A Stereographic Projection example using a Photo Sphere as source

projection_equirectangular_to_stereographic.fx

float4 mainPS(VsToPs inp) : SV_Target
{
	float2 scale = float2(4.0f / 3.0f, 1) * 3;	// scale & aspect ratio
	float2 plane = inp.screenPos.xy * scale;
	float3 dir = float3(plane.x * 2, plane.y * 2, -1 + dot(plane, plane)) / (1 + dot(plane, plane));

	dir = dir.xzy;	// y is upper

	float longitude = atan2(dir.x, dir.z) * (180 / 3.14159265f);
	float latitude = asin(dir.y) * (180 / 3.14159265f);
	return gTexture.Sample(samplerState, float2(longitude, latitude) / float2(360, -180) + 0.5);
}