Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/96

Such lines as differ in direction by ${\textstyle 360^{\circ },}$ or by a multiple of ${\textstyle 360^{\circ },}$ have, therefore, precisely the same direction, and may, generally speaking, be regarded as the same. However, in such cases where the manner of describing a variable angle is taken into consideration, it may be necessary to distinguish carefully angles differing by ${\textstyle 360^{\circ }.}$

If, for example, we have decided to measure the arcs from left to right, and if to two straight lines ${\textstyle l,}$ ${\textstyle l'}$ correspond the two directions ${\textstyle L,}$ ${\textstyle L',}$ then ${\textstyle L'-L}$ is the angle between those two straight lines. And it is easily seen that, since ${\textstyle L'-L}$ falls between ${\textstyle -180^{\circ }}$ and ${\textstyle +180^{\circ },}$ the positive or negative value indicates at once that ${\textstyle l'}$ lies on the right or the left of ${\textstyle l,}$ as seen from the point of intersection. This will be determined generally by the sign of ${\textstyle \sin(L'-L).}$

If ${\textstyle aa'}$ is a part of a curved line, and if to the tangents at ${\textstyle a,}$ ${\textstyle a'}$ correspond respectively the directions ${\textstyle \alpha ,}$ ${\textstyle \alpha ',}$ by which letters shall be denoted also the corresponding points on the auxiliary circles, and if ${\textstyle A,}$ ${\textstyle A'}$ be their distances along the arc from the origin, then the magnitude of the arc ${\textstyle \alpha \alpha '}$ or ${\textstyle A'-A}$ is called the amplitude of ${\textstyle aa'.}$

The comparison of the amplitude of the arc ${\textstyle aa'}$ with its length gives us the notion of curvature. Let ${\textstyle l}$ be any point on the arc ${\textstyle aa',}$ and let ${\textstyle \lambda ,}$ ${\textstyle \Lambda }$ be the same with reference to it that ${\textstyle \alpha ,}$ ${\textstyle A}$ and ${\textstyle \alpha ',}$ ${\textstyle A'}$ are with reference to ${\textstyle a}$ and ${\textstyle a'.}$ If now ${\textstyle \alpha \lambda }$ or ${\textstyle \Lambda -A}$ be proportional to the part ${\textstyle al}$ of the arc, then we shall say that ${\textstyle aa'}$ is uniformly curved throughout its whole length, and we shall call

$${\frac {\Lambda -A}{al}}$$

the measure of curvature, or simply the curvature. We easily see that this happens only when ${\textstyle aa'}$ is actually the arc of a circle, and that then, according to our definition, its curvature will be ${\textstyle \pm {\frac {1}{r}},}$ if ${\textstyle r}$ denotes the radius. Since we always regard ${\textstyle r}$ as positive, the upper or the lower sign will hold according as the centre lies to the right or to the left of the arc ${\textstyle aa'}$ (${\textstyle a}$ being regarded as the initial point, ${\textstyle a'}$ as the end point, and the directions on the auxiliary circle being measured from left to right). Changing one of these conditions changes the sign, changing two restores it again.

On the contrary, if ${\textstyle \Lambda -A}$ be not proportional to ${\textstyle al,}$ then we call the arc non-uniformly curved and the quotient

$${\frac {\Lambda -A}{al}}$$