Constructing a spherical circle
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Recall from the construction of a spherical line that we use stereographic projection to transfer the geometry of the sphere onto the plane.
A circle in spherical geometry is the intersection of a plane with the sphere. The stereographic projection of a circle on the sphere is a circle in the plane. The orange circle in the diagram below shows an example of such a circle passing through the point \(Q\). The centre of the circle on the sphere projects to a point \(P\) in the plane which is inside the circle. \(P\) is called the spherical centre of the spherical circle.
Notice that \(P\) is different to the Euclidean centre of the circle, since if you centre your compass at \(P\) and draw a circle passing through \(Q\), you will not reproduce the orange circle. Instead, the spherical centre has a different property: every spherical line through \(P\) will intersect the orange circle at right angles. This is true for great circles on the sphere passing through the spherical centre, and therefore it is true in the plane, since stereographic projection preserves angles.
The diagram below contains two points \(R\) and \(S\), as well as the equatorial circle with its centre \(O\). Your task is to construct the spherical circle passing through \(S\) with spherical centre at \(R\).
The diagram now contains two tools to help you: one will construct a spherical line through two points, and the other will construct a spherical circle centred at one point and passing through another given point (you can use this to check your answer).
Hint: You can use the property that every spherical line through \(R\) must perpendicularly intersect the spherical circle centred at \(R\). This will help you find the Euclidean centre of the circle, from which you can construct the spherical circle using a compass.
