The Klein model (which the applet does not show) is an orthographic projection of the hemisphere model to an equatorial disk. In the hemisphere model, where the "lines" are semicircles in vertical planes over the lines in the Klein model. Projecting from a point on the equator to the vertical plane tangent at it andipodal point produces the standard UHP model, while projecting from the North Pole to the plane of the Klein disk produces a disc twice as large where the vertical semicircles go to the circles orthogonal to the boundary of the Poincaré Disk.
This applet shows the stereographic projection of the hemisphere model to the Poincaré Disk model of the hyperbolic plane. Choosing an angle c will rotate a vertical circle in the hemisphere giving a corresponding rotation in the disc. The projection onto the Poincare disk give geodesics in the Poincaré Disk. Changing the value of a will change the radius of the circles on the hemisphere and in the disc.