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3D Game Engine Design, 2nd Edition
book

3D Game Engine Design, 2nd Edition

by David Eberly
November 2006
Intermediate to advanced content levelIntermediate to advanced
1040 pages
26h 31m
English
CRC Press
Content preview from 3D Game Engine Design, 2nd Edition
16.3 Least-Squares Fitting 729
where
¯x =
1
m
m
i=1
x
i
, ¯y =
1
m
m
i=1
y
i
,
¯
L =
1
m
m
i=1
L
i
,
¯
L
a
=
1
m
m
i=1
a x
i
L
i
,
¯
L
b
=
1
m
m
i=1
b y
i
L
i
Fixed-point iteration can be applied to solving these equations: a
0
x, b
0
y, and
a
i+1
= F(a
i
, b
i
) and b
i+1
= G(a
i
, b
i
) for i 0.
16.3.6 Fitting a Sphere to 3D Points
Given a set of points {(x
i
, y
i
, z
i
)}
m
i=1
, m 4, fit them with a sphere (x a)
2
+
(y b)
2
+ (z c)
2
= r
2
,where(a, b, c) is the sphere center and r is the sphere
radius. An assumption of this algorithm is that not all the points are coplanar. The
error function to be minimized is
E(a, b, c, r) =
m
i=1
(L
i
r)
2
where L
i
=
-
(x
i
a)
2
+ (y
i
b)
2
+ (z
i
c). Take the partial derivative ...
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Publisher Resources

ISBN: 9781482267303