XC. Curves

1.Many of these can be described by ‘degree’:

a. Straight line

b. c. Circle

e. Ellipse [Note: the word ‘oval’ is often used in common parlance as similar to an ellipse. It is rarely used in mathematics. We might want to use ‘oval’ for any ‘egg-shaped’ curve: it would thus have a broader usage than ellipse, for which there is a precise mathematical definition.]

h. hyperbola

p. parabola

Various curves of degrees 3, 4, 5, and 6 have been identified and named, often for the discoverer. These can be designated c,d,e, and f. Cutter numbers could be used for these.

g. Curves of variable degree. Again Cutter numbers could be used.

2. Curves of genus greater than 0. [The above curves have genus 0. The word ‘genus’ is not easy to define, but is defined precisely within analytical geometry.] Again Cutter numbers can be employed.

a. Curves of genus 1

b. Curves of genus greater than 1.

c. Curves of variable genus.

3. Transcendental curves

s. Spiral [deserves special treatment]

4. Fractals

5. Piece-wise Constructions

6. Three dimensional curves

h. Helix [deserves special treatment] [Note that this and others are 3-dimensional spirals]

Curves generated by other curves can be captured synthetically (though Cutter numbers may yet again be necessary).

Note that many curves are identified within particular fields, such as the ‘supply curve’ in Economics. Such curves should be classified as curves associated with that field (as precisely as possible), with recourse to Cutter numbers as necessary.