Graphics Reference
In-Depth Information
Note:
As we just indicated, to get a well-defined geometric definition of curvature
for a space curve F(t) we need to assume that F≤(t) does not vanish. For that reason
and the need for that assumption in other formulas related to geometric properties
of curves, the condition F≤(t) π 0 is often assumed implicitly in discussions about
curves, just like the condition of regularity. On a related point of terminology, given
a space curve F(t), a point
p
= F(t) with the property that F≤(t) = 0 is often referred
to as a point at which the curve is
locally flat
. The justification for this is that a curve
whose second derivative vanishes in a neighborhood of a point is in fact a straight
line in that neighborhood.
For an arbitrary regular curve F(t) in
R
3
9.4.1. Proposition.
¢
()
¥≤
()
¢
()
3
Ft F t
Ft
()
=
k t
.
Proof.
See [Spiv70b]. Compare this formula to the one for plane curves in Proposi-
tion 9.3.4.
Definition.
A point of a curve where the curvature vanishes is called an
inflection
point
.
Definition.
If F(s) is an arc-length parameterization and if T(s) = F¢(s), then the
prin-
cipal normal
of F at s, denoted by N(s), and the
binormal
of F at s, denoted by B(s),
are defined by
¢
()
=
() ()
()
=
()
¥
()
T s
k
s N s
and
B s
T s
N s
,
where k(s) is the curvature.
Clearly, both N(s) and B(s) are unit vectors and the orthonormal basis
(T(s),N(s),B(s)) determines the standard orientation of
R
3
.
Definition.
The tuple (T(s),N(s),B(s)) is called the
Frenet frame
or
moving trihedron
to the curve F(s) at s or the point F(s). The Frenet frame or moving trihedron at a
point of an arbitrary regular parameterization is the Frenet frame or moving trihe-
dron at that point of the induced arc-length parameterization.
Definition.
The
osculating plane
of F at s is the plane at F(s) generated by T(s) and
N(s). The
normal plane
of F at s is the plane at F(s) generated by N(s) and B(s). The
rectifying plane
of F at s is the plane at F(s) generated by T(s) and B(s).
9.4.2. Proposition.
B¢(s) = -t(s)N(s) for some function t(s).
Proof.
First, B • B = 1 implies that
BB
¢∑
=0.
