4.6: Osculating Spheres of Space CurvesIn our discussion of the curvature of a plane curve, we considered the osculating circle at a point, which was defined to be the circle that best approximates the curve at the point. We found several equivalent descriptions of this osculating circle and we examined the relationship between properties of the osculating circles and properties of the original curve. For space curves, we have also defined the osculating circle at each point, and we can also introduce a new notion, the osculating sphere at a point. If the curve lies on a sphere, then this sphere will be the osculating sphere at all points of the curve. In this case there is a point such that the distance function from the point to the points of the curve is constant, so all derivatives of the function are zero at the point. Just as the first two derivatives of the distance function from a point are zero at a point of the curve if and only if the given point is the center of the osculating circle at the point of the curve, we may define the osculating sphere at a point by finding the center such that the first three derivatives of the distance function to that center are zero at the point of the curve. If the curve, X(t), lies on the sphere centered at C, the squared distance,The condition on the first derivative at s0 is If the second derivative at s0 is also zero, then The third derivative is This demo will allow you to input a curve in the usual fashion in the control panel. In Osculating Spheres there is also a checkbox that toggles the osculating sphere and a tapedeck that determines the point on the curve at which the sphere is to be found. The tapedeck scales between the minimum and maximum of the domain it is given. Hence, if you change the domain, the osculating sphere might shift also. This demo draws the osculating sphere at a point on a space curve.
In the case of a plane curve, if the curvature is monotonic over a
portion
of the domain, then the osculating circles are "nested" so that no two
distinct osculating circles intersect. Is the analogous property true
for
osculating spheres of space curves?
What happens to the osculating
sphere
at a point where the torsion is zero? Where the torsion has a critical
point? When will the osculating sphere at a point of a space curve be a
plane?
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