The Circle of Curvature

A new construction of the Circle of Curvature

        What determines a circle?  A center and a radius.  The formula for the radius of curvature is well established.  What idea could we use to help understand the situation.  We could use the fact that three points determine a circle and see where this leads.  

Derivation of the Radius of Curvature

    The standard derivation of the formula for radius curvature involves the rate of change of the unit tangent vector.  This new derivation starts with the collocation the collocation circle to go through the three points  ,  ,  and    on the curve  .  The limit as is the osculating circle to the curve   at the point .  The radius of curvature and formulas for the location of its center are simultaneously derived.  The computer algebra software Mathematica is used to assist in finding the limits.   

    Start with the equation ,  of a circle.  Then write down three equations that force the collocation circle to go through the three points  ,  ,  and    on the curve  .  Enter the equations into Mathematica

Solve the equations for and extract the formula for the radius of the collocation circle.  Since it depends on we will store it as the function .    

The formula looks bewildering and one may wonder if it is of any value.  
However, we can demonstrate the power of Mathematica and see if it can take the limit.  

The computer gets the correct formula, but leaves out all the human understanding.  The formula for   can be rewritten so that human insight and inspiration is involved, but the computer cannot think of the formulas we desire to see.  So human intervention must be called upon and the simplifications must be typed in by hand.   

Notice that when we take the limit as , the limiting value for each of the radicals in the numerator is  .  Mathematica is capable of finding them, we illustrate this with the third one.

Therefore, the limit in the numerator is .  The difference quotient in the denominator is recognized as the numerical approximation formula for the second derivative, hence the is  .  

We already knew that Mathematica knows the rules for taking limits of functions when the formula is given.  Now we know that it has the "artificial intelligence" to rearrange quantities involving an arbitrary function and can identify difference quotients for approximating derivatives and find their appropriate limits.  It takes quite a bit of trust to let Mathematica do our thinking.  From the steps we filled in, we can gain trust for the computer's way of doing it.  

The Osculating Circle

We now show that the limit of the collocation circle as is the osculating circle.
Now we want to find the center  (a, b)  of the osculating circle.

The abscissa for the center of the collocation circle is

Take the limit as  ,  to obtain the abscissa for the center of the circle of curvature.  

If we want to see "what's happening" in this limit, then we must rearrange the formula for  .  A little finesse permits us to write it as follows

The numerators involve three difference quotients, all of which tend to when   , and the difference quotient in the denominators tends to when  .  

Thus we have established the formula   for the abscissa of center of the circle of curvature.

The Abscissa the Easy Way

Subtract from   the radius of curvature times .

The Ordinate for the Center of the Circle of Curvature

The ordinate for the center of the collocation circle is

If we want to see "what's happening" in this limit, then we must rearrange the formula for  .  A little finesse permits us to write it as follows

The limit of   as   is merely ,  and we have already observed that the limit in the denominator is and both quotients in the numerator tend to   as   .  

Thus we have established the formula   for the ordinate of center of the circle of curvature.
 

The Ordinate of the Circle the Easy Way

Add to    the radius of curvature times

The Osculating Circle

At the point   on the curve  , the center and radius of the osculating circle are given by the limits calculated in the preceding discussion.

Generalizations for 2D

    In two dimensions, a curve can be expressed with the parametric equations   and  .   Similarly, the formulas for the radius of curvature and center of curvature can be derived using limits.  At the point    the center and radius of the circle of convergence is
    
          
        
          

Remark.  The absolute value is necessary, otherwise the formula would only work for a curve that is positively oriented.