ABSTRACT
I , [ du dv
[ du dv
] = Edu2 + 2Fdudv +Gdv2 (6.32)
are defined by the symmetric matrix
MI , [ E F F G
] ,
with determinant98
det(MI) (6.33)240
= EG− F 2 (6.34)
= ∥∥∥∥∂~p(u, v)∂u
= ∥∥∥∥∂~p(u, v)∂u × ∂~p(u, v)∂v
∥∥∥∥2 = ‖~n(u, v)‖2 (6.35) > 0 (6.36)
is by
s = ∫ S
ds
√ (dx)2 + (dy)2 + (dz)2
√(dx dt
)2 + (dy
dt
)2 + (dz
dt
)2 dt (6.37)
√(dx du
du dt +
dx dv
dv dt
)2 + (dy
du du dt +
dy dv
dv dt
)2 + (dz
du du dt +
dz dv
dv dt
)2 + (dy
du
)2 + (dz
du
)2 + 2
(dx du
dx dv +
dy du
dy dv +
dz du
dz dv
) du dt
dv dt
+ ((dx
dv
)2 + (dy
dv
)2 + (dz
dv
√ Eu′(t)2 + 2Fu′(t)v′(t) +Gv′(t)2dt (6.39)
and the surface area by
A = ∫ Du
∫ Dv ‖~n(u, v)‖dudv (6.35) 240=
√ det(MI)dudv (6.40)
Similarly, the coefficients of the second fundamental form
II , [ du dv
[ du dv
] = Ldu2 + 2Mdudv +Ndv2 (6.41)
are defined by the symmetric matrix
MII , [ L M M N
] ,
〉 〈 nˆ(u, v), ∂
〉 (6.42) = −
〉 (6.43) with determinant 98
= LN −M2 (6.44)
=
〈 ~n(u, v), ∂
〉〈 ~n(u, v), ∂
〉 − 〈 ~n(u, v), ∂
〉2 ‖~n(u, v)‖2 (6.45)
The partial derivatives of the unit normal vector nˆ(u, v) = ~n(u,v)‖~n(u,v)‖ may be explicitly
∂nˆ(u, v) ∂u
= ∂ ~n(u,v)‖~n(u,v)‖ ∂u
= ∂~n(u,v) ∂u ‖~n(u, v)‖ − ∂‖~n(u,v)‖∂u ~n(u, v)
‖~n(u, v)‖2 (6.30)240
‖~n(u, v)‖ (6.46)
∂nˆ(u, v) ∂v
= ∂ ~n(u,v)‖~n(u,v)‖ ∂v
= ∂~n(u,v) ∂v ‖~n(u, v)‖ − ∂‖~n(u,v)‖∂v ~n(u, v)
‖~n(u, v)‖2 (6.31)240
‖~n(u, v)‖ (6.47)
or by the Weingarten equations[ ∂nˆ(u, v) ∂u
, ∂nˆ(u, v) ∂v
] = [ ∂~p(u, v) ∂u
, ∂~p(u, v) ∂v
] (−S) (6.48)
where the shape operator , also called Weingarten map, reads
S , M−1I MII (6.49) (6.33)240
1 EG− F 2
[ G −F −F E
] [ L M M N
= 1 EG− F 2
[ GL− FM GM − FN EM − FL EN − FM
] (6.50)
The symmetric Jacobian matrix of the R2 → R3 transformation is then defined as
(6.48)242 = (−ST )
] (−S)
(6.33)240 = (−ST )MI(−S) (6.51)
with Jacobian determinant98 √
= √
det(−ST ) det(MI) det(−S) (6.35)
‖~n(u, v)‖|det(S)| (6.52)
Considering a normal plane (i.e., a plane containing the normal vector nˆ and a given tangent direction) at a point ~p on the surface, the parametrization-independent curvature is defined as the signed reciprocal radius, so-called radius of curvature, of the osculating circle (with center ~c) of the normal section (i.e., the curve resulting from the intersection of the surface and the normal plane), where the sign of the curvature sgn(〈~c − ~p, nˆ〉) is negative whenever the curve is convex, and positive if it is concave. The principal directions (given as the eigenvectors of the shape operator) are defined as the two orthogonal directions along which the curvature reaches a maximum and a minimum, so-called principal curvatures
κ1 and κ2 of the quadratic equation 28
det(MII − κMI) (6.42) 241
∣∣∣∣∣L− κE M − κFM − κF N − κG ∣∣∣∣∣
= (L− κE)(N − κG)− (M − κF )2 = (LN − κLG− κEN + κ2EG)− (M2 − 2κMF + κ2F 2) = (LN −M2) + κ(2MF − LG− EN) + κ2(EG− F 2) = 0 (6.53)
from which the mean curvature H and the Gaussian curvature K are, respectively, defined as
2H , κ1 + κ2 (2.99) 29
= EN + LG− 2MF EG− F 2
(6.50) 242= tr(S) (6.54)
K , κ1 × κ2 (2.100)
= LN −M 2
det(MII) det(MI)
det(S) (6.55)
〈 ~n(u, v), ∂
〉〈 ~n(u, v), ∂
〉 − 〈 ~n(u, v), ∂
〉2 ‖~n(u, v)‖4 (6.56)
Equivalently rewriting the quadratic equation as 28
K − 2Hκ+ κ2 (6.53) 243= 0 (6.57) additionally yields
= H ± √ H2 −K (6.58)
where
( κ1 + κ2
)2 − κ1κ2 = κ
4 − 4κ1κ2
4 = ( κ1 − κ2
)2 ≥ 0 (6.59)
As illustrated in Figure 6.1, several cases are then distinguished depending on the value 244 of
= sgn(κ1)× sgn(κ2) (6.55) 243
sgn (
243= sgn (
det(S) )
(6.60)
and a point on the surface is said to be an:
• umbilic point if K = κ21 = κ22, i.e., κ1 = κ2, such that any tangent vector of the surface is a principal direction
• elliptic point if K > 0, i.e., κ1 and κ2 have the same sign, such that the surface is either locally convex or locally concave along both principal directions
• parabolic point if K = 0, i.e., κ1 or κ2 is zero, such that the surface is locally flat along one principal direction
• hyperbolic point if K < 0, i.e., κ1 and κ2 have different signs, such that the saddleshaped surface is locally convex along one principal direction and locally concave along the other
Figure 6.1: Gaussian Curvature: Illustration of an elliptic, a parabolic and a hyperbolic point (from left to right, respectively).
