Stereographic projections: A common way of displaying a sphere (such as Earth) on a plane (such as a map) is to use a ?stereographic proje ? ction. Here is the two-dimensional version of the method, which maps a circle to a line. Let ? P be a point on the right half of a circle of ra ? dius ? identified by the ?angle ??. Find the fu?nctio ? n?? = ? (? )that give? s the? -coordin ? ate (?x ? 0) corresponding to ? ?? fo? 0? ? ??.

Step-by-step solution Step 1 Given that P is a point on the right half of a circle of radius R identified by the angle as shown in the figure below, we need to find the function x = F() that gi es th -coordinate (x0) of the point P for 0 < . Step 2 To find F(), we disregard the circle and just consider the following part of the diagram: In the right triangle ACD, we know that: AB = BP = BC = R ABP = CD = x Looking at the figure, we see that AC constitutes the diameter of the circle with radius R , so: AC = 2R In triangle ABP , the sides AB and BP are equal that is: AB = BP Therefore, triangle ABP is an isosceles triangle.