## 曲率とは curvature　 †

この値はｙ″の正負によって正、または負となるが、曲率が正とはｘが増加するとき曲線が左側（すなわち正の方向）に曲がっていくこと、負とは右側に曲がっていくことを意味する。

`κ　=　lim(p->p0)　θ/Δｓ`

という式で与えることもできる。曲率は曲線の形状を特徴づける数である。 一般に κを曲率、κの逆数 を曲率半径と言う。

• The meaning of curvature

Suppose that a particle moves on the plane with unit speed. Then the trajectory of the particle will trace out a curve C in the plane. Moreover, taking the time as the parameter, this provides a natural parametrization for C. The instantaneous direction of motion is given by the unit tangent vector P and the curvature measures how fast this vector rotates. If a curve keeps close to the same direction, the unit tangent vector changes very little and the curvature is small; where the curve undergoes a tight turn, the curvature is large.

• For a plane curve given parametrically as c(t) = (x(t),y(t)), the curvature is
For the less general case of a plane curve given explicitly as y = f(x) the curvature is
If a curve is defined in polar coordinates as r(θ), then its curvature is

where here the prime refers to differentiation with respect to θ.

## 曲率半径　the radius of curvature †

Last-modified: 2009-09-22 (火) 13:15:00 (4019d)