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Math 314 Lecture #6 §12.6: Cylinders and Quadric Surfaces Outcome A: Identify and sketch (by hand) a surface that is a cylinder. Recall that an equation in x, y, and z in R3 represents a surface. Example. Here are the surfaces, or graphs, for the equations z = y 2 and x2 + z 2 = 1. Observe that each of these surfaces consist of lines (called rulings) that are parallel to a given line and pass through a given plane curve. Such a surface is called a cylinder. For the first surface, the given plane curve is the parabola z = y 2 in the yz-plane; this surface is called a parabolic cylinder. For the second surface, the given plane curve is the circle x2 + z 2 = 1 in the xz-plane; this surface is called a circular cylinder. Observe that in both of the equations one of the three variables is missing. This is typical of a cylinder whose rulings are parallel to one of the coordinate axes. However, not all cylinders have rulings that are parallel to a coordinate axis: just rotate the above cylinders about the z-axis a bit. Outcome B: Use traces to identify and sketch (by hand) a quadric surface. A quadric surface is the graph of an equation in x, y, and z of the form Ax2 + By 2 + Cz 2 + Dxy + Eyz + F xz + Gx + Hy + Iz + J = 0, for constants A, B, C, D, E, F , G, H, I, and J. By translations and rotations, the above equation can be brought (through lots of algebra) into one of two forms: Ax2 + By 2 + Cz 2 + J = 0, or Ax2 + By 2 + Iz = 0, where the variables x, y, and z can be interchanged. A trace of a surface is the intersection of the surface with a plane parallel to one of the coordinate planes. Setting one of the variables to a constant gives a trace. Various traces of a surface help identify the surface and sketch it by hand. Example. Identify and sketch the quadric surface 25x2 + 4y 2 + z 2 = 100. Setting x = k and moving all the constants to the right side of the equation gives the intersection of the surface with the plane x = k: 4y 2 + z 2 = 100 − 25k 2 , −2 ≤ k ≤ 2. The inequalities on k occur because 4y 2 + z 2 ≥ 0; the plane x = k does not intersect the surface when k > 2 or k < −2. The trace of the surface in the plane x = k is the ellipse 4y 2 + z 2 = 100 − 25k 2 . (see picture below on the left for these traces) Setting y = k in the equation and moving all the constants to the right side of the equations gives the intersection of the surface with plane y = k: 25x2 + z 2 = 100 − 4k 2 , −5 ≤ k ≤ 5. These traces are ellipses in the plane y = k. (See picture below in the center for these traces.) Setting z = k gives the elliptical traces 25x2 + 4y 2 = 100 − k 2 , −10 ≤ k ≤ 10. Here are all of these traces (with the z = k traces on the right). From these traces we identify the quadric surface as an ellipsoid. Example. Identify and sketch by hand the quadric surface 4x2 + 9y 2 − z = 0. Setting z = k gives the trace 4x2 + 9y 2 = k for k ≥ 0 which is an ellipse on the horizontal plane z = k > 0. Here are these traces for the values k = 1, 4, 9, 16, 25, 39, where the larger ellipses correspond to the larger values of k. From these traces for different heights k, we see that this quadric surface is an elliptic paraboloid. What are the traces of this surface when x = k or y = k? They are parabolas. Example. Identify and sketch by hand the surface z = x2 − y 2 . Setting z = k gives the trace x2 − y 2 = k which is a hyperbola in the z = k 6= 0 plane. Here are these traces for k = −2, −1, 0, 1, 2. The X-looking trace passing through the origin corresponds to k = 0. Which ones correspond to k = −2, −1? To k = 1, 2? From these traces for varying heights k, this quadric surface is hyperbolic paraboloid, or saddle. What are the traces corresponding to x = k? to y = k? They are parabolas opening up or opening down. Remaining Quadric Surfaces. These are the cone, hyperboloid of one sheet, and the hyperboloid of two sheets. What are their traces? Can you sketch them by hand? (See Table on p.854 in text.)