DIFFERENTIAL GEOMETRY COURSE

6.*THE OSCULATING SPHERE*

If a curve * x*(t) lies on a surface F(

Also we find that for all t

We say the curve has in each point a contact of order ∞ with the surface.

*Definition 52 :* We say the surface F(* x*)=0 and the curve

*Explanation 53 :* Contact of order 0 means that the curve intersects the surface in * x*(t

*Problem 54 :* Show that definition 52 is independent of the choice of the parameter.

*Problem 55 :* Show that the plane that in a point * x*(t

*Example 56 :* We seek the sphere that has a maximal contact with * x*(s) in

Let the equation of this sphere be ||

Then g(s) = (

i) The requirement g(s_{o})=0 expresses that * x*(s

ii) The requirement g'(s

iii) The requirement g"(s

this requirement expresses that the projection of the center of the osculating sphere onto the principal normal must coincide with the center of the osculating circle,

So, if we write

iv) g

Using

*Definition 57 :* The osculating sphere in the point * x*(s

*Proposition 58 :* The curves that lie on a sphere are exactly the curves that satisfy a natural equation of the form
κ^{-2} + κ ^{.}^{2}τ^{-2}κ^{-4} = c (c constant), except that curves with constant κ need not lie on a sphere.

(Give an example of a curve with constant curvature that lies on a sphere, and two examples of curves with constant curvature that don't lie on a sphere.)

*Proof :* If a curve lies on a sphere, then in each point the osculating sphere coincides with the sphere where the curve lies on, so that
κ^{-2} + κ ^{.}^{2}τ^{-2}κ^{-4} is constant and equal to the square of the radius of that sphere.

On the other hand, from κ^{-2} + κ ^{.}^{2}τ^{-2}κ^{-4} = c we find that the centre of the osculating sphere is a fixed point
(see problem 59),
so the distance between any point on the curve and this fixed point is equal to √c.

*Problem 59 :* Prove by differentiation of the centre of the osculating sphere that this is a fixed punt if κ^{-2} +
κ ^{.}^{2}τ^{-2}κ^{-4} = c and κ is not constant.

*Problem 60 :* Determine the curvature as a function of the arc length for a planar curve with the property that the centers *M* of the segments *P**K*, where *P* is on the
curve and *K* is the curvature centre at *P* all lie on one straight line.

(Hint: calculate using R as shorthand for κ^{-1}; it turns out the curve is a cycloid.)

*Problem 61 :* Let φ be a differentiable function of t. Prove that the curve * x*(t) = (t cos(φ(t)), t sin(φ(t)), √(1-t

Which differential equation must hold for φ if in each point of the curve the angle between the tangent and the x

*Problem 62 :* Prove that the following assertions hold:

a) If the osculating planes of a curve all go through a fixed point, then the curve is planar.

b) If the osculating planes of a curve all are parallel to a fixed straight line, then the curve is planar.

*Problem 63 :* Prove that a curve whose curvature centres all lie on a fixed straight line, must be a circle.

*Problem 64 :* In each point of a curve *C* there is a normal that goes through a fixed point *A*. Prove that *C* lies on a sphere whose centre is *A*.

*Problem 65 :* The straight lines connecting a fixed point with a point *P* running through a given curve are all perpendicular to this curve whilst the angles between these straight lines
and the principal normal in *P* are all equal. Prove this curve must be a circle.

*Problem 66 :* Let *V*(s) be the rectifying plane in the point *P*(s) of a curve whose parameter s is arc length.

Let the curvature κ(s) in *P*(s) be positive. The distance between a fixed point *M* and *P*(s) is r(s). The distance between *M* and *V*(s) is a(s).

Prove that 2 a κ = |(d^{2}/ds^{2})(r^{2}-s^{2})|.