Great and Small Circles

Excerpts

• Great and Small Circles

  A sphere is a solid bounded by a surface every point of which is equally distant from a fixed point which is called the *centre* of the sphere.

• Great and Small Circles

  The section of the surface of a sphere by a plane is called a *great circle* if the plane passes through the centre of the sphere, and a *small circle* if the plane does not pass through the centre of the sphere.

• Great and Small Circles

  When only one great circle can be drawn through two given points, the great circle is unequally divided at the two points; we shall for brevity speak of the shorter of the two arcs as *the* arc of a great circle joining the two points.

• Great and Small Circles

  Then $PO$ is at right angles to the plane $ABC$, because $P$ is the pole of $ABC$, therefore $POA$ is a right angle, and the arc $PA$ is a quadrant.

• Great and Small Circles

  Thus the distance of a pole of a circle from every point of the circumference of the circle is constant, whether that distance be measured by the straight line joining the points, or by the arc of a great circle intercepted between the points.

Equations

• Great and Small Circles

  CD=\surd(OD^2-OC^2)

  The radius CD of the plane section of a sphere is the square root of the sphere's radius squared minus the squared distance from the centre of the sphere to the plane.

• Great and Small Circles

  PD=\surd(PC^2+CD^2)

  The straight-line distance from a pole P of a circle to a point D on its circumference is the square root of the sum of the squares of PC (pole to centre of the circle) and CD (radius of the circle).

• Great and Small Circles

  AOB = AOM - BOM = BON - BOM = MON

  The angle AOB subtended at the centre of the sphere by the arc joining two poles equals the inclination MON of the two great circles, since AOB is found by subtracting equal parts.

• Great and Small Circles

  \frac{\operatorname{arc} ab} {\operatorname{radius} Ca}=\frac{\operatorname{arc} AB} {\operatorname{radius} OA}

  The ratio of arc to radius is the same for the small-circle arc ab and the great-circle arc AB, since both subtend the same angle at their centres.

• Great and Small Circles

  \frac{\operatorname{arc} ab}{\operatorname{arc} AB}=\frac{Ca}{OA} =\frac{Ca}{Oa}=\sin POa

  The ratio of the small-circle arc ab to the great-circle arc AB equals the sine of the angle POa, the ratio of the radii of the two circles.

Problems

No exercises in this chapter.