Consider the point P with coordinates (a, b, c) on the surface of the hyperbolic

Question

Consider the point P with coordinates (a, b, c) on the surface of the hyperbolic paraboloid described by z = y2 ? x2.
(a) Determine z(t) such that the straight line r(t) = (a + t) i + (b + t) j + z(t) k lies entirely on the surface and passes through the point P. Simplify your result so that only the terms a, b, and c appear. That is, terms such as a2, or a3, do not appear.
(b) A ruled quadric surface is a quadric surface that can be generated by the motion of a straight line. Clearly, based on your results from part (a), any point on a hyperbolic paraboloid has at least one straight line that passes through it, but is the hyperbolic paraboloid a ruled quadric surface? If so, what motion of a straight line will generate the hyperbolic paraboloid?
(c) Are there any other ruled quadric surfaces? If so, which surfaces are they, and how are they generated?

Explanation / Answer

Three common methods for the analytical representation of a surface. Three methods are commonly used to represent surfaces. They are 1] Surface representation 1. An equation of form z = f(x, y) where f is a single-valued continuous function defined on a region R of the xy-plane. 2] Surface representation 2. An equation of the form f(x, y, z) = 0 3] Surface representation 3. Parametric equations of the form x = f(u, v) 1) y = g(u, v) z = h(u, v) where f, g, h are continuous functions defined on a connected region R of the uv-plane. Equations 1) of Surface Representation 3] do not necessarily map in a one-to-one fashion. That is, more than one point (u, v) may map into the same point (x, y, z). Because of this fact the “surface” produced by the system may sometimes be something quite different from the usual conception of a surface. To avoid this, a stipulation that it is one-to-one needs to be appended. We do this in the concept of a simple surface element. Simple surface element. Let R be a simply connected region (i.e. a bounded region such as a rectangular or circular region without “holes”) of the uv-plane. Let equations x = x(u, v) y = y(u, v) z = z(u, v)

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