Here we have an interactive construction of the hyperbola and ellipse in the Hyperbolic Upper Half-Plane. Points A and C are the foci and the sum/difference of the focal radii is equal to the distance from points A to B. If point C is in the the circle centered at A passing through B, then we have an ellipse, otherwise we have a hyperbola. Feel free to move all these points around the space above the x-axis to see what these conic sections look like in hyperbolic geometry.