Cuspidal cubic parametrisation

$ y^2 = x^3 $

$ [x,y] = [a^2, a^3] $

line: $ y = ax $

Nodal cubic parametrisation

$ y^2 = x^2(x+1) $

$ [x,y] = [a^2-1, a(a^2-1)] $

line: $ y = ax $

Bi-nodal quintic parametrisation

$ y^2 = (x+2)(x+1)^2(x-1)^2 $

$ [x,y] = [a^2-2, a(a^2-1)(a^2-3)] $

parabola: $ y = a(x^2 - 1) $

Cusp-node quintic parametrisation

$ y^2 = (x+1)^3 x^2 $

$ [x,y] = [a^2-1, a^3(a^2-1)] $

parabola: $ y = a(x^2 + x) $

Tacnode quintic parametrisation

$ y^2 = x^4(x+1) $

$ [x,y] = [a^2-1, a(a^2-1)^2] $

parabola: $ y = ax^2 $

Twisted bi-nodal quintic parametrisation

$ (y-x^2)^2 = (x+1.4)(x+1)^2(x-1)^2 $

$ [x,y] = [(a-1)^2-1.4, a((a-1)^2-1.4)^2 + (1-a)] $

parabola: $ y = ax^2 + (1-a) $

Tri-nodal septic parametrisation

$ y^2 = x^2(x+2)(x+1)^2(x-1)^2 $

$ [x,y] = [a^2-2, a(a^2-1)(a^2-2)(a^2-3)] $

cubic: $ y = a(x^3 - x) $

Cusp-bi-nodal septic parametrisation

$ y^2 = (x+1)^3 x^2 (x-1)^2 $

$ [x,y] = [a^2-1, a^3(a^2-1)(a^2-2)] $

cubic: $ y = a(x^3 - x) $

$ a = $ 0