In this project we will look at an interesting construction involving the curvature of a plane curve. The goal of this project is for students to investigate the idea of curvature.
1. In the window below we have graphed the function f(x) =
sin(x) (in black). Also, we have found the derivative of
f(x) (in red). Also, point A is attached to the x-axis,
point B tracks
point A on the graph of f(x), and point C tracks point A on the graph
of f'(x). We have constructed the tangent line a to
f(x) and to the derivative (line b) and measured the slopes of these
(slope(a) and slope(b))
Point G is created by dilating point B in a perpendicular direction from the tangent a by an amount equal to the measurement of 1/slope(b), that is the reciprocal of the value of f''(x). The circle shown is then constructed from points B and G.
2. Move point A back and forth on the x-axis and notice what happens to the circle. It appears to be curved just as the function is curved. That is, the circle is "flat" where the function flattens out and is highly curved where the function is curving rapidly. The circle appears to just touch or "kiss" the function. What is actually happening here is we have constructed a circle that has the same curvature as the function. This circle is called the osculating circle to the function at the given point.