4.2: Distance Functions to Space Curves

In the Curvature Quest in lab two , we defined a distance function from a plane curve to a given point and we saw how the critical behavior of such functions was related to the evolute curve of the original curve. For space curves, we may define a similar distance function for each point, and once again we may relate the critical point behavior of such functions to the geometry of the curve. For each point C , let

f_C(t)=(X(t) -C)²

In the Distance Function window a red line parallel to the t -axis is drawn at height r² .

Choose the point C by clicking with your middle mouse button in the Curve window. When you choose a point in 3d with the mouse, the point you choose is in the plane of the window. Rotating the graph will help you see where you are placing your point.

This demo shows a space curve along with a sphere of radius r centered at the point C . It also shows the graph of the distance function from the point to the curve.

For various C, how many local maxima and how many local minima will the function f_C(t) have? For which points C and curves X(t) does the function have a horizontal inflection? For which points C does the function have a value at which the first three derivatives of the function f_C(t) equal zero?

How does the circle of radius r with center C intersect the curve, and how is this related to the intersection of the graph of f_C with the line whose height is r²?