. Plane and solid analytic geometry . Fig. 6 554 ANALYTIC GEOMETRY It is never a surface of revolution, no matter what values areassigned to a and b. The sections of the surface by the vertical coordinate planesare the parabolas x = 2u% » y^ = -2b% of which the first opens upwards and the second, downwards. The section by the (x, 2/)-planeconsists of the two lines,. OA: OB: X + 1 = 0, a 0 X a b Fig. 7 A section parallel to and abovethe (x, 2/)-plane is a hyperbolawhose vertices are on the parabolaopening upwards, whereas a sec-tion parallel to and below the (x, 2/)-plane is a hyperbolawhose ve
. Plane and solid analytic geometry . Fig. 6 554 ANALYTIC GEOMETRY It is never a surface of revolution, no matter what values areassigned to a and b. The sections of the surface by the vertical coordinate planesare the parabolas x = 2u% » y^ = -2b% of which the first opens upwards and the second, downwards. The section by the (x, 2/)-planeconsists of the two lines,. OA: OB: X + 1 = 0, a 0 X a b Fig. 7 A section parallel to and abovethe (x, 2/)-plane is a hyperbolawhose vertices are on the parabolaopening upwards, whereas a sec-tion parallel to and below the (x, 2/)-plane is a hyperbolawhose vertices are on the parabola opening downwards. It isseen, then, that the surface is saddle-shaped; it rises along theparabola which opens upwards, and falls along the parabolawhich opens downwards. The (z, ic)-plane contains the pommeland the (y, 2;)-plane, thestirrups. The surface can bestbe plotted by drawingthe sections parallel to avertical coordinate plane,for example, the (y, z)-plane. These sectionsare all parabolas openingdownwards and havingtheir vertices on the parabola in the {z, a;)-plane. Figure 8shows part of the surface constructed by means of them. Vertex, Axis, Principal Planes. Each paraboloid is sym-metric in only one line, the axis of z, and in only two planes,
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