Differential Geometry has the following important elements which form
the basic for studying the elementary differential geometry, these are
as follows:

**Length of an arc**: This is the total distance
between the two given points, made by an arc of a curve or a surface,
denoted by C (u) as shown below:

**Tangent to a curve**:
The tangent to a curve C (u) is the first partial derivative of the
curve at a fixed given point u and is denoted by C ‘(u) or its also
denotes as a ‘ (s), where the curve is represented by a (s), as shown
below:

Hence, a ‘(s) or C ‘ (u) or T are the similar notations used for denoting tangent to a curve.

**Curvature: **Curvature
is defined as any direction in any normal degree, (denoted as n) into
its side which is empty or blank. In mathematics, we can find the
curvature of any surface or curve by calculating the ratio of the rate
of change of the angle made by the tangent that is moving towards a
given arc to the rate of change of the its arc length, that is, we can
define a curvature as follows:

**C ‘’ (s) or a’’(s) = k (s) n (s),
where k (s) is the curvature, which can be understood better by looking
at the following diagram:**

We can now prove that if a’(s) * a ‘(s) = 1, then this would definitely imply that:

a’’(s) * a ‘(s) = 0.

Thus
a curvature is basically the capability of changing of a curve form a ‘
(s) to a ‘ (s + $\Delta$ s) in a given direction as shown below:

Once,
we have calculated the tangent T to a given cure, its easy to find out
the value of normal N and binormal B of a given curve, which gives us
the elements of a famous formula in differential geometry, which is
known as Frenet Frames, which is a function of F (s) = (T(s), N (s),
B(s)), where C (s) is any given curve in the space.

**For example:** **consider the following diagram of a circle, and then we can differentiate its various elements as follows:**.

- The length of the arc, s is given to be equal to r * $\theta$, this
implies, $\theta$ = s/ r, whose coordinates will be as follows:

a (s) = [r cos (s/r), r sin (s/r)],- The tangent would be calculated by taking the first partial differentiation of a (s), which would be:

a ‘ (s) or T = [- sin (s/r), cos (s/r)], and

- The curvature of the circle would by calculated by taking the second partial differentiation of a (s) as shown below:

a '' (s) or k = [- cos (s/r) / r, - sin (s/r) / r ], which will give k = - 1 / r

^{2} a (s), on further calculation, thus, mod of k would be equal to 1 / r.

**The course of Differential Geometry can be better understood by reading the below mentioned summarized notes:**The concept of Curve in Differential Geometry: Any curve can be represented by C (u) at a point u = u

_{o},
which can be further examined for its parametrization, by depicting its
length of the arc, its tangent, normal and bi normal. These derivatives
if combined together makes jointly an important formula known as The
Frenet Serret Formula, because they gives us the properties related to
the kinematic field of any curve or surface in a three dimensional
space.

As we know, that differential geometry is basically the
concept which is widely applied to find out the dimensions in any moving
images. In a one dimensional space, we find the differential geometry
of a curve, which is calculated by finding its curvature and torsion
along its curve.

**Torsion of a curve:** is the rate of change of the curve’s plane which is osculating as shown below:

We
can see from the above diagram that the whole plane is moving in a
particular direction, which is termed as Torsion and is denoted as t.

Torsion
t is positive if the plane goes in the direction of positive x axis and
it becomes negative if the plane goes in the negative direction of the x
axis. t is calculated as t = - N * B’. Note that 1 / t is known as the
radius of Torsion and is denoted by phi.

**Curvature and its Lines:** The
principal direction (PD) of any curvature is that direction which is
the resulting amount of the maximum and the minimum of a normal
curvature.

**There are basically two kinds of principal curvature that can be possessed by any given curve. These are:**

- The Gaussian Curve: This principal curvature is denoted by K, where K = K
_{1} * K_{2}. Note that K_{1} and K_{2} are the principal curvatures, where a principal curvature is defined as the maximum and the minimum of the normal curvature.

- The Mean Curve: This kind of principal curvature is denoted by H, which is the average mean of two curvatures, as H = ( K
_{1} + K_{2}) / 2

By the definition of curvature, we can conclude that:

- The principal curvature of parabolic point will be K = 0

- The principal curvature of elliptical point will be K > 0

- The principal curvature of hyperbolic point will be K < 0.

These principal curvatures are applied widely in case of the mapping of tangents.