Animating Impossible Objects
Peter Kovesi and Chih Khoh
School of Computer Science & Software Engineering
The University of Western Australia
An impossible figure is a two-dimensional image
that is interpreted to give the impression of some
three-dimensional object that cannot exist.
vision
graphics
3D model
image
!
vision
3D model
?
graphics
image
Impossible
shading...
Uccello: The Battle of San Romano ~1430
Uccello: The Hunt ~1460
Uccello
Drawing of a Chalice
False Perspective, William Hogarth (1753)
Giovanni Battista Piranesi 14th Prison (1760)
Swedish artist Oscar Reutesvard was the first to
intentionally construct impossible figures.
He devised this version of the impossible tri-bar in 1934
Oscar Reutesvard
Oscar Reutesvard
In 1958 Penrose independently
devised the impossible tri-bar and
published a paper (with his dad) in
the British Journal of Psychology.
Correspondence between Penrose
and Escher resulted in this image
Waterfall (1961)
Penrose also devised the
impossible staircase
Ascending Descending (1960)
Why are Objects Impossible?
Line Labeling Inconsistency
+
^
^^
Convex edge.
Concave edge.
Occluding edge (surface to the right).
Apparent contour (surface to the right).
Shigeo Fukuda
(Huffman: Impossible Objects as Nonsense Sentences, 1971)
But some impossible objects can be labeled consistently…
The Aspect Graph
(Koenderink and van Doorn 1979)
Nodes: Generic views, or aspects of an object.
Edges: Possible transitions between aspects.
Aspect graph of a tetrahedron
Aspect Graph of a Cube
An impossible object can result from the simultaneous
presentation of two distant aspects of an object.
“An impossible figure is a two-dimensional image
that is interpreted to give the impression of some
three-dimensional object that cannot exist.”
But some impossible 3D objects are possible…
Impossible triangle by
Mathieu Hamaekers
A 3D model must be handcrafted to suit the viewpoint.
A computer model has an advantage in that it can be
continuously adjusted to suit the viewpoint …
Constructing Impossible Figures via Complementary Halves
An impossible rectangle and its two halves,
each of which are globally consistent
One complementary half can be obtained from the
other via reflections across two orthogonal axes
An impossible rectangle can also be created by reversing the
visibility of the faces on one half of a possible rectangle
The Necker Cube and its two interpretations
Donald Simanek’s Ambiguous Ring
The Impossible Stall:
The basis of Escher’s Belvedere
Model of Belvedere by
Shigeo Fukuda
Model of Belvedere by
Shigeo Fukuda
Model of Waterfall by
Shigeo Fukuda
The Crazy Crate
Mathieu Hamaekers and his model of an impossible crate
Rotating the
Impossible Rectangle
Animation Requires Continuous Modification of the 3D Model
Failure to adjust thickness during rotation produces halves that cannot be joined
Note how the bars of this crazy crate must be non-square to allow joining
Algorithm
1.
2.
3.
4.
5.
6.
7.
Construct 3D model of one half of the object (origin at the centre point of join).
Orient it to the desired view.
Project into the image plane (orthographic projection).
Calculate projected widths of surfaces to be joined.
Rescale widths of corresponding surfaces on the 3D model to allow joining in 2D.
Construct second half by negating X and Y coordinates (Z values unchanged).
Add lines to the 2D image to ‘fix’ the join as necessary.
Challenges…
Impossible stereo/autostereograms.
Impossible shading/lighting.
Impossible motion.
Non-even symmetry
Line labeling inconsistency
Model by Shigeo Fukuda
A computer model of Esher’s
“High Low” by Sascha Ledinsky
rendered in POV-ray.