Lecture 6 - Surfaces Vaguequestion zeroloci...

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Surfaces
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Vague question How can we relate equations to the shapes of their zero loci?
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A menagerie of shapes Cone: + = x 2 y 2 z 2
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A menagerie of shapes Freaky cylinder: = ( x J 1) y 2 x 2
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A menagerie of shapes Ellipsoid: + + = 1 1 2 x 2 1 3 y 2 z 2
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A menagerie of shapes Hyperbolic paraboloid: J = z 1 9 x 2 1 4 y 2
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A menagerie of shapes Elliptic paraboloid: + = z 1 9 x 2 1 4 y 2
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Key idea Slice the shape with planes and reassemble the pieces. This idea recurs throughout the study of geometry (even by professionals!). Subdividing, solving, reassembling is also how computers graph these things.
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Example: We can make a horizontal trace (horizontal slice) at . + = z 1 9 x 2 1 4 y 2 z = 6
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Example: We can make a horizontal trace (horizontal slice) at . + = z 1 9 x 2 1 4 y 2 z = 6
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Example: We can make a horizontal trace (horizontal slice) at . Equation for the curve in the horizontal plane: What shape is this? What shape will a general horizontal trace have? + = z 1 9 x 2 1 4 y 2 z = 6 + = 6.
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Lecture 6 - Surfaces Vaguequestion zeroloci...

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