4.6: Osculating Spheres of Space Curves

In 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,
    fC(s)=(X(s) -C)·(X(s) -C)
from the point to the curve will be constant for all t, implying that all of its derivatives will be zero. (In the following calculations, it will be helpful to assume that the curve is parametrized by arclength.) For a more general curve, however, the more derivatives of fC(s) we can make zero for a given value of the parameter, t0, by changing C, the closer our approximation will get at X(t0), i.e. the order of contact will be higher. However, as you will see if you follow the link, after three derivatives, the center of the sphere (and therefore the sphere) is completely determined. This is a link to a somewhat involved calculation of the location of the center of the osculating sphere to a space curve at a point. While some of the vector algebra is a bit weighty, this is representative of a type of computation common in differential geometry.

The condition on the first derivative at s0 is

    0=f'C(s0)=2(X(s0)-C)·X '(s0)
which means that (X(s 0)-C) is perpendicular to T(s0) and therefore
    (X( s0)-C)=uP( s0)+vB(s0)
for some numbers u and v .

If the second derivative at s0 is also zero, then

    0=f''C(s0)
    f''C(s) =2(X(s )-C) ·X''( s)+2X' (s)·X '(s)
    =2(X(s 0)-C) ·k(s0)P(s0) +2
    =2(uP( s0)+vB(s0)) ·k(s0)P(s0) +2
    =2(uk(s0) +1)
In order for both the first and second derivatives to be zero at s0 we must have
    u=-1k(s0).

The third derivative is

    f'''C( s)=2X' (s)·k( s)P( s)
    +2(X( s)-C) ·k'(s)P(s)
    +2(X( s)-C) ·k(s)-k( s)T( s)+t(s)B(s)
    =2(X( s)-C) ·(-k(s) 2T(s) +k'(s)P(s) +k(s)t( s)B( s)).
If this is also zero at s0 , we have
    0=f'''C( s0)=2(-1k(s0)P(s0) +vB(s0))·
    (-k(s0) 2T(s0) +k'(s0)P(s0) +k(s0)t(s0)B(s0))
    =2(-k'( s0)k( s0)+vk( s0)t(s 0))=0
Thus
    v=-(1t( s0)( 1k(s0))')
and we end up with a specific description of the unique point for which the distance function has three derivatives equal to zero at s0 , namely
    C=X(s0)+1k(s0)P(s0) +1t(s0)(1k( s0)) 'B(s 0)
This C will be the center of the osculating sphere at the point X(s0) . In the case of a plane curve, the torsion is zero, so this reduces to the equation for the center of the osculating circle.

Need to put an APPLET here! Osculating Sphere Demo Need to put an IMG here!

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?

Top of Page