Previous: Non-Inflectional Curves and the Frenet Formulas
6. Local Equations of a CurveWe can "see" the shape of a curve more clearly in the neighborhood of a point X(t0) when we consider its parametric equations with respect to the Frenet frame at the point. For simplicity, we will assume that t0 = 0, and we may then write the curve as X(t) = X(0) + x1(t)T(0) + x2(t)N(0) + x3(t)B(0). On the other hand, using the Taylor series expansion of X(t) about the point t = 0, we obtain X(t) = X(0) + t X'(0) + t2/2 X"(0) + t3/6 X"'(0) + higher order terms. From our earlier formulas, we have X'(0) = s'(0)T(0), X"(0) = s"(0)T(0) + k(0)s'(0)2N(0), and X"'(0) = s"'(0)T(0) + s"(0)s'(0)k(0)N(0) + (k(0)s'(0)2)'N(0) + k(0)s'(0)2(-k(0)s'(0)T(0) + w(0)s'(0)B(0)). Substituting these equations in the Taylor series expression, we find:
X(t) = X(0) + (t s'(0) + t2/2 s"(0) + t3/6 [s"'(0) - k(0)2s'(0)3] + . . . )T(0)
+ ( t2/2 k(0)s'(0)2 + t3/6 [s"(0)s'(0)k(0) + (k(0)s'(0)2)'] + . . .)N(0)
+ ( t3/6 k(0)w(0)s'(0)3 + . . . )B(0)
| |
|
If the curve is parametrized by arclength, this representation is much simpler:
X(s) = X(0) + ( s - [k(0)2/6] s3 + . . .)T(0)
+ ([k(0)/2]s2 + [k'(0)/6]s3 + . . .)N(0)
+ ([k(0)w(0)/6]s3 + . . . )B(0).
| |
|
Relative to the Frenet frame, the plane with equation x1 = 0 is the normal plane; the plane with x2 = 0 is the rectifying plane, and the plane with x3 = 0 is the osculating plane. These planes are orthogonal respectively to the unit tangent vector, the principal normal vector, and the binormal vector of the curve.
|