surface in space

10.6 SURFACES IN SPACE
EXAMPLE 6.1

Sketching a Surface

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Slide 1

10.6 SURFACES IN SPACE
EXAMPLE 6.1

Sketching a Surface

Solution
Since there are no x’s in the equation, the trace of the graph in the plane x = k is the same for every k.
This is then a cylinder whose trace in every plane parallel to the yz-plane is the parabola z = y2.

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Slide 2

10.6 SURFACES IN SPACE
EXAMPLE 6.1

Sketching a Surface

Solution
Draw the trace in the yz-plane and then make several copies of the trace, locating the vertices at various points along the x-axis.
Finally, we connect the traces with lines parallel to the x-axis to give the drawing its three-dimensional look. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Slide 3

10.6 SURFACES IN SPACE
Quadric Surfaces
The graph of the equation

in three-dimensional space (where at least one of a, b, c, d, e or f is nonzero) is referred to as a quadric surface.

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Slide 4

10.6 SURFACES IN SPACE
Quadric Surfaces
The most familiar quadric surface is the sphere of radius r centered at the point (a, b, c).

A generalization of the sphere is the ellipsoid:

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Slide 5

10.6 SURFACES IN SPACE
EXAMPLE 6.3

Sketching an Ellipsoid

Graph the ellipsoid

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Slide 6

10.6 SURFACES IN SPACE
EXAMPLE 6.3

Sketching an Ellipsoid

Graph the ellipsoid
Solution
Draw the traces in the three coordinate planes: yz-plane (x = 0):

xy-plane (z = 0): xz-plane (y = 0):
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