푎, 푦) with 푦 ≥ 푎 have the same distance to the x-axis as to F, because we can use minimal paths through the corner points (±푎, 푎). The parabola extends indefinitely upward (or down, if F is below the x-axis), its width is constrained to −푎 ≤ 푥 ≤ 푎. If the directix is a vertical line the figure is rotated by 90°. Figure 3 shows a parabola with directrix of slope m=1. There is no unique shortest segment from F to l, the parabola contains a quarter-circle segment centered at F with radius 푑푖푠푡(퐹,푙) . The parabola continues with a horizontal and a vertical ray, because for P as shown we 2 can take a minimal path through corner point Q. If the directrix has slope 푚 = −1 the figure is rotated by 90°. A parabola with directrix of slope |푚| ≠ 1 will have three corner points: a vertex point V in the middle of the shortest segment from F to l , and two corner points Q, see figure 4. 푑푖푠푡(퐹,푙) If |푚| < 1 the points Q are at a horizontal distances of from F, and starting at Q the 1±푚 parabola contains vertical rays. If |푚| > 1 (remember to measure distances to the line by going horizontally!) the points Q are 푑푖푠푡(퐹,푙) at a vertical distances of 1 from F, and starting at Q the parabola contains horizontal rays. 1± 푚
If the parabola includes many grid points it is easier for students to find points by trial and error. You can insure that this is the case by carefully choosing the distance of grid point F to the directrix. If dist(F,l) is even then V is also a grid point. If we want the Q to be grid points, we need their distance from F to be integers. For a directrix of slope m=a/b, choose a focus location such that dist(F,l) is a multiple of |푏2 − 푎2|. If a and b are both odd, then half of that 32−12 suffices. In figure 4, for m=1/3 both Q are grid points because = 4 = 푑𝑖푠푡(퐹, 푙). For m=2 2 both Q are grid points because |12 − 22| = 3 = 푑𝑖푠푡(퐹, 푙). Choosing 푑𝑖푠푡(퐹, 푙) = 2 ∗ 3 = 6 would also make the vertex a grid point.
Circles and parabolas in Taxicab Geometry can be obtained as sections of a square pyramid, see [Laatsch]. What about the other conic sections? In Euclidean geometry there are two equivalent definitions for ellipses and hyperbolas, respectively, but they lead to very different figures in taxicab geometry. [Laatsch] shows that they are obtained as the other sections of a square pyramid, but only if the definition based on a focus point and directrix is used. If we use the more popular high school definition based on two focus points, we obtain the ellipse and hyperbola shapes found more commonly in the taxicab geometry literature. A great introduction to taxicab geometry with many problems to work through is Krause’s classic book [Krause].
Summary: Working with the taxicab metric forces students to pay careful attention to definitions. In taxicab geometry circles are diamond shapes, and the shape of a parabola depends on the slope of its directrix.
References
Krause Eugene, Taxicab Geometry: an Adventure in Non-Euclidean Geometry, New York: Dover Publications, 1986. Laatsch Richard, Pyramidal Sections in Taxicab Geometry, Math. Mag. 55 (1982) 205- 212.