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Let H be the surface with equation x2+y2-z2 = -1. Since x2+y2= z2-1, the points on H have z2 ≥ 1, i.e. z ≥ 1 or z ≤ -1. Thus, the surface H consists of two disjoint parts - the sheets of H. If |k| > 1, H meets the plane z = k in the circle x2+y2 = k2-1, z = k. Also from this rearrangement, H has radial symmetry about the z-axis. Now, H meets the xz-plane (y=0) in the curve x2-z2 = -1. If we write this in the form z2-x2 = 1, we see that the curve of intersection is a hyperbola. With the radial symmetry, we observe that H is obtained by rotating this hyperbola about the z-axis. The sheets of H derive from the branches of the hyperbola. Also, the asymptotes x±z =0, y=0 of the hyperbola generate the cone C : x2+y2-z2 = 0. |
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