Differential geometry notes 5
by Emin Orhan
Definition 19.1: Let a curve be represented by and be represented by where and are the arc length parameters of the respective curves. is said to have contact of order with at a point if at point :
and ( and )
and if also derivatives of order at exist and .
Definition 19.2: A curve has contact of order with a surface at a point if there exists at least one curve on which has a contact of order with at and there does not exist a curve on which has a contact of order greater than with at .
In this definition, a surface can be informally thought of as a point set satisfying .
Theorem 19.1: At any of its points, a curve has a contact of second order with its corresponding osculating plane.
Lemma 19.2: Let be a curve with the allowable representation of class with arc length parameter . Let have a point in common with a surface which has a representation of class . Then, has a contact of order with at point iff the function:
and its derivatives with respect to up to and including the -th derivative vanish at :
, (), .
Theorem 19.3: Let be a curve of class which has contact of order with a surface of class at point . If is even, pierces at . If is odd, then in a sufficiently small neighborhood of , lies on one side of .
Contact between a sphere and a curve: A sphere of radius and center can be represented as:
(1) For a curve to have a first-order contact with at a point :
Thus the center lies in the normal plane and can be written as
(because the normal plane is spanned by and ).
Here and are real numbers. Thus, there is a two-parameter family of spheres satisfying the first-order contact condition.
(2) For to have a second-order contact with at , we must in addition have:
In this case, where is the radius of curvature and is an arbitrary real number. Thus, there is a single-parameter family of spheres satisfying the second-order contact condition.
(3) For to have a third-order contact with at , we must have (in addition to the conditions for first- and second-order contact):
. It can be shown that in this case has to be:
This unique sphere is called the osculating sphere of at point . The radius of the osculating sphere is given by: .