Are there known practical algorithms or methods to calculate the bitangent lines of a quartic defined by $f(u,v,t)=0$ in terms of the 15 coefficients? Theoretically you can set up $f(u,v,-au-bv)=(k_0u^2+k_1uv+k_2v^2)^2$ and then compare the coefficients, but the system of equations seem pretty difficult to solve, especially when I just keep the coefficients as parameters rather than realizing them to be specific numbers.
Another question that is related: in THIS paper, under section three after equation (3.9), the author says "eliminating d, we obtain...", but (3.9) is a system of degree 3 and 4 equations in $d$ and $c$, which have really complicated root expressions, and I couldn't simplify it using Mathematica. The author also says "It is not practical to print these expressions here". Then how did the author know the relation is of degree 56, and it is the universal polynomial? He also seemed to know (at least some) coefficients of the relation.
