Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 60', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Theory of Curves 381 At a regular point Mo the equation of the normal plane has the form x - xo x t to y - yo y t to z - Zo z t to 0 or r - ro rt to 0 and for the curve r obtained as the intersection of two planes we have x - xo y - yo F1 X F1 y F2 x F y z - zo Fi z o F2 z where all derivatives are evaluated at x xo and y yo . 1 . Consider a curve defined via its natural parameter s r r s . Unlike the derivatives with respect to an arbitrary parameter the derivatives with respect to s will be denoted by primes. Consider the vectors r and r . It follows from that r 1. Thus the first derivative with respect to the natural parameter s of the position vector of a point on a curve is the unit vector tangent to the curve. 2 . The second derivative with respect to the natural parameter s of the position vector of a point of a curve is equal to the first derivative of the unit vector rzs . of a vector of constant length and hence it is perpendicular to this vector. But since the vector of the first derivative is tangent to the curve the vector of the second derivative with respect to the natural parameter s is normal to the curve. This normal is called the principal normal to the curve. At a regular point Mo the equation of the principal normal has the form f Z or r ro Art X rt X 4 xtm - ytl m ztxtt- zttxt n xtytt- xuy t x - xo _ y - yo y tn - z tm z tl - x tn l y z y z where all derivatives are evaluated at t to . . Osculating plane. Any plane passing through the tangent line to a curve is called a tangent plane. A tangent plane passing through a principal normal to the curve is called the osculating plane. A curve has exactly one osculating plane at each of its points assuming that the vectors rt and rtt are linearly independent . This plane passes through the vectors rt and rtt drawn from the point of tangency. The osculating plane is independent of the choice of the parameter t on the curve. Remark. If t is time .
