Mathematics of Physics and Engineering

32 Curves in Space

At every point P of the curve, the vector triple (v., p, b) is a right-

handed coordinate system with origin at P. We will call this coordi-

nate system Frenet' s trihedron at P. The choice of initial conditions

r(to), r '(to) means setting up a coordinate system in the frame with origin

at PQ, where OPQ = r(to). The coordinate planes spanned by the vectors

(u, p), (p, b), and (b, u) are called, respectively, the osculating, normal,

and rectifying (binormal) planes. The word osculating comes from

Latin osculum, literally, a little mouth, which was the colloquial way of say-

ing "a kiss". Not surprisingly, of all the planes that pass through the point

P, the osculating plane comes the closest to containing the curve C.

EXERCISE 1.3.10? A curve is called planar if all its points are in the same
plane. Show that a planar curve other than a line has the same osculating
plane at every point and lies entirely in this plane (for a line, the osculating
plane is not well-defined).
The curvature and torsion uniquely determine the curve, up to its posi-
tion in space. More precisely, if K(S) and r(s) are given continuous functions
of s, we can solve the corresponding equations (1.3.15)-(1.3.17) and obtain
the vectors u(s),p(s), b(s) which determine the shape of a family of curves.
To obtain a particular curve C in this family, we must specify initial values
(u(so),p(so), b(so)) of the trihedron vectors and an initial value r(so) of a
position vector at a point PQ on the curve. These four vectors are all in
some frame with origin O. To obtain r(s) at any point of C we solve the
differential equation dr/ds = u(s), with initial condition r(so), together
with (1.3.15)-(1.3.17). Note that the curvature is always non-negative, and
the torsion can be either positive or negative.

EXERCISE 1.3.11.A For the right circular helix (1.3.13) compute the curva-

ture, torsion, and the Frenet trihedron at every point. Show that the right

circular helix is the only curve with constant curvature and constant positive

torsion.

As the point P moves along the curve, the trihedron executes three ro-
tations. These rotations about the unit tangent, principal unit normal, and
unit binormal vectors are called rolling, yawing, and pitching, respec-
tively. Rolling and yawing change direction of the unit binormal vector 6,
rolling and pitching change the direction of the principal unit normal vector
p, yawing and pitching change the direction of the unit tangent vector u.
To visualize these rotations, consider the motion of an airplane. Intuitively,