MATHEMATICAL SOLUTION

Elementary Curves

Ellipse, Circle, Hyperbola, Parabola - Parallel, Intersecting and Coincident Lines 

The simplest curve is the straight line. The next complexity are the second degree or conics curves.

Non-degenerate Conics

Ellipse

x²/a² + y²/b² = 1        (1)
where
a, b = intrinsic parameters

Circle

Circle, a special ellipse, can be expressed as

x² + y² = a²       (2)
where
a = radius

Hyperbola

x²/a² - y²/b² = 1        (3)
where
a, b = intrinsic parameters

Parabola

y² - 2 l x = 0       (4)
where
l = intrinsic parameter

Degenerate Conics

Parallel Lines

x² - a² = 0        (5)
where
a = half the distance between the parallel lines

Intersecting Lines

x²/a² - y²/b² = 0        (6)
where
tan^-1(b/a) = half the angle between the intersecting lines

Coincident Lines

x² = 0        (7)