Previous: Moving Frames

5. Non-Inflectional Curves and the Frenet Formulas

A curve X is called non-inflectional if the curvature k(t) is never zero. By our earlier calculations, this condition is equivalent to the requirement that X'(t) and X"(t) are linearly independent at every point X(t), i.e. X'(t) x X"(t) 0 for all t. For such a non-inflectional curve X, we may define a pair of natural unit normal vector fields along X.
Let B(t) = [X'(t) x X"(t)]/|X'(t) x X"(t)|, called the binormal vector to the curve X(t). Since B(t) is always perpendicular to T(t), this gives a unit normal vector field along X.
We may then take the cross product of the vector fields B(t) and T(t) to obtain another unit normal vector field N(t) = B(t) x T(t), called the principal normal vector. The vector N(t) is a unit vector perpendicular to T(t) and lying in the plane determined by X'(t) and X"(t). Moreover, X"(t)N(t) = k(t)s'(t)2, a positive quantity.
Note that if the parameter is arclength, then X'(s) = T(s) and X"(s) is already perpendicular to T(s). It follows that X"(s) = k(s)N(s) so we may define N(s) = X"(s)/k(s) and then define B(s) = T(s) x N(s). This is the standard procedure when it happens that the parametrization is by arclength. The method above works for an arbitrary parametrization.
We then have defined an orthonormal frame X(t) T(t) N(t) B(t) called the Frenet frame of the non-inflectional curve X.
By the previous section, the derivatives of the vectors in the frame can be expressed in terms of the frame itself, with coefficients that form an anti-symmetric matrix. We already have X'(t) = s'(t)T(t), so p1(t) = s'(t), p2(t) = 0 = p3(t). Also T'(t) = k(t)s'(t)N(t) so q12(t) = k(t)s'(t) and q13(t) = 0. We know that B'(t) = q31(t)T(t) + q32(t)N(t), and q31(t) = - q13(t) = 0. Thus B'(t) is a multiple of N(t), and we define the torsion w(t) of the curve by the condition B'(t) = -w(t)s'(t)N(t), so q32(t) = -w(t)s'(t) for the Frenet frame. From the general computations about moving frames, it then follows that N'(t) = q21(t)T(t) + q23(t)B(t) = -k(t)s'(t)T(t) + w(t)s'(t)B(t). The formulas for T'(t), N'(t), and B'(t) are called the Frenet formulas for the curve X.
If the curve X is parametrized with respect to arclength, then the Frenet formulas take on a particularly simple form:
X'(s) = T(s)
T'(s) =k(s)N(s)
N'(s) = -k(s)T(s)+ w(s)B(s)
B'(s) =-w(s)B(s)
The torsion function w(t) that appears in the derivative of the binormal vector determines important properties of the curve. Just as the curvature measures deviation of the curve from lying along a straight line, the torsion measures deviation of the curve from lying in a plane. Analogous to the result for curvature, we have:
Proposition 1: If w(t) = 0 for all points of a non-inflectional curve X, then the curve is contained in a plane.
Proof: We have B'(t) = -w(t)s'(t)N(t) = 0 for all t so B(t) = A, a constant unit vector. Then T(t)A = 0 for all t so (X(t)A)' = X'(t)A = 0 and X(t)A = X(a)A, a constant. Therefore (X(t) - X(a))A = 0 and X lies in the plane through X(a) perpendicular to A.
If X is a non-inflectional curve parametrized by arclength, then w(s) = B(s)N'(s) = [ T(s), N(s), N'(s) ]. Since N(s) = X"(s)/k(s), we have N'(s) = X"'(s)/k(s) + X"(s)(-k'(s)/k(s)2), so w(s) = [ X'(s), X"(s)/k(s), X"'(s)/k(s) + X"(s)(-k'(s)/k(s)2) ] = [ X'(s), X"(s). X"'(s) ]/k(s)2.
We can obtain a very similar formula for the torsion in terms of an arbitrary parametrization of the curve X. Recall that X"(t) = s"(t)T(t) + k(t)s'(t)T'(t) = s"(t)T(t) + k(t)s'(t)2 N(t), so X"'(t) = s"'(t)T(t) + s"(t)s'(t)k(t)N(t) + [k(t)s'(t)2 ]'N(t) + k(t)s'(t)2 N'(t). Therefore X"'(t)B(t) = k(t)s'(t)2N'(t)B(t) = k(t)s'(t)2w(t)s'(t), and X"'(t)X'(t) x X"(t) = k2(t)s'(t)6w(t). Thus we obtain the formula w(t) = X"'(t)X'(t) x X"(t)/|X'(t) x X"(t)|2, valid for any parametrization of X.
Notice that although the curvature k(t) is never negative, the torsion w(t) can have either algebraic sign. For the circular helix X(t) = ( r cos(t), r sin(t), pt) for example, we find w(t) = p/[r2 + p2], so the torsion has the same algebraic sign as p. In this way, the torsion can distinguish between a right-handed and a left-handed screw.
