# Elementary Curves

## Ellipse, circle, hyperbola, parabola, parallel and intersecting and coincident lines

The simplest of all curves are the straight lines. The next in complexity are the second degree or conics curves.

### Non-degenerate Conics

#### Ellipse

x^{2 }/ a^{2}+ y^{2 }/ b^{2}= 1 (1)

where

a, b = intrinsic parameters

#### Circle

Circle, a special ellipse, can be expressed as

x^{2}+ y^{2 }= a^{2}(2)

where

a = radius

#### Hyperbola

x^{2 }/ a^{2}- y^{2 }/ b^{2}= 1 (3)

where

a, b = intrinsic parameters

#### Parabola

y^{2}- 2 l x = 0 (4)

where

l = intrinsic parameter

### Degenerate Conics

#### Parallel Lines

x^{2}- a^{2}= 0 (5)

where

a = half the distance between the parallel lines

#### Intersecting Lines

x^{2 }/ a^{2}- y^{2 }/ b^{2}= 0 (6)

where

tan^{-1}(b/a) = half the angle between the intersecting lines

#### Coincident Lines

x^{2}= 0 (7)