Osculating Circles in the Plane  

This sheet creates animations showing plots of planar curves along with their osculating circles and normal lines: Osculating circles (.mw).
 

Initialization   initialization code (.html) 

Example 1: The osculating circles for sine: and  

>

# Consider the sine wave curve_eq := y=sin(x): curve_eq; # We also need a parametrized version of the sine wave # Please note: Maple uses e_x and e_y to denote the i and j unit vectors r := t -> <t,sin(t),0>: 'r(t)' = r(t); # We will plot the osculating circles and normal lines for the sine wave # between x=-8 and 8. "5000" is the number of points used by implicitplot # to make the graphs look smooth. "true" tells our procedure to show the # normal lines. "-10,10,-10,10" are x and y ranges for the plot. osculatingCircles(curve_eq,r,-8,8,5000,100,true,-10,10,-10,10); # Click on the plot below and use the buttons above to play the animation.

 

Circles for (and without normal lines). 

>	osculatingCircles(curve_eq,r,0.5,2.5,5000,50,false,-5,10,-10,5);

 

Example 2: The osculating circles for the parabola and . 

>	curve_eq := y=x^2: curve_eq; r := t -> <t,t^2,0>: 'r(t)' = r(t); osculatingCircles(curve_eq,r,-3,3,5000,100,true,-10,10,-10,20);

 

Example 3: The osculating circles for a bell-like curve: and . 

>	curve_eq := y=exp(-x^2): curve_eq; r := t -> <t,exp(-t^2),0>: 'r(t)' = r(t); osculatingCircles(curve_eq,r,-4,4,5000,100,true,-5,5,-5,5);