Nativity Scene Space

When crafting a Christmas nativity scene, one often desires a way to depict distant figures and objects at a reduced scale. When looking through a window, distant objects appear smaller. If one marks a colored dot on the windowpane where the line of sight passes through, a pleasing image is created. This image corresponds to central perspective, yet it suffers from the disadvantage of being merely two-dimensional. Let b represent the distance between the eye and the window. Let the point where the perpendicular from the eye meets the windowpane serve as the origin:

x'= bx/(z+b), y'= by/(z+b)

Here, *z* represents the distance to the pane. By reflecting the viewpoint and combining the central projection with a suitable parallel projection, a third dimension can be generated:

z' = bz/(z+b)

Since bz/(z+b) = b-bē/(z+b), the point at infinity is mapped onto the reflected viewpoint. While this mapping is not conformal, line segments are mapped to line segments. The half-space of the world is mapped onto a flat plane. Parallel lines angled relative to the pane converge at a horizon circle on the rear pane. This method allows for the creation of a realistic-looking 3D model.

Ludwig Resch