Sunday 17 March 2019

Curvature

Curvature is the rate of change of direction of a curve with respect to distance along the curve. It express curve state of being curved. At any point on a curve, the curvature is the reciprocal of the radius, for other curve the curvature is the reciprocal of radius of the circle that mostly closed conforms to the curve.

Radius of curvature

Definition of curvature


Let P and Q be any neighbouring point on a curve AB such that arc AP=s and arc AQ=  so that arc PQ=. Let the tangent to the curve at P and Q makes angle  and  with x-axis so that . Then

Curve

   
     (1).  , measured in radius, is called the total curvature or total bending of arc PQ,

     (2).  The ratio   is called the average curvature of the arc PQ,

     (3).  , if it exist, if it exist, is called the curvature of the curve at P and is denoted by .

     (4). The reciprocal of curvature at any point P is called the radius of curvature and is denoted by Greek letter  
             

Circle, center and chord of curvature


Curvature of a circle


     The center of curvature of a curve at a point P is the point C which lies on the position direction of the normal at P and which is at a distance   from it.

     The circle with center C and radius CP=  is called circle of curvature of the curve at P.

     Any chord of the circle of curvature at P passing through P is called chord of curvature through P.

Radius of curvature



We know that the curvature of the curve at any point depends only upon its shape and so ut is independent of the coordinate system. Therefore, interchanging x and y, we get,


Center of curvature


The center of curvature for the point (x,y) is   .
The locus of the centre of curvature of a curve is called its evolute and the curve itself us called involute.

Example 

Q1. Find the radius of curvature of the parabola   at the point (x,y).
Sol.
      The equation of the parabola  is  
Differentiating both side w.r.t x, we get,
             
    

Now  

         

    = 

    

Q2. Prove that the curvature of a straight line is zero.
Sol.
   Let the equation of the straight line be

             

Differentiating both side w.r.t x, we get,

               

              

           
         
          = 0 
  Hence the result. 


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