Surfaces in Three-Space

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7 Consider a surface whose equation is x2 + 2x + 1 + y2 2y + 1 z2 + 6z 9 = 36 What sort of object is this What are the coordinates of the center How is the axis oriented 8 Write down a generalized equation for an elliptic cone whose axis is parallel to the coordinate y axis, and whose vertex is at ( 2,3,4) 9 Suppose we slice the elliptic cone described in Problem 8 straight through with the coordinate xz plane The cone s surface intersects the xz plane in a curve Derive a generalized equation of that curve in the variables x and z What sort of curve is it Here s a hint: At every point in the xz plane, y = 0 10 Suppose we slice the elliptic cone described in Problem 8 straight through with the coordinate xy plane The cone s surface intersects the xy plane in a curve Derive a generalized equation of that curve in the variables x and y What sort of curve is it Here s a hint: At every point in the xy plane, z = 0

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CHAPTER

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Lines and Curves in Three-Space

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In Chap 16, we learned how parametric equations can define curves that are difficult to portray as conventional relations Parametric power becomes more apparent when we graduate to three dimensions

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Straight Lines

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Finding an equation for a straight line in Cartesian three-space is harder than it is in the Cartesian plane The extra dimension makes expressing the line s location and orientation more complicated There are at least two ways we can do it: the symmetric method and the parametric method

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Symmetric method A straight line in Cartesian xyz space can be represented by a three-part symmetric-form equation Suppose that (x0,y0,z0) are the coordinates of a known point on the line, and a, b, and c are nonzero real-number constants Given this information, we can represent the line as

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(x x0)/a = (y y0)/b = (z z0)/c If a = 0 or b = 0 or c = 0, then we get a zero denominator somewhere, and the system becomes meaningless

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Direction numbers In the symmetric-form equation of a straight line, the constants a, b, and c are known as the direction numbers Imagine a vector m whose originating point is at the origin (0,0,0) and whose terminating point has coordinates (a,b,c) Under these circumstances, the vector m either lies right along, or is parallel to, the line denoted by the symmetric-form equation

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Lines and Curves in Three-Space

+y Line L and vector m are parallel m = (a, b, c)

P = (x0, y0, z0)

Figure 18-1

We can uniquely define a line L in Cartesian xyz space on the basis of a point P on L and a vector m = (a,b,c) parallel to L

(In three-space, a vector m and a straight line L are parallel if and only if the line containing m occupies the same plane as L but does not intersect L) We have m = ai + bj + ck where m is the three-dimensional equivalent of the slope of a line in the Cartesian plane Figure 18-1 shows a generic example

Parametric method Given any particular line L in Cartesian xyz space, we can find infinitely many vectors to play the role of the direction-defining vector m If t is a nonzero real number, then any vector

t m = (ta,tb,tc) = ta i + tb j + tc k works just as well as m = ai + bj + ck for the purpose of defining the direction of L, so we have an alternative way to describe a straight line using the following equations: x = x0 + at y = y0 + bt z = z0 + ct