Curves

 

It Takes Two

It takes two, baby
It takes two, baby
Make a dream come true
It just takes two

Not Four,  Two.

It is a standard concept that there are four fundamental curves:

1.      a circle,

2.      an ellipse,

3.      a parabola, and

4.      a hyperbola. 

I would suggest that there are really only two curves: an ellipse and a hyperbola. A circle is a special, perfect, case of an ellipse, and a parabola is a special case between a hyperbola and an ellipse.

A curve is defined by its major axis, a, and its minor axis, b.  An ellipse is a curve where the major and minor axes are both less than infinity.  There is no other restriction on a or b.  A circle is the special case of an ellipse where a = b, which is why it is said to be perfect.

A hyperbola is defined by having both axes, a and b equal to infinity.  A parabola is that case where also a ≠b.  But this is between an ellipse and an hyperbola.  Saying that at least one of the axes, a or b, is less than infinity makes it a partial  ellipse.  Saying that at least one of the axes is equal to infinity makes it a partial hyperbola.  A parabola is a mathematical curve where one of the axes is very, very large but NOT infinity.  It looks like it is different than a hyperbola and an ellipse, but actually it is partially both at the same time.