Differential Geometry/Tangent Line, Unit Tangent Vector, and Normal Plane

The arc length can be used as a derivative for the vector function f, which is denoted t(x):

$$t(x)=\frac{df(x)}{ds}$$.

Since

$$\frac{df(x)}{ds} = \frac{f'(x)}{s'(x)}= \frac{f'(x)}{|f'(x)|}$$

which confirms the fact that it is a unit vector since the dot product with itself is 1. This also verifies a useful formula for the unit tangent vector

$$t(x)= \frac{f'(x)}{|f'(x)|}$$.

The tangent line goes through f(x) and is spanned by the vector t(x). Thus, it is equal to the line spanned by

$$f(x)+a\frac{f'(x)}{|f'(x)|}$$

where a is any real number.

The normal plane at the point f(x) is the plane that is normal to the tangent line, and thus the unit tangent vector. Therefore, its equations is given by

$$|z-f(x)|\cdot t(x)=0$$

where z is any element of the surface, since it must be orthogonal.