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Computing in Geographic Information Systems
book

Computing in Geographic Information Systems

by Narayan Panigrahi
July 2014
Intermediate to advanced content levelIntermediate to advanced
303 pages
8h 34m
English
CRC Press
Content preview from Computing in Geographic Information Systems
Computational Geometry and Its Application to GIS 149
To solve for γ we know that the plane passes through (a, b, a
2
+ b
2
) so by
solving it gives
a
2
+ b
2
= 2a.a + 2b.b + γ
γ = (a
2
+ b
2
) (7.14)
Thus the plane equation is
z = 2ax + 2by (a
2
+ b
2
)
If we shift the plane upwards by some positive amount r
2
we get the plane
z = 2ax + 2by (a
2
+ b
2
) + r
2
How does this plane intersect the paraboloid? Since the paraboloid is de-
fined by z = x
2
+ y
2
we can eliminate z giving:
x
2
+ y
2
= 2ax + 2by (a
2
+ b
2
) + r
2
which after some simple rearrangements is equal to
(x a)
2
+ (y b)
2
= r
2
(7.15)
This is just a circle (Figure 7.16). Thus, we have shown that the intersection
of a plane with the paraboloid produces a space curve (which turns out to be
an ellipse), which when pro
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Publisher Resources

ISBN: 9781482223149