Generalizations for 2D

    In two dimensions, a curve can be expressed with the parametric equations  [Graphics:Images/CurvatureMod_gr_213.gif] and  [Graphics:Images/CurvatureMod_gr_214.gif].   Similarly, the formulas for the radius of curvature and center of curvature can be derived using limits.  At the point  [Graphics:Images/CurvatureMod_gr_215.gif]  the center and radius of the circle of convergence is
    
        [Graphics:Images/CurvatureMod_gr_216.gif]  
        
        [Graphics:Images/CurvatureMod_gr_217.gif]  

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

Details

[Graphics:../Images/CurvatureMod_gr_218.gif]


[Graphics:../Images/CurvatureMod_gr_219.gif]

[Graphics:../Images/CurvatureMod_gr_220.gif]

 

 

Now let Mathematica find the limit of the collocation circles and get the osculating circle.

[Graphics:../Images/CurvatureMod_gr_221.gif]


[Graphics:../Images/CurvatureMod_gr_222.gif]

 

 

[Graphics:../Images/CurvatureMod_gr_223.gif]


[Graphics:../Images/CurvatureMod_gr_224.gif]

 

 

[Graphics:../Images/CurvatureMod_gr_225.gif]


[Graphics:../Images/CurvatureMod_gr_226.gif]

 

 

[Graphics:../Images/CurvatureMod_gr_227.gif]


[Graphics:../Images/CurvatureMod_gr_228.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004