Curvature

Japanese: 曲率 - きょくりつ

A quantity that expresses the degree of curvature of a curve or curved surface. (1) If the length of the arc of a curve from a point P on a plane curve to a nearby point Q is Δs, and the angle between the tangent to the curve at P and the tangent to the curve at Q is Δθ (radians), the limit value of the ratio Δθ/Δs when Q approaches P infinitely is called the curvature. When this is expressed as κ, ρ = 1/|κ| is called the radius of curvature, and the larger ρ is, the more gentle the curve is. A circle that is tangent to the curve at point P, has a radius equal to the radius of curvature, and is on the same side of the curve with respect to the common tangent is called the circle of curvature of the curve at point P, and its center is called the center of curvature. Similarly, curvature can be defined for curves in space, but since the angle between the two tangents cannot be signed, it is considered as |Δθ/Δs|. (2) For curved surfaces, it is considered as the curvature at a point P on it. If a surface is cut with a plane containing the normal, the radius of curvature at point P of the cut curve is R. If the orientation of the plane is changed in various ways, R also changes, taking on a maximum value _R1 and a minimum value _R2 . 1/ _R1 and 1/ _R2 are called the principal curvatures, (1/ _R1 + 1/ _R2 )/ ₂ is the mean curvature, and 1/ _R1R2 is called the total curvature or Gaussian curvature.
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Source : Heibonsha Encyclopedia About MyPedia Information

Japanese:

曲線または曲面の曲がる度合を表す量。（１）平面曲線上の一点Pから近くの点Qまでの曲線の弧の長さをΔsとし，Pにおける曲線の接線とQにおける曲線の接線がなす角をΔθ（ラジアン）とするとき，QがPに無限に近づいたとき比Δθ/Δsがとる極限値を曲率という。これをκで表すとき，ρ＝1/｜κ｜を曲率半径といい，曲線の曲がり方はρが大きいほどゆるやかである。点Pでこの曲線に接する円で，半径が曲率半径に等しく，共通接線に関し曲線と同側にあるものを，点Pにおけるこの曲線の曲率円といい，その中心を曲率中心という。同様に空間内の曲線についても曲率を定義できるが，２接線のなす角に符号がつけられないので，｜Δθ/Δs｜として考える。（２）曲面ではその上の１点Pにおける曲率として考える。法線を含む平面で曲面を切り，切口の曲線の点Pにおける曲率半径をRとする。平面の向きをいろいろに変えれば，Rも変化し，最大値R₁と最小値R₂をとる。1/R₁，1/R₂を主曲率，（1/R₁＋1/R₂）/2を平均曲率，1/R₁R₂を全曲率またはガウスの曲率という。
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