Plane and solid analytic geometry; an elementary textbook . coordinates of the point wherethe line touches the directing curve, while the z coordi-nate may have any value whatever. The equation in xand y of the directing curve is, therefore, the only necessaryrelation between the coordinates of any point on the sur-face, and as it is not satisfied by any point not on thesurface, it is (when interpreted as an equation in threedimensions) the equation of the surface. In a similar manner, it may be shown that the equationsof cylindrical surfaces, whose elements are parallel to theJT-axis, contain
Plane and solid analytic geometry; an elementary textbook . coordinates of the point wherethe line touches the directing curve, while the z coordi-nate may have any value whatever. The equation in xand y of the directing curve is, therefore, the only necessaryrelation between the coordinates of any point on the sur-face, and as it is not satisfied by any point not on thesurface, it is (when interpreted as an equation in threedimensions) the equation of the surface. In a similar manner, it may be shown that the equationsof cylindrical surfaces, whose elements are parallel to theJT-axis, contain only y and z ; parallel to the !F-axis, onlyx and z. 13. Surfaces of revolution. — Surfaces generated by therevolution of a plane curve about one of the coordinateaxes form another class of surfaces whose equations canbe determined easily. 222 ANALYTIC GEOMETRY OF SPACE [Ch. II, § 13 For example, let it be required to determine the equationof the surface generated by the revolution of the ellipse 1 about the X-axis. Let P (V, y\ z) be anyz a? r a2_h62. Fig. 10. point on the surface, and through P pass a plane perpen-dicular to the X-axis. The section of the surface madeby this plane is evidently a circle. Hence LP = LK. But LP = V/M^T2 and OL = x\ The coordinates of iT in the X-P-plane are, therefore,x and Vy2 + z2, and since K is a point on the ellipse 1, these coordinates must satisfy that equation, ^2 fa2_h62 or + y> + z> = 1. Dropping primes, we have as the equation of an ellipsoidof revolution about the X-axis, a2^2^2 Ch. II, § 14] LOCI 223 A general rule for finding the equation of a surface ofrevolution, formed by revolving a plane curve about oneof the coordinate axes, may be stated thus: Replace in theequation of the plane curve the coordinate perpendicular tothe axis of revolution by the square root of the sum of thesquares of itself and of the third coordinate. PROBLEMS 1. Find the equation of the surface generated by a linemoving parallel to the Z-axis al
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