Parametric Curves and Calculus with Parametric Equations

The equation $y = f(x)$ tells you the shape of a curve but not how a particle travels along it. Parametric equations $x = f(t)$, $y = g(t)$ restore the missing information: for each moment $t$, the particle sits at $(f(t), g(t))$. In finance, price trajectories are naturally parametric — the "parameter" is time and the "curve" is the path of the asset price (or option value) in $(S, V)$-space. The payoff to mastering parametric calculus is access to tangent lines (risk sensitivities), arc length (realised path variation), and speed (the rate at which the state vector moves) — all computable from the first derivatives $\dot{x} = dx/dt$ and $\dot{y} = dy/dt$ via the chain rule.

• parametrise standard curves and eliminate the parameter to identify the Cartesian form;
• compute $dy/dx$ and $d^2y/dx^2$ from parametric equations;
• find tangent lines (horizontal and vertical) from the signs of $\dot{x}$ and $\dot{y}$;
• compute the arc length and surface area of a parametric curve.

1. Parametric curves
2. Tangent lines to parametric curves
3. The cycloid
4. Arc length of a parametric curve