**Table of content**

**Introduction to Radius of Curvature**

The curvature is something that relates to the geometry Also, it is a scalar quantity. That one may also define as curvature vector. Besides, in this topic, we will discuss what is curvature and radius of curvature?

**Curvature**

According to mathematics curvature is any of the number loosely related concepts in different areas of geometry. Naturally, it is the amount by which geometric surfaces derivate themselves from being flat plane and also from a curve being straight like a line. However, it is defined differently for a different context.

Furthermore, the curvature is a scalar quantity that one can also define as a curvature vector that takes into explanation the direction of the curve and its magnitude. Moreover, the curvature of the complex object is distinct by complex objects from linear algebra.

**Radius of Curvature**

The radius of curvature ‘R’ in differential geometry is the reciprocal of the curvature. Besides, the radius of the circular arc is the best approximate the curve at that point. Also, for surfaces, the radius of the curvature is the radius of the circle that fits best in a normal section or combination.

**Formula of the Radius of Curvature**

Normally the formula of curvature is as:

R = 1 / K’

Here K is the curvature. Also, at a given point R is the radius of the osculating circle (An imaginary circle that we draw to know the radius of curvature). Besides, we can sometimes use symbol ρ (rho) in place of R for the denotation of a radius of curvature.

**Application of Radius of Curvature**

In differential geometry, it is used in Cesàro equation which tells that a plain curve is an equation that relates the curvature (K) at a point of the curve to the arc length (s) from the start of the curve to a given point. Also, it is an equation relating to the radius of curvature (R) to the arc length.

Also, it can help to find the radius of curvature of the earth along a course at an azimuth.

Besides, the radius of curvature also uses three parts equation for bending of beams.

Moreover, it has a specific meaning and a sign convention in optical design. Also, spherical lenses have a center of curvature.

**Difference Between Radius and Radius of Curvature**

Radius refers to the distance between the center of a circle or any other point on the circumference of the circle and surface of the sphere. While on the other hand, the radius of curvature is the radius of the circle that touches the curve at a given point. Also, it has the same tangent and curvature at that point.

Moreover, the radius is of a real figure or shape whereas the radius of curvature is an imaginary circle.

**Types of Curvature**

In general, there are two types of curvature namely extrinsic curvature and intrinsic curvature. In this topic, we will discuss these two types.

**1. Extrinsic Curvature**

It a curvature that is a submanifold of a manifold that depends on its particular inserting. Besides, its examples include the torsion of curves in three-space and the curvature. Also, it includes the mean curvature of surfaces in three-space.

**2. Intrinsic Curvature**

It is also a curvature such as Gaussian curvature that is detectable to 2-D (two dimensional) the “populations” of a surface and not just outside observers. Also, they cannot study 3-D (three dimensional).

**Solved Question for You**

**Question. What is the curvature of the earth?**

- 7.98 inch
- 16.68 inch
- 42.6 inch
- 0.21 inch

**Answer.** The correct answer is option A because the curvature of the earth is 7.98 inch.