Graphics Reference
In-Depth Information
both the surface and the parameterization of the curve on the surface, which can yield tangents
with one degree of freedom (revolution around surface normal), the two principal curvatures
are a sufficient description of the local surface around a point. In fact, the local surface can be
locally approximated using the principal curvatures up to the rigid transformations using the
paraboloid
z
1
2
(
+
κ
2
y
2
).
The principal curvatures are given in terms of the first and second fundamental forms as
follows (starting from the Eq. 2.31):
κ
1
x
2
=
2
γ
˙
κ
n
=
γ
·
¨
n
.
(2.32)
d
d
t
=
γ
·
˙
n
.
(2.33)
d
d
t
d
u
d
t
+
σ
v
d
v
d
t
=
σ
u
·
n
.
(2.34)
d
u
d
t
σ
u
d
2
u
d
t
2
+
σ
v
d
2
v
d
t
2
σ
uu
d
u
d
t
+
σ
uv
d
v
=
+
d
t
d
v
d
t
σ
uv
d
u
d
t
+
σ
vv
d
v
+
·
n
.
(2.35)
d
t
γ
In Equation 2.34, the tangent vector (to the curve) ˙
is expressed as a weighted combination
d
t
and
d
d
t
. Some terms in Equation
2.35 are tangential, so their dot product with the surface normal reduces to zero. Equation 2.35
is expressed in terms of the first and second fundamental forms,
d
u
σ
u
and
σ
v
, where the weights are
of the surface tangents
F
1
and
F
2
, and the weight
[
d
d
t
d
v
d
t
]
.
vector
w
=
w
σ
uu
·
n
σ
uv
·
n
2
w
γ
˙
κ
n
=
γ
·
˙
γκ
n
=
˙
.
(2.36)
σ
uv
·
n
σ
vv
·
n
w
σ
u
d
u
σ
u
d
u
d
t
+
σ
v
d
v
d
t
+
σ
v
d
v
σ
uu
·
n
σ
uv
·
n
w
·
κ
n
=
.
(2.37)
d
t
d
t
σ
uv
·
σ
vv
·
n
n
w
w
σ
u
·
σ
u
σ
u
·
σ
v
σ
uu
·
n
σ
uv
·
n
w
w
κ
n
=
.
(2.38)
σ
u
·
σ
v
σ
v
·
σ
v
σ
uv
·
n
σ
vv
·
n
EF
FG
w
LM
MN
w
w
w
κ
n
=
.
(2.39)
w
F
1
w
w
F
2
w
κ
n
=
.
(2.40)
Using the Equation 2.40, the normal curvature can be computed in any tangential direction
specified by
w
,
w
F
2
w
w
F
1
w
. Also, on the basis of Equation 2.40, Equation 2.41 holds indicating
that the maximum and minimum principal curvatures are the eigenvalues of
κ
n
=
F
1
−
1
F
2
.
κ
n
=
F
1
−
1
w
F
2
w
.
(2.41)
