If the coordinate functions of a parametric curve (x(t), y(t)) are
differentiable at t = t0, then (x’(t0),y’(t0))
is called the velocity vector.
At a point that is not critical,
the velocity vector will be non-zero. The “tangent line to the
parametric curve (x(t),y(t)) at (x(t0),y(x0)) is
then given by (x(t0) + x’(t0)(t-(t0)),
y(t0) + y’(t0)(t-t0) = (x(t0),y(t0))
+ (x’(t0),y’(t0))(t-t0).
The normal vector at t0, (-y '(t0), x'(t0)),
is perpendicular to the tangent line, and the normal line is then given
by by (x(t0) -y’(t0)(t-(t0)), y(t0)
+ x’(t0)(t-t0) = (x(t0),y(t0))
+ (-y’(t0),x’(t0))(t-t0).