Parametric Surfaces

Section 3.5 Parametric Surfaces

Previous sections introduced parametric curves. These are topologically one dimensional relations produced by a single parameter. Section 1.3 introduced planes, a type of surface which is a topologically two dimensional relation that are produced by two parameters. Here we develop other topologically two dimensional relations (surfaces) that are produced by two parameters.

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Subsection 3.5.1 Recognizing Surfaces in Parametric Form

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Activity 57.

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The purpose of this activity is to use the idea that surfaces are “wide” curves (consist of a set of curves) to express them in parametric rather than as a function.

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Graph the following parametric surfaces using technology then answer the questions.

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(a)

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Consider

P (t, u) = (1, 2, 2) t + (- 1, 1, - 1) u + (7, 2, 4) .

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(i)

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What is a part of

P (t, u)

that is a line?

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(ii)

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Why does adding

(1, 2, 2) t

(- 1, 1, - 1) u + (7, 2, 4)

turn a line into a plane?

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(b)

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Consider

S (t, θ) = (\cos θ, \sin θ, 0) + (- t, t, - t) + (7, 2, 4) .

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(i)

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Which part of

S (t, θ)

is a circle?

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(ii)

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Which part of

S (t, θ)

is a line?

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(iii)

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Why does adding

(\cos θ, \sin θ, 0)

to the line

(- t, t, - t) + (7, 2, 4)

turn a line into a (leaning) cylinder?

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(c)

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F (t, θ) = (0, \cos θ, \sin θ) + (t, t^{2}, 0) .

Why does adding

(0, \cos θ, \sin θ)

(t, t^{2}, 0)

turn this parabola into a pipe segment (parabolic cylinder)?

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Subsection 3.5.2 Tangent Planes for Parametrized Surfaces

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In Section 2.3 we learned to calculate planes tangent to a surface at a point when the surface was expressed as a function. Here we use an idea from that section to calculate tangent planes when the surface is parameterized.

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