Parametric curve

A parametric curve separates two things that the graph of a function mixes together: the set of points (the trace) and the way it is traversed. With \(\mathbf r(t)=\langle\cos t,\sin t\rangle\) and with \(\mathbf r(t)=\langle\cos2t,\sin2t\rangle\) the trace is the same circle, but the second one traverses it at double speed. That is why a curve in the plane or in space is described by a vector function rather than by an equation \(y=f(x)\), which cannot represent full circles or self-crossing paths.

To go from parametric equations to a Cartesian equation one eliminates the parameter: from \(x=t+1,\ y=2t\) we solve \(t=x-1\) and get \(y=2x-2\); from \(x=3\cos t,\ y=2\sin t\) we use \(\cos^2t+\sin^2t=1\) and get an ellipse. Elimination can lose information, since it does not tell the direction of travel or which part of the graph is covered: \(\langle t^2,t^4\rangle\) satisfies \(y=x^2\) but traces only the half with \(x\ge0\).

Calculus is done component by component: \(\mathbf r'(t)=\langle x'(t),y'(t),z'(t)\rangle\) is the velocity vector, tangent to the curve and pointing in the direction in which \(t\) increases. The tangent line at \(t_0\) passes through \(\mathbf r(t_0)\) with direction \(\mathbf r'(t_0)\), as long as that derivative is not the zero vector. A segment from \(P\) to \(Q\) is parametrized as \(\mathbf r(t)=P+t(Q-P)\) with \(0\le t\le1\), and the helix \(\langle\cos t,\sin t,t\rangle\) is the classic example of a curve that does not fit in a plane.

A common mistake is to identify the curve with its Cartesian equation and ignore the parameter interval: the same pair of equations with a different interval gives different curves. With the velocity \(\mathbf r'(t)\) the distance traveled can be measured right away, the topic of the next concept, arc length.