The rollercoaster has to fit the following criteria:
- Starts and finishes at ground level (does not go underground)
- Maximum height is 100m and maximum horizontal displacement is 1000m
- The cross-section profile must include 2-3 different functions (preferably beginning with a cubic, then parabola, then straight line).
- At least one linear equation and a polynomial of degree 2-3.
then label the parts that are differentiable.
Example design:
You are asking for piecewise curves pasted together. If you are going to use $f_i(x)$ and $f_{i+1}(x)$ adjacently and paste them at $x=x^*$ then choose $x^*$ such that $f_i(x^*) = f_{i+1}(x^*)$ (naturally for pasting) and $\frac{\partial^k f_i(x^*)}{\partial^k x} = \frac{\partial^k f_{i+1}(x^*)}{\partial^k x}$ for $k = [1,K]$ with $K$ as large as possible without making $f_i= f_{i+1}$. for example if you are pasting linear and quadratic together i.e., $f_i(x) = ax$ and $f_{i+1}(x) = bx^2$ then $K = 1$ inother words $K = \max(\deg(f_i),\deg(f_{i+1})) - 1$. If these equalities are satisfied, just come up with $f_i's$ which fit your area and keep pasting them.
This is a theoretical roller coaster where criteria is to make the transition between curves as smooth as possible. Include your practical constraints and design them with safety measures.