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points on a surface.
Hence, if one of the surfaces be a plane, and the tween two trace on the other be a geodetic line, the trace on the plane is a straight line. Conversely, if the trace on the plane be a straight line, that on the surface is a geodetic line.
Area of spherical polygon.
And, quite generally, if the given trace be a geodetic line, the other trace is also a geodetic line.
134. The area of a spherical triangle (on a sphere of unit radius) is known to be equal to the "spherical excess," ie., the excess of the sum of its angles over two right angles, or the excess of four right angles over the sum of its exterior angles. The area of a spherical polygon whose n sides are portions of great circles-i.e., geodetic lines-is to that of the hemisphere as the excess of four right angles over the sum of its exterior angles is to four right angles. (We may call this the 'spherical excess" of the polygon.)
For the area of a spherical triangle is known to be equal to
Divide the polygon into n such triangles, with a common vertex, the angles about which, of course, amount to 2π.
Reciprocal polars on a
Given an open or closed spherical polygon, or line on the
sphere. surface of a sphere composed of consecutive arcs of great circles.
Take either pole of the first of these arcs, and the corresponding poles of all the others (all the poles to be on the right hand, or all on the left, of a traveller advancing along the given great circle arcs in order). Draw great circle arcs from the first of these poles to the second, the second to the third, and so on in order. Another closed or open polygon, constituting what is called the polar diagram to the given polygon, is thus obtained. The sides of the second polygon are evidently equal to the exterior angles in the first; and the exterior angles of the second are equal to the sides of the first. Hence the relation between the two diagrams is reciprocal, or each is polar to the other. The polar figure to any continuous curve on a spherical
surface is the locus of the ultimate intersections of great circles Reciprocal equatorial to points taken infinitely near each other along it.
polars on a sphere.
The area of a closed spherical figure is, consequently, according to what we have just seen, equal to the excess of 2π above the periphery of its polar, if the radius of the sphere be unity.
135. If a point move on a surface along a figure whose Integral sides are geodetic lines, the sum of the exterior angles of this direction in polygon is defined to be the integral change of the direction in the surface.
In great circle sailing, unless a vessel sail on the equator, or on a meridian, her course, as indicated by points of the compass (true, not magnetic, for the latter change even on a meridian), perpetually changes. Yet just as we say her direction does not change if she sail in a meridian, or in the equator, so we ought to say her direction does not change if she moves in any great circle. Now, the great circle is the geodetic line on the sphere, and by extending these remarks to other curved surfaces, we see the connexion of the above definition with that in the case of a plane polygon (§ 10).
Note. We cannot define integral change of direction here by Change of any angle directly constructible from the first and last tangents a surface, to the path, as was done (§ 10) in the case of a plane curve or traced on it. polygon; but from §§ 125 and 133 we have the following statement:—The whole change of direction in a curved surface, from one end to another of any arc of a curve traced on it, is equal to the change of direction from end to end of the trace of this arc on a plane by pure rolling.
136. Def. The excess of four right angles above the inte- Integral gral change of direction from one side to the same side next time in going round a closed polygon of geodetic lines on a curved surface, is the integral curvature of the enclosed portion of surface. This excess is zero in the case of a polygon traced on a plane. We shall presently see that this corresponds exactly to what Gauss has called the curvatura integra.
Def. (Gauss.) The curvatura integra of any given portion Curvatura of a curved surface, is the area enclosed on a spherical surface
of unit radius by a straight line drawn from its centre, parallel to a normal to the surface, the normal being carried round the boundary of the given portion.
The curve thus traced on the sphere is called the Horograph of the given portion of curved surface.
Curvatura integra, and horograph.
The average curvature of any portion of a curved surface is the integral curvature divided by the area. The specific curvature of a curved surface at any point is the average curvature of an infinitely small area of it round that point.
Change of direction
137. The excess of 27 above the change of direction, in a surface, of a point moving round any closed curve on it, is equal to the surface, the area of the horograph of the enclosed portion of surface.
together with area of the horograph, equals four rightangles: or "Inte
gral Curvature" equals "Curvatura Integra."
