. Differential and integral calculus, an introductory course for colleges and engineering schools. e of the simpler equations, somevery elementary considerations may enable us to get a fairlygood idea of the shape of the surface represented. The surfacerepresented by an equation of the second degree in x, y, z is termeda quadric surface. Some special cases of quadric surfaces follow. 1. The Sphere, x2 + y2 + z2 = r2. This is the equation of the surface of a sphere. For it assertsthat every point on the surface is at the constant distance r fromthe origin (Art. 181). The equation of the sphere
. Differential and integral calculus, an introductory course for colleges and engineering schools. e of the simpler equations, somevery elementary considerations may enable us to get a fairlygood idea of the shape of the surface represented. The surfacerepresented by an equation of the second degree in x, y, z is termeda quadric surface. Some special cases of quadric surfaces follow. 1. The Sphere, x2 + y2 + z2 = r2. This is the equation of the surface of a sphere. For it assertsthat every point on the surface is at the constant distance r fromthe origin (Art. 181). The equation of the sphere whose centeris at (a, /3, 7) and whose radius is r is (x - aY + (y - /3)2 + O - 7)2 = r\ 2. The Ellipsoid, n2 T 7,2 T ^2 -1- 272 GEOMETRY OF THREE DIMENSIONS 185 Setting x, y, z successively equal to 0, we get as the equationsof the traces of the surface in the coordinate planes + ~o=h C2 + *_ = 1 and the traces are seen to be ellipses whose axes coincide with the axes of coordinates and whose centersare at the origin. Writing the given equation in theform ^ 4- yl = 1 _ zl n2 ~T~ 7>2 L „2. we note that, because the first mem-ber is the sum of two squares, thesecond member can never be negative for real values of x,y, and z, and that therefore | z \ = \c\.* Hence the surface lieswholly between the two planes z = — c and z= +c. In a simi-lar manner it can be shown that the surface lies between theplanes y = —b and y =-\-b, and between the planes x = — a andx = +o. Hence the surface lies entirely within the rectangularparallelopiped bounded by these six planes. We shall next deter-mine the curves of section of the given surface by planes parallelto the coordinate planes. Consider first the plane x = k. Wesubstitute k for x in the given equation, and get, after reduction, (a) y b2 l -. + <-5) 1. This may be regarded as the equation of the right cylinder whichprojects upon the yz-plane the curve of section S, made withthe surface by the plane x = k, or, as the equation
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
