### A) Historical Curves

An extraordinary new look at the “Historical Curves.” The list of historical curves presented, for example, here has over sixty listings. With the ideas presented in this book there are entirely new equations for dozens of them and a fresh, three dimensional look at a large majority of them. These are mostly found in section 3.0.

### B) Cusps

Many ‘cusps’ that appear in two dimensions are actually transitions of the graph on or off the front view. For example, using an * base helix function * when *projected* to the front view shows the cusp at (x, y) = (0, 1).

The curve is actually a 3D helix on the positive side and an asymptotic curve *exactly on the* front real plane for negative

### C) Second Level Exponents – “Airy Zeros”

Using second level exponents on the helix and adjusting coefficients, good, differentiable approximations to Airy function zeros and Bessel J_{0} and J_{1} function zeros and derivatives are possible in elementary function form.

For example, in two dimensions the zeros for Bi(x) can be approximated by:

1 | 2 | 3 | 4 | 5 | 6 | 7 | |

Bi (b) | -1.1737 | -3.2711 | -4.8307 | -6.1699 | -7.3768 | -8.4920 | -9.5381 |

Helix Re + Im | -1.1657 | -3.2639 | -4.8227 | -6.1668 | -7.3820 | -8.5076 | -9.5657 |

### D) Helix Functions

With the 3Di coordinate system, Euler’s Formula becomes a helix and amounts to giving rise to a whole new class of elementary functions, i.e. helix functions.

One example from section 7.63 are two versions of a “Cornucopia Function.”

In 3Di:

And in front and top view projections: