A parametrized curve describes a path by giving its position as a function of a parameter:

$$\mathbf r(t)=(x(t),y(t),z(t)).$$

As $t$ varies through an interval, the point $\mathbf r(t)$ traces the curve.

Example: a circle

The parametrization

$$\mathbf r(t)=(\cos t,\sin t),\qquad0\le t\le2\pi,$$

traces the unit circle once counterclockwise.

The same geometric curve can have many parametrizations. Changing the parameter may change how quickly the curve is traced or even reverse its orientation without changing the set of points on the curve.

Tangent vector

Differentiating component by component gives

$$\mathbf r'(t)=(x'(t),y'(t),z'(t)).$$

When $\mathbf r'(t)\ne\mathbf0$, this vector is tangent to the curve and points in the direction of increasing parameter.

For the unit circle,

$$\mathbf r'(t)=(-\sin t,\cos t).$$

Speed along the curve

The magnitude

$$|\mathbf r'(t)|$$

measures how quickly the parametrization moves along the curve per unit change in $t$. Two parametrizations can trace the same curve with different speeds.

Parametrization therefore separates the geometric path from the particular rule used to travel along it. This distinction becomes important when integrating quantities along curves.