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Surfaces in space sketching cylinders and the 7

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Unformatted text preview: nders and the 7 surfaces using trace curves Identifying the type of surface from the equation Notes for 12.1 General Second Degree Equation in TWO variables You have already studied these (quadric) curves but we referred to them as Conic Sections Circles , Ellipses , Parabolas , Hyperbolas and the Degenerate Conics: Point , Line , Pair of Intersecting Lines , Parallel Lines , No Graph A conic section may more formally be defined as the locus of a point fixed line constant if called the conic section directrix (with called the eccentricity. If not on Possible Cases ) such that the ratio of the distance of , the conic is a circle, if , it is a hyperbola. Example Circle x^2 + y^2 == 4 Parabola y^2 == 9 x Ellipse 4 x^2 + 9 y^2 == 36 that moves in the plane of a fixed point , the conic is an ellipse, if from called the focusand a to its distance from is a , the conic is aparabola, and Hyperbola x^2 - y^2 == 1 One Line x^2 == 0 Intersecting Lines (x - 1) (y + 1) == 0 Parallel Lines (x - 1) (x - 2) == 0 A Point x^2 + y^2 == 0 No Graph x^2 == -1 Quadric Surfaces: General Second Degree Equation in THREE variables The TRACE of a surface is the intersection curve of a plane and the surface. 1) Ellipsoid If a = b = c then we have a sphere 2) Elliptic Paraboloid Important to Note that one of the variables is NOT squared. Good hint that your dealing with a paraboloid. This is a paraboloid opening upward on the z-axis The trace curves are parabolas for x=0 and y=0. For z=# we have trace curves that are ellipses or circles depending on if a = b 3) Elliptic Cone Notice how close this resembles the paraboloid equation (this is a cone along the z-axis) Also note this equation can be rewritten: Important to Note that it is easy to see that the point ( 0 , 0 , 0 ) is a solution to the equation! See the negative in front of the z . So you can use that to note if the negative is in front of the x then the cone is along the x-axis. Also it is very important to notice that this is equal to 0 since we will...
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