Page:General Investigations of Curved Surfaces, by Carl Friedrich Gauss, translated into English by Adam Miller Hiltebeitel and James Caddall Morehead.djvu/101

or also

$$1=r(x'y''-y'x''),$$

so that

$$x'y''-y'x''$$

represents the curvature, and

$${\frac {1}{x'y''-y'x''}}$$

the radius of curvature.

7.

We shall now proceed to the consideration of curved surfaces. In order to represent the directions of straight lines in space considered in its three dimensions, we imagine a sphere of unit radius described about an arbitrary centre. Accordingly, a point on this sphere will represent the direction of all straight lines parallel to the radius whose extremity is at this point. As the positions of all points in space are determined by the perpendicular distances ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z}$ from three mutually perpendicular planes, the directions of the three principal axes, which are normal to these principal planes, shall be represented on the auxiliary sphere by the three points ${\textstyle (1),}$ ${\textstyle (2),}$ ${\textstyle (3).}$ These points are, therefore, always ${\textstyle 90^{\circ }}$ apart, and at once indicate the sense in which the coordinates are supposed to increase. We shall here state several well known theorems, of which constant use will be made.

1) The angle between two intersecting straight lines is measured by the arc [of the great circle] between the points on the sphere which represent their directions.

2) The orientation of every plane can be represented on the sphere by means of the great circle in which the sphere is cut by the plane through the centre parallel to the first plane.

3) The angle between two planes is equal to the angle between the great circles which represent their orientations, and is therefore also measured by the angle between the poles of the great circles.

4) If ${\textstyle x,}$ ${\textstyle y,}$ ${\textstyle z;}$ ${\textstyle x',}$ ${\textstyle y',}$ ${\textstyle z'}$ are the coordinates of two points, ${\textstyle r}$ the distance between them, and ${\textstyle L}$ the point on the sphere which represents the direction of the straight line drawn from the first point to the second, then

$${\begin{alignedat}{2}x'&=x&&+r\cos(1)L,\\y'&=y&&+r\cos(2)L,\\z'&=z&&+r\cos(3)L.\end{alignedat}}$$

5) It follows immediately from this that we always have

$$\cos ^{2}(1)L+\cos ^{2}(2)L+\cos ^{2}(3)L=1$$