Curvature and Torsion -- curvature osculating circle torsion       The Frenet Trihedron -- tangent, normal and binormal vectors: Let a: I = (a,b) --> R3, be a curve parametrized by arc length s. Let t(s) = a'(s). Then, t(s) is a unit tangent vector at s since |a'(s)| = 1. Recall, a"(s) is normal to a'(s). Usually |a"(s)| does not equal 1. We let n(s) = a"(s) / |a"(s)| . We call n(s) the normal vector at s. Finally, let b(s) = t(s) x n(s). It is easy to see that |b(s)| = 1. We call b(s) the binormal vector at s. osculating, rectifying and normal planes The tn plane determined by the unit tangent t(s) and normal vector n(s) is called the osculating plane at s. The tb plane is called the rectifying plane, and the nb plane is called the normal plane. Frenet frame (or Frenet Trihedron) Let a: I = (a,b) --> R3 be a curve parametrized by arc length s. To each value of the parameter s, we have associated three orthogonal unit vectors t(s), n(s) and b(s). The frame (or trihedron) thus formed is referred to as the frenet frame (or frenet trihedron ) at s.   Curvature and Torsion -- Curvature: Let a: I = (a,b) --> R3, be a curve parametrized by arc length. As you may observe, the larger |a"(s)| is, the more the curve is "curved" at s. You may also observe that since the the tangent vector a'(s) has unit length, the norm |a"(s)| of the second derivative measures the rate of change of the angle which the neighboring tangents make with the tangent at s. |a"(s)| gives, therefore, a measure of how rapidly the curve pulls away from the tangent line at s, in the neighborhood of s. This suggests the following definition Definition: Let a: I --> R3 be a curve parametrized by arc length s in I. The number |a"(s)| = k(s) is called the curvature of a at s. The Oscillating Circle: junk Torsion: Let a: I --> R be a curve parametrized by arc length without a singular point of order 1 (i.e a"(s) does not equal 0). As you may observe, since the binormal b(s) is a unit vector, the length of |b'(s)| measures the rate of change of the neighboring osculating planes with respect to the osculating plane at s; that is, b'(s) measures how rapidly the curve pulls away from the osculating plane at s, in the neighborhood of s. Definition: Let a: I --> R3 be a curve parametrized by arc length s such that a"(s) does not equal 0, s in I. The number t(s) defined by b'(s) = t(s)n(s) is called the torsion of a at s.   The Frenet Formula -- Frenet formulas: You may have observed that the derivatives t'(s) = kn, b'(s) = tn of the vectors t(s) and b(s), when expressed in the basis {t, n, b}, yield geometrical entities (curvature k and torsion t) which gives us information about the behavior of a in a neighborhood of s. The search for other local geometric entities would lead us to compute n'(s). It is easy to check n'(s) = - tb - kt. Definition: We call the equations t' = kn, n' = - kt - tb b' = tn the frenet formulas. Fundamental theorem: Physically, we can think of a curve in R as being obtained from a straight line by bending (curvature) and twisting (torsion). After reflecting on this construction, you may guess roughly that k and t describe completely the local behavior of the curve. Indeed, we have Fundamental theorem of the local theory of curves Given differentiable functions k(s) > 0 and t(s), s in I, there exists a regular parametrized curve a: I --> R3 such that s is the arc length, k(s) is the curvature and t(s) is the torsion of a. Moreover, any other curve b satisfying the same conditions, differs from a by a rigid motion; that is, there exists an orthogonal linear map r of R3, with positive determinant, and a vector c such that b = r.a + c. (proof) Local canonical form: We have successfully found a natural coordinate system, namely the frenet trihedron at s. We want to refer the curve to this trihedron. Let a: I --> R3 be a curve parametrized by arc length without a singular point of order 1. After we have successfully found a natural coordinate system, namely the frenet frame, we shall write the equations of the curve, in a neighborhood of s0, using the trihedron t(s0), n(s0), b(s0) as a basis of R3. Without loss of generality, we may assume s0 = 0. Let us now take the system Oxyz in such a way that the origin O agrees with a(0) and that t = (1,0,0), n = (0,1,0), b = (0,0,1). Under these conditions, we can show that a(s) = (x(s), y(s), z(s)) is given by ( x(s), y(s), z(s)). This representation is called the local canonical form of a, in a neighborhood of s = 0. projection of trace of a: junk Moving the Frame: Letting the frenet frame vary with t provides a good idea of the curve's behavior in space. It is a fundamental idea in differential geometry to express the local change of the frame in terms of the frame itself.