George Polya --How to Solve It, 1945 [75]
Goal of the applets
These applets visually illustrates the problem of the project. We are given a line drawing D on a plane, which is supposed to be a projection of a spatial polyhedron. We develop algorithms to solve these two problems:
In this example, we have a drawing of a pyramid(left applet) and a drawing of a truncated pyramid(right applet) -a pyramid whose apex has been removed and replaced by a triangular face. The pyramid case is rather simple, its line drawing is always correct; The truncated-pyramid case is much more complex, the given drawing is correct if and only if the three edges joining the two triangles of the truncated pyramid meet at a common point (the apex of the initial pyramid). You can visualize all possible spatial realizations (or liftings) by cliking the "animate" buttons. Try it now. Each button animates the height of a vertex within a modifyable range. In this case the space of spatial liftings can be fully explored by giving heights to four vertices: P, R, Q and U. The definition of these four vertices(P, R, Q and U) is shown in the following figure:
- Realizability: decide whether D corresponds to the projection of some polyhedron in 3D-Space ?
- Reconstruction: if so, ... what are all the possible shapes for this polyhedron ?
Note that if you slightly move a vertex in the drawing, the above mentioned
concurrence condition of three edges no longer holds and the spatial
realization "breaks" with a gap in the height of a vertex of the upper
triangle. Try to "repair" the drawing again by fulfilling this concurrence.
You can move all red points.
Instructions
- Use mouse to drag the object to change viewpoint
- Click all "Animate" buttons below to see all liftings
Now, it's time to play ! :-)
Play these applets to get a feeling of 3D scene reconstruction
from line drawings, enjoy it ....
