How to select parameters for elliptic curves not found in standards (Hessian, Jacobi Intersection, Jacobi Quartic, etc)? I am currently in the process of researching different forms of elliptic curves defined over prime fields. In many curve standards, such as NIST, Brainpool, etc, there exist a list of curve equations for Weierstrass, Montgomery, and (twisted) Edwards curves.
However, I wish to also investigate various other curve forms, such as Hessian, Doche-Icart-Kohel, Jacobi intersections, Jacobi quartics, etc, but unfortunately I have been unable to find any such standards covering the selection of these curves. I'd like to know if any such papers / websites exist that detail some curve equations for each form. A list of curve equations with different parameters for these forms would be ideal.
Weierstrass models can be converted to Hessian and Jacobi models (where such models exist) and in the days before Edwards curves were considered for ECC-based on implementation and side-channel properties.
The conversion to Jacobi model seems to be covered on mathoverflow.
To convert to Hessian, compute the $j$ invariant in $\mathbb F_p$ and then solve
$$j=\frac{c^3(c^3+216)}{(c^9-81c^6+2187c^3-19683)}$$
for $c\in\mathbb F_p$ (if the corresponding degree 9 polynomial has a root in $\mathbb F_p$). Your curve is then birationally equivalent to
$$X^3+Y^3+Z^3=cXYZ.$$
The Handbook of Elliptic and Hyperelliptic Curve Cryptography section 13.1.5.b shows how to explicitly calculate the binational maps if you want to find the image of specific points on the Hessian.
Thus you could take a standard Weierstrass model (e.g. NIST P256, Curve25519) and likely convert it to an alternative form if you so wished.