Distance-based shapes
A few weeks ago I wrote about how ellipses and hyperbolas have multiple definitions, which in Euclidean geometry are equivalent but in taxicab geometry produce different shapes. Ellipses can be defined using the sum of the distances to two foci, or as a specific range of ratios of the distances to a focus and a directrix (a line). Similarly, hyperbolas can be defined using the difference of the distances to two foci, or as another range of ratios of the distances to a focus and a directrix. Definitions based on ratio of the distances to a point and a line produce conic section. Since ovals of Cassini use the product of the distances to two foci, we can start to fill in a table of distance-based shapes:
| □ | dist(p, f₁) □ dist(p, f₂) | dist(p, f) □ dist(p, d) |
|---|---|---|
| + | Ellipse • circle when f₁ = f₂ | |
| − | Hyperbola | |
| × | Oval of Cassini | |
| ÷ | Conic section • ellipse when ratio < 1 • parabola when ratio = 1 • hyperbola when ratio > 1 |
Filling in the lower-left gap, I learned that the shape based on a ratio of the distances two foci is called a circle of Apollonius. However, I haven't found names for the shapes based on the sum, difference, or product of the distances to a focus and a directrix. (AI has suggested limaçon, which as far as I can tell is unrelated.) If you have suggestions, please email me.
| □ | dist(p, f₁) □ dist(p, f₂) | dist(p, f) □ dist(p, d) |
|---|---|---|
| + | Ellipse • circle when f₁ = f₂ | ? |
| − | Hyperbola | ? |
| × | Oval of Cassini | ? |
| ÷ | Circle of Apollonius | Conic section • ellipse when ratio < 1 • parabola when ratio = 1 • hyperbola when ratio > 1 |
Names or not, in the next few posts I'll illustrate each shape in both Euclidean and taxicab geometries.