Graphics Reference
In-Depth Information
n
p
n
p
c
max
p
n
p
p
p
c
max
c
max
c
min
c
min
(a) Elliptic point
(b) Hyperbolic point
(c) Parabolic point
Figure 9.19.
Local curvature properties of surfaces.
X
u
X
u
(a)
n
p
p
u
S
g
Figure 9.20.
Meusnier's theorem.
surface lies on both sides of the tangent plane. The point
p
in Figure 9.19(c) is called
a
parabolic point
because the surface lies to one side of the tangent plane and meets
the tangent plane in a line locally.
Definition.
The points
c
min
and
c
max
are called the
foci
of the normal line at
p
or
the
focal points
of
S
at
p
. The set of focal points of
S
is called the
surface of centers
,
the
focal surface
, or the
evolute
of
S
.
See [HilC99] or [Gray98] for more about focal points. We shall have more to say
about the surface of centers and evolutes in Sections 9.12 and 9.13.
Euler's theorem was soon generalized in 1776 by Meusnier. Keeping the same nota-
tion as above, let
X
u
(a), 0 £a<p/2, be a plane through
p
that contains
u
and makes an
angle a with
X
u
. Consider the parametric curve g parameterized by arc-length that
parameterizes the intersection of
X
u
(a) and
S
in a neighborhood of
p
and which has
tangent vector
u
at
p
. Let k
u
(a) denote the curvature of that curve at
p
. See Figure 9.20.
Meusnier's Theorem.
k
u
(a) cos a=k
u
.
Both Euler's and Meusnier's theorem will be proved shortly. Although interesting,
these two theorems still deal with curves and are only indirectly about surfaces. It was
