| x² | + | y² | = | γ² | − 1, |
| a² | b² | c² |
provided γ² > c², and the surface, consisting of two distinct portions or sheets, may be considered as generated by a variable ellipse moving parallel to itself along the hyperbolas as directrices.
38. Differential Geometry of Curves.—For convenience consider the coordinates (x, y, z) of a point on a curve in space to be given as functions of a variable parameter θ, which may in particular be one of themselves. Use the notation x′, x″ for dx/dθ, d²x/dθ², and similarly as to y and z. Only a few formulae will be given. Call the current coordinates (ξ, η, ζ).
The tangent at (x, y, z) is the line tended to as a limit by the connector of (x, y, z) and a neighbouring point of the curve when the latter moves up to the former: its equations are
(ξ − x)/x′ = (η − y)/y′ = (ζ − z)/z′.
The osculating plane at (x, y, z) is the plane tended to as a limit by that through (x, y, z) and two neighbouring points of the curve as these, remaining distinct, both move up to (x, y, z): its one equation is
(ξ − x) (y′z″ − y″z′) + (η − y) (z′x″ − z″x′) + (ζ − z) (x′y″ − x″y′) = 0.
The normal plane is the plane through (x, y, z) at right angles to the tangent line, i.e. the plane
x′(ξ − x) + y′ (η − y) + z′ (ζ − z) = 0.
It cuts the osculating plane in a line called the principal normal. Every line through (x, y, z) in the normal plane is a normal. The normal perpendicular to the osculating plane is called the binormal. A tangent, principal normal, and binormal are a convenient set of rectangular axes to use as those of reference, when the nature of a curve near a point on it is to be discussed.