Curvature

Definition (Curvature) Suppose the smooth curve is the graph of the vector function , parametrized in terms of the arc length Then the curvature of is the function where is the unit tangent vector.

Example (Curvature) Find the curvature of a circle.

    Solution. A circle can be parametrized by where is the radius. We have



Solving for we find the component functions to be Thus,



and



where


Thus the curvature of a circle is

Example (Unit Tangent and Curvature) Let be the curve given by



Find the unit tangent vector to and the curvature

    Solution. We have,
    


and









thus,



Then,



because we have



therefore,

Proposition (Cross Product Formula for Curvature) Suppose the smooth curve is the graph of the vector function Then the curvature is given by



    Proof. Given two vector functions and with the same smooth graph and where is a parametrization vector function in terms of the arc length of , then according to the rule


we have

Therefore,















Since



and



we have,






Therefore,



Finally,

Example (Cross Product Formula for Curvature) Given the curve defined by find a unit tangent vector at the point on the curve where , the curvature at , and find the length of the curve from to .

    Solution. We have,
    
,

,

and




Therefore,   
  
  
  
  and  
  
   
   

Proposition (Curvature of Planar Curve) The graph of the function has curvature



where and all exist.

    Proof. Given the vector function and defined by we have Using the formula , we have

Example (Curvature of Planar Curve) Find the maximum curvature on the curve

    Solution. We have and



So,



Applying the first derivative test we have the maximum curvature at

Example (Curvature of One Variable Functions) Find the maximum curvature on the curve

    Solution. We have and



So,



Applying the first derivative test we have the maximum curvature at with curvature of   

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