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 .

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:

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:

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.