Tangent line

to a curve $ \newcommand{\vect}[1]{\mathbf{#1}} $

Figure 1. The tangent $MT$ (in red) to the line $L$ at the point $M$ (pdf)

A straight line representing the limiting position of the secants. Let $M$ be a point on a curve $L$ (Fig. 1). A second point $M_1$ is chosen on $L$ and the straight line $M_1$ is drawn. The point $M$ is regarded as fixed, and $M_1$ approaches $M$ along the curve $L$. If, as $M_1$ goes to $M$, the line $MM_1$ tends to a limiting line $MT$, then $MT$ is called the tangent to $L$ at $M$.

Figure 2. A point $M$ on a curve $L$ without a tangent(pdf)

Not every continuous curve has a tangent, since $MM_1$ need not tend to a limiting position at all, or it may tend to two distinct limiting positions as $M_1$ tends to $M$ from different sides of $L$ (Fig. 2). If a curve in the plane with rectangular coordinates is defined by the equation $y=f(x)$ and $f$ is differentiable at the point $x_0$, then the slope of the tangent at $M$ is equal to the value of the derivative $f^\prime(x_0)$ at $x_0$; the equation of the tangent at this point has the form $$ y - f(x_0) = f^\prime(x_0)(x - x_0). $$ The equation of the tangent to a curve $\vect{r} = \vect{r}(t)$ in space is $$ \vect{t}(\lambda) = \vect{r} + \lambda \frac{\mathrm{d}\vect{r}}{\mathrm{d}t}, \quad \lambda \in \R. $$

By a tangent to a surface $S$ at a point $M$ one means a straight line passing through $M$ and lying in the tangent plane to $S$ at $M$.

References

[BeGo]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces", Springer (1988) (Translated from French)
[Co]	H.S.M. Coxeter, "Introduction to geometry", Wiley (1961)
[Gu]	H.W. Guggenheimer, "Differential geometry", McGraw-Hill (1963)
[HiCo]	D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination", Chelsea (1952) (Translated from German)