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11.6 - Standard Equation 2 2 2 2 2 2 = c z b y a x 5...

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11.6 Surfaces in Space
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Definition of a Cylinder Let C be a curve in a plane and let L be a line not in a parallel plane. The set of all lines parallel to L and intersecting C is called a cylinder. C is called the generating curve (or directrix), and the parallel lines are called rulings. Note: If one of the variables is missing from the equation of a cylinder, its rulings are parallel to the coordinate axis of the missing variable.
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Examples: 4 2 = + z y 0 = - x e z
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Quadric Surfaces The equation of a quadric surface in space is a second-degree equation of the form There are six basic types of quadric surfaces: 0 2 2 2 = + + + + + + + + + J Iz Hy Gx Fyz Exz Dxy Cz By Ax
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1) Ellipsoid Standard Form 1 2 2 2 2 2 2 = + + c z b y a x
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2) Hyperboloids of One Sheet Standard Equation: 1 2 2 2 2 2 2 = - + c z b y a x
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3) Hyperboloid of Two Sheets Standard equation 1 2 2 2 2 2 2 = + - - c z b y a x
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4) Elliptic Cone
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Unformatted text preview: Standard Equation: 2 2 2 2 2 2 =-+ c z b y a x 5) Elliptic Paraboloid • Standard Equation: 2 2 2 2 a x b y z-= 6) Hyperbolic Paraboloid • Standard Equation: 2 2 2 2 a x b y z-= To Sketch a Quadric Surface 1) Write the surface in standard form. 2) Determine the traces in the coordinate planes by setting each variable =0 For example: To get the trace in the xy-plane, set z=0. To get the trace in the xz-plane, set y=0, etc. 3) If needed, find the traces in planes that are parallel to coordinate planes by holding a variable constant. Examples: Identify and Sketch: 1) 2) 4 4 2 2 = +-z y x 1 4 / 2 2 2 = + + z y x #42 • Sketch the region bounded by the graphs of the equations. X=0, y=0, z=0 2 4 x z-= 2 4 x y-=...
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