Space Curves

A space curve describes the path followed by a particle moving through three-dimensional space. Unlike plane curves, which are confined to two coordinates, space curves require three coordinate functions. If t is a parameter, the position of the particle is represented by the vector function

\begin{equation*}\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle,\end{equation*}

where x(t), y(t), and z(t) are differentiable functions of t. As t varies over an interval, the endpoints of the position vectors trace the curve.

A standard example of a space curve is the circular helix, defined by\begin{equation*}\mathbf{r}(t)=\langle \cos (t),\sin (t),t\rangle.\end{equation*}At t = 0, the position vector is\begin{equation*}\mathbf{r}(0)=\langle 1,0,0\rangle,\end{equation*}So the curve begins at the point (1, 0, 0). As t increases, the x- and y-coordinates are determined by cos(t) and sin(t). These functions satisfy the identity\begin{equation*}x^2+y^2=\cos^2 (t)+\sin^2 (t)=1,\end{equation*}showing that the projection of the curve onto the xy-plane is a circle of radius 1.

While the x- and y-coordinates produce circular motion, the z-coordinate increases linearly as z = t. This steady vertical rise causes the curve to spiral upward rather than remain in a plane. The resulting path wraps around the cylinder, forming a smooth three-dimensional helix.

Space curves are useful for modeling physical motion, such as the flight path of an aircraft, because they describe position in all three spatial directions simultaneously.