曲率

曲面は曲がり具合で三分類できます。

①どの方向にも凹、あるいはどの方向にも凸な、お椀型の曲面

②一方向にのみに曲がった、単曲面

③凹と凸が両方含まれた、馬の鞍型の曲面

一般に曲面はそれぞれの場所で異なる曲がり具合を持ちます。身の回りの曲面のどの場所がどの種類か観察してみましょう。

数学では、曲がり具合は「曲率」で表し、凹を正、真っすぐを０、凸を負とした数値で表現します。曲面の場合、ある点での曲率は方向によって変わるので、曲率の最大値と最小値を使って表現することになります。この最大値と最小値を主曲率と呼び、その積をガウス曲率と呼びます。椀型曲面はガウス曲率が正（正×正、負×負）、単曲面はガウス曲率が０（正×０、負×０）、鞍型曲面はガウス曲率が負（正×負）です。

Curvature

Curved surfaces can be classified into three categories based on their curvature:

1. Bowl-shaped surfaces: Concave in all directions or convex in all directions.

2. Singly curved surfaces: Curved in only one direction.

3. Saddle-shaped surfaces: Containing both concave and convex portions.

In general, curved surfaces have varying curvature depending on location. Observe the curved surfaces around you and identify which category each location belongs to.

In mathematics, the curvature is expressed as a numerical value where concave is positive, straight is 0, and convex is negative. In curved surfaces, the curvature at a given point varies depending on the direction, so it is represented by the maximum and minimum values of curvature. These maximum and minimum values are called principal curvatures, and their product is called the Gaussian curvature. Bowl-shaped surfaces have positive Gaussian curvature (positive × positive — or negative × negative), singly curved surfaces have zero Gaussian curvature (positive × 0 — or negative × 0), and saddle-shaped surfaces have negative Gaussian curvature (positive × negative).