Be aware that curves can describe the motion of a fluid, gas, light rays, lines current. The radius of curvature for a plane curve at a point is the radius of the tangent circle at that point. In some cases the curve is given by equations, and the radius of curvature is calculated according to the formulas. Accordingly, in order to know the radius of curvature, it is necessary to know the radius of the circle relating to a certain point.
Define a plane curve point And, near it, take another point B. Construct a tangent to an existing curve that pass through points A and B.
Swipe through point A and the line perpendicular to the constructed tangent, extend them to the intersection. Label the point of intersection as O. the Point O is the center of the tangent circle at a given point. So OA is the radius of the circle, i.e. the curvature in a given point A.
Note that when a point moves along any curved path at any point in the motion it moves in some circles, which varies from point to point.
If the point in space to define curvature in two mutually perpendicular directions, these curvature will be referred to the principal. The direction of the main curvatures must be 900. For the calculations often use the average curvature is equal to the sum of the main curvatures, and Gaussian curvature, equal to their work. There is also a notion of curvature of a curve. It is the reciprocal of the radius of curvature.
Acceleration is an important factor in the motion of a point. The curvature of the trajectory directly affects the acceleration. Acceleration occurs when a point with a constant speed begins to move along the curve. Changes not only the absolute value of the velocity, but its direction, there is a centripetal acceleration. I.e. in reality, the point begins to move along the circumference, which at this point in time.