Changing the orientation of the curve from s to -s changes T to -T, and choosing the opposite sign for k(s) changes N to -N. With different choices, then, we can obtain four different right-handed orthonormal frames, XTNB, X(-T)N(-B), XT(-N)(-B), and X(-T)(-N)B. Under all these changes of the Frenet frame, the value of the torsion w(t) remains unchanged.
A circular helix has the property that its curvature and its torsion are both constant. Furthermore the unit tangent vector T(t) makes a constant angle with the vertical axis. Although the circular helices are the only curves with constant curvature and torsion, there are other curves that have the second property. We characterize such curves, as an application of the Frenet frame.
Proposition 2: The unit tangent vector T(t) of a non-inflectional space curve X makes a constant angle with a fixed unit vector A if and only if the ratio w(t)/k(t) is constant.
Proof: If T(t)A = constant for all t, then differentiating both sides, we obtain T'(t)A = 0 = k(t)s'(t)N(t)A, so A lies in the plane of T(t) and B(t). Thus we may write A = cos()T(t) + sin()B(t) for some angle . Differentiating this equation, we obtain 0 = cos()T'(t) + sin()B'(t) = cos()k(t)s'(t)N(t) - sin()w(t)s'(t)N(t), so w(t)/k(t) = sin()/cos() = tan(). This proves the first part of the proposition and identifies the constant ratio of the torsion and the curvature.
Conversely, if w(t)/k(t) = constant = tan() for some , then, by the same calculations, the expression cos()T(t) + sin()B(t) has derivative 0 so it equals a constant unit vector. The angle between T(t) and this unit vector is the constant angle .
Curves with the property that the unit tangent vector makes a fixed angle with a particular unit vector are called generalized helices. Just as a circular helix lies on a circular cylinder, a generalized helix will lie on a general cylinder, consisting of a collection of lines through the curve parallel to a fixed unit vector. On this generalized cylinder, the unit tangent vectors make a fixed angle with these lines, and if we roll the cylinder out onto a plane, then the generalized helix is rolled out into a straight line on the plane.
We have shown in the previous section that a moving frame is completely determined up to an affine motion by the functions pi(t) and qij(t). In the case of the Frenet frame, this means that if two curves X and X have the same arclength s(t), the same curvature k(t), and the same torsion w(t), then they curves are congruent, i.e. there is an affine motion of Euclidean three-space taking X(t) to X(t) for all t. Another way of stating this result is:
The Fundamental Theorem of Space Curves: Two curves parametrized by arclength having the same curvature and torsion at corresponding points are congruent by an affine motion.

Exercises

  1. Compute the torsion of the circular helix. Show directly that the principal normals of the helix are perpendicular to the vertical axis, and show that the binormal vectors make a constant angle with this axis.
  1. Prove that if the curvature and torsion of a curve are both constant functions, then the curve is a circular helix (i.e. a helix on a circular cylinder).
  1. Prove that a necessary and sufficient condition for a curve X to be a generalized helix is that X"(t) x X"'(t) Xiv(t) = 0.
  1. Let Y(t) be a curve on the unit sphere, so that |Y(t)| = 1 and Y(t) Y'(t) x Y"(t) 0 for all t. Show that the curve X(t) = c Y(u) x Y"(u) du with c 0 has constant torsion 1/c.
  1. (For students familiar with complex variables) If the coordinate functions of the vectors in the Frenet frame are given by T = (e11, e12, e13), N = (e21, e22, e23), and B = (e31, e32, e33), then we may form the three complex numbers zj = ( e1j + ie2j )/(1 - e3j ) = (1 + e3j )/( e1j - i e2j). Then the functions zj satisfy the Riccati equation zj' = -ik(s)zj + (i/2)w(s)( -1 + zj2 ). This result is due to S. Lie and G. Darboux.
Next: Local Equations of a Curve