Game Development Reference
In-Depth Information
After projection of
α
on the y- and
β
on the x-axis we get d and e , respectively:
2
2
α
(
α
/
β
)
d
=
+
φ
2
φ
2
cos(
)
sin(
)
2
2
α
(
α
/
β
)
e
=
+
2
2
sin(
φ
)
cos(
φ
)
For three perpendicular surfaces, namely surf1, surf2, and surf3, the diagonal
components
λ
and off-diagonal components
γ
are calculated for the 2x2 matrices
representing the sectional ellipses:
2
2
2
2
λ
=
1/ 2 (
α
(cos(
φ
))
+
β
(sin(
φ
)) )
+
1/ 2(
α
(cos(
φ
))
+
β
(sin(
φ
)) )
surf
1
surf
2
surf
2
surf
2
surf
2
surf
3
surf
3
surf
3
surf
3
λ
=
α
φ
2
+
β
φ
2
+
α
φ
2
+
β
φ
2
1/ 2 (
(cos(
))
(sin(
)) )
1/ 2 (
(cos(
))
(sin(
)) )
surf
2
surf
1
surf
1
surf
1
surf
1
surf
3
surf
3
surf
3
surf
3
λ
=
1/ 2 (
α
(cos(
φ
))
2
+
β
(sin(
φ
)) )
2
+
1/ 2 (
α
(cos(
φ
))
2
+
β
(sin(
φ
)) )
2
surf
3
surf
1
surf
1
surf
1
surf
1
surf
2
surf
2
surf
2
surf
2
γ
=
α
sin(
φ
)
cos(
φ
)
β
sin(
φ
)
cos(
φ
)
surf
1
surf
1
surf
1
surf
1
surf
1
surf
1
surf
1
γ
=
α
sin(
φ
)
cos(
φ
)
β
sin(
φ
)
cos(
φ
)
surf
2
surf
2
surf
2
surf
2
surf
2
surf
2
surf
2
γ
=
α
sin(
φ
)
cos(
φ
)
β
sin(
φ
)
cos(
φ
)
surf
3
surf
3
surf
3
surf
3
surf
3
surf
3
surf
3
is doubly defined. To get an initial estimate
we average the two doubly defined terms. To get a better best-fit estimate we
define a matrix P and calculate the normalized eigenvalues Π and eigenvectors
V of the sectional ellipses by using singular value decomposition.
Note that the diagonal component
λ
P
=
[(
λ
,
γ
,
γ
)
(
γ
,
λ
,
γ
)
(
γ
,
γ
,
λ
)]
1
3
2
3
2
1
2
1
3
surf
surf
surf
surf
surf
surf
surf
surf
surf
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