Let a tangent plane roll without spinning on the surface over every point of the bounding line. (Its instantaneous axis will always lie in it, and pass through the point of contact, but will not, as we have seen, be at right angles to the given bounding curve, except when the twist of a narrow ribbon of the surface along this curve is nothing.) Considering the auxiliary sphere of unit radius, used in Gauss's definition, and the moving line through its centre, we perceive that the motion of this line is, at each instant, in a plane perpendicular to the instantaneous axis of the tangent plane to the given surface. The direction of motion of the point which cuts out the area on the spherical surface is therefore perpendicular to this instantaneous axis. Hence, if we roll a tangent plane on the spherical surface also, making it keep time with the other, the trace on this tangent plane will be a curve always perpendicular to the instantaneous axis of each tangent plane. The change of direction, in the spherical surface, of the point moving round and cutting out the area, being equal to the change of direction in its own trace on its own tangent plane (§ 135), is therefore equal to the change of direction of the instantaneous axis in the tangent plane to the given surface reckoned from a line fixed relatively to this plane. But having rolled all round, and being in position to roll round again, the instantaneous axis of the fresh start must be inclined to the trace at the same angle as in the beginning. Hence the change of direction of the instantaneous axis in either tangent plane is equal to the change of direction, in the given surface, of
a point going all round the boundary of the given portion of it Curvatura (135); to which, therefore, the change of direction, in the horograph. spherical surface, of the point going all round the spherical area is equal. But, by the well-known theorem (§ 134) of the "spherical excess," this change of direction subtracted from 2 leaves the spherical area. Hence the spherical area, called by Gauss the curvatura integra, is equal to 2 wanting the change of direction in going round the boundary.
It will be perceived that when the two rollings we have considered are each complete, each tangent plane will have come back to be parallel to its original position, but any fixed line in it will have changed direction through an angle equal to the equal changes of direction just considered.
Note. The two rolling tangent planes are at each instant parallel to one another, and a fixed line relatively to one drawn at any time parallel to a fixed line relatively to the other, remains parallel to the last-mentioned line.
If, instead of the closed curve, we have a closed polygon of geodetic lines on the given surface, the trace of the rolling of its tangent plane will be an unclosed rectilineal polygon. If each geodetic were a plane curve (which could only be if the given surface were spherical), the instantaneous axis would be always perpendicular to the particular side of this polygon which is rolled on at the instant; and, of course, the spherical area on the auxiliary sphere would be a similar polygon to the given one. But the given surface being other than spherical, there must (except in the particular case of some of the geodetics being lines of curvature) be tortuosity in every geodetic of the closed polygon; or, which is the same thing, twist in the corresponding ribbons of the surface. Hence the portion of the whole trace on the second rolling tangent plane which corresponds to any one side of the given geodetic polygon, must in general be a curve; and as there will generally be finite angles in the second rolling corresponding to (but not equal to) those in the first, the trace of the second on its tangent plane will be an unclosed polygon of curves. The trace of the same rolling on the spherical surface in which it takes place will generally be a spherical polygon, not of great circle arcs, but of other curves. The sum of the exterior angles of this polygon, and of the changes of direction from one end to the other of each of its sides, is the whole change of direction considered, and is, by the proper
Curvatur integra, and horograph.
application of the theorem of § 134, equal to 2 wanting the spherical area enclosed.
Or again, if, instead of a geodetic polygon as the given curve, we have a polygon of curves, each fulfilling the condition that the normal to the surface through any point of it is parallel to a fixed plane; one plane for the first curve, another for the second, and so on; then the figure on the auxiliary spherical surface will be a polygon of arcs of great circles; its trace on its tangent plane will be an unclosed rectilineal polygon; and the trace of the given curve on the tangent plane of the first rolling will be an unclosed polygon of curves. The sum of changes of direction in these curves, and of exterior angles in passing from one to another of them, is of course equal to the change of direction in the given surface, in going round the given polygon of curves on it. The change of direction in the other will be simply the sum of the exterior angles of the spherical polygon, or of its rectilineal trace. Remark that in this case the instantaneous axis of the first rolling, being always perpendicular to that plane to which the normals are all parallel, remains parallel to one line, fixed with reference to the tangent plane, during rolling along each curved side, and also remains parallel to a fixed line in space.
Lastly, remark that although the whole change of direction of the trace in one tangent plane is equal to that in the trace on the other, when the rolling is completed round the given circuit; the changes of direction in the two are generally unequal in any part of the circuit. They may be equal for particular parts of the circuit, viz., between those points, if any, at which the instantaneous axis is equally inclined to the direction of the trace on the first tangent plane.
Any difficulty which may have been felt in reading this Section will be removed if the following exercises on the subject be performed.
(1) Find the horograph of an infinitely small circular area of any continuous curved surface. It is an ellipse or a hyperbola according as the surface is synclastic or anticlastic (§ 128). Find the axes of the ellipse or hyperbola in either case.
(2) Find the horograph of the area cut off a synclastic surface by a plane parallel to the tangent plane at any given point of it, and infinitely near this point. Find and interpret the corresponding result for the case in which the surface is anticlastic in the neighbourhood of the given point.