Tangent lines to parametrized curves
Recall that for a curve parametrized by c(t), the derivative c'(t) is tangent to the curve.
Fixed a time t0. We want to find the tangent line to the curve that goes through the point c(t0). The direction of the line is given by the tangent to the curve ath that point, which is c(t0).
Recall that we can write down a parametrization of a line through a point a and parallel to the vector v by l(t) = a + tv. In this case, the point is a = c(t0) and the direction vector is v = c'(t0). Therefore, a parametrization of the tangent line could be l(t) = c(t0) + tc'(t0).
However, we typically want the line given by l(t) to pass through c(t0) when t = t0. So we usually change the parametrization slightly to
| l(t) = c(t0) + (t - t0)c'(t0). |
You can see some examples here.
