Graphics Reference
In-Depth Information
1
R
, where
R
is the radius of the osculating circle (that
locally fits the curve). For a parametric curve embedded in a 3D space,
point. Curvature
κ
is quantified as
±
[
x
(
t
)
y
(
t
)
z
(
t
)]
,
γ
=
the curvature is described as
˙
γ
−
¨
γ
·
¨
γ
κ
=
(2.28)
γ
˙
2
γ
×
γ
˙
¨
=
,
(2.29)
3
γ
˙
where ˙
are the first and second derivatives of the curve with respect to the parameter
t
. The unit vector ˙
γ
and ¨
γ
is the curve tangent, ˙
˙
is orthogonal to
the tangential vector, and when normalized to a unit magnitude, it defines the principal normal
of the curve. In theory, 3D curves can be (re)-parameterized so they are unit-speed, that is,
γ
γ
=
γ/
˙
γ
˙
. The vector ¨
γ
−
γ
·
¨
γ
, and the principal normal to ¨
.
It should be noted that both the osculated circle and the principal normal are coplanar and the
curvature
γ
=
˙
1. The curvature of unit speed curves simplifies to
γ
¨
γ
κ
is invariant to rigid transformations.
Curvatures of Curves on 3D Surfaces
For curves on 3D surfaces, the second derivative vector of the parametric curve defines two
types of curvatures: the geodesic and the normal curvatures. Let
v
)]
be a 3D surface parameterized by
u
and
v
. A further parameterization of the surface param-
eters, with respect to a single parameter
t
, yields a 3D curve on the surface
σ
=
[
x
(
u
,
v
)
y
(
u
,
v
)
z
(
u
,
σ
, that is,
σ
u
×
σ
v
σ
u
×
σ
v
v
(
t
)) ]
. The surface normal
n
γ
=
[
x
(
u
(
t
)
,
v
(
t
))
y
(
u
(
t
)
,
v
(
t
))
z
(
u
(
t
)
,
=
, where
σ
u
and
σ
v
are the derivatives of the surface with respect to
u
and
v
(the two vectors span the
tangential plane). The tangent vector of the curve is
t
˙
, differentiated with respect to
t
.
The geodesic and the normal curvature are shown in Equations 2.30 and 2.31.
=
γ
γ
¨
κ
g
=
2
·
(
n
×
t
)
.
(2.30)
γ
˙
γ
¨
κ
n
=
2
·
n
.
(2.31)
γ
˙
The curvatures
κ
n
are a quantification of how the curve is bent within and away from
the tangential plane of the surface. The normal curvature.
κ
g
and
κ
n
. has a particular importance in
3D face recognition because it is a characteristic of the surface in the tangential direction of
the curve,
t
. While
κ
g
is solely a characteristic of the 3D curve (as opposed to the surface).
From the preceding two equations (Eqs. 2.30 and 2.31) a
nd Equa
tions 2.28 and 2.29, the total
curvature
g
2
.
κ
of the curve is related to
κ
g
and
κ
n
by
κ
=
κ
+
κ
Principal curvatures:
The principal curvatures are two unique normal curvatures (Eq. 2.31)
corresponding the curvature of the curve resulting from the intersection of the 3D surface with
a normal plane (orthogonal to the tangential plane) in the tangential directions that give the
minimum
κ
1
and maximum
κ
2
normal curvatures. While normal curvatures have pertinence to
