Calculus with Parametric Curves

Parametric equations are equations which depend on a single parameter. You can rewrite $y=x$ such that $x=t$ and $y=t$ where $t$ is the parameter.

A common example occurs in physics, where it is necessary to follow the trajectory of a moving object. The position of the object is given by $x$ and $y$, signifying horizontal and vertical displacement, respectively. As time goes on the object flies through its path and $x$ and $y$ change. Therefore, we can say that both $x$ and $y$ depend on a parameter $t$, which is time.

Trajectories

A trajectory is a useful place to use parametric equations because it relates the horizontal and vertical distance to the time.

This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves term-wise. Thus, one can describe the velocity of a particle following such a parametrized path as:

$v(t)=r'(t) \\ \, \quad =(x'(t),y'(t),z'(t)) \\ \,\quad =(-a\sin(t),a \cos(t),b)$

where $v$ is the velocity, $r$ is the distance, and $x$, $y$, and $z$ are the coordinates. The apostrophe represents the derivative with respect to the parameter.

The acceleration can be written as follows with the double apostrophe signifying the second derivative:

$a(t)=r''(t) \\ \, \quad =(x''(t),y''(t),z''(t)) \\ \, \quad =(-a\cos(t),-a \sin(t),b)$

Writing these equations in parametric form gives a common parameter for both equations to depend on. This makes integration and differentiation easier to carry out as they rely on the same variable. Writing $x$ and $y$ explicitly in terms of $t$ enables one to differentiate and integrate with respect to $t$. The horizontal velocity is the time rate of change of the $x$ value, and the vertical velocity is the time rate of change of the $y$ value. Writing in parametric form makes this easier to do.