Example 7.  Consider the cardioid  ,  .  Draw the circle of curvature at   .  

Solution 7.

The radius of curvature formula    can be used provided that the curve is positively oriented.  Loosely speaking the curve must be oriented in the "counterclockwise direction".  Let's investigate the situation at hand.  

The formula for the radius of curvature computes a negative value.  The correct the situation, change the sign and use .  

Caveat.  The formula for the radius of curvature should include an absolute value.  Be careful!  In Mathematica it could be written as

        .  

(c) John H. Mathews 2004