Animating Impossible Objects

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Animating Impossible Objects
Peter Kovesi and Chih Khoh
School of Computer Science Software
Engineering The University of Western Australia
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An impossible figure is a two-dimensional image
that is interpreted to give the impression of
some three-dimensional object that cannot exist.
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vision
graphics
image
3D model
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!
vision
3D model
?
graphics
image
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Impossible shading...
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Uccello The Battle of San Romano 1430
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Uccello The Hunt 1460
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Uccello Drawing of a Chalice
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False Perspective, William Hogarth (1753)
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Giovanni Battista Piranesi 14th Prison (1760)
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Swedish artist Oscar Reutesvard was the first to
intentionally construct impossible figures. He
devised this version of the impossible tri-bar in
1934
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Oscar Reutesvard
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Oscar Reutesvard
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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)
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Penrose also devised the impossible staircase
Ascending Descending (1960)
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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)
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But some impossible objects can be labeled
consistently
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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
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Aspect Graph of a Cube
An impossible object can result from the
simultaneous presentation of two distant aspects
of an object.
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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
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Impossible triangle by Mathieu Hamaekers
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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
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Constructing Impossible Figures via Complementary
Halves
An impossible rectangle and its two halves, each
of which are globally consistent
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One complementary half can be obtained from the
other via reflections across two orthogonal axes
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An impossible rectangle can also be created by
reversing the visibility of the faces on one
half of a possible rectangle
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The Necker Cube and its two interpretations
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Donald Simaneks Ambiguous Ring
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The Impossible Stall The basis of Eschers
Belvedere
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Model of Belvedere by Shigeo Fukuda
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Model of Belvedere by Shigeo Fukuda
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Model of Waterfall by Shigeo Fukuda
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The Crazy Crate
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Mathieu Hamaekers and his model of an impossible
crate
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Rotating the Impossible Rectangle
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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
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Algorithm

1. Construct 3D model of one half of the object
   (origin at the centre point of join).
2. Orient it to the desired view.
3. Project into the image plane (orthographic
   projection).
4. Calculate projected widths of surfaces to be
   joined.
5. Rescale widths of corresponding surfaces on the
   3D model to allow joining in 2D.
6. Construct second half by negating X and Y
   coordinates (Z values unchanged).
7. Add lines to the 2D image to fix the join as
   necessary.

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Challenges
Impossible stereo/autostereograms. Impossible
shading/lighting. Impossible motion.
Non-even symmetry
Model by Shigeo Fukuda
Line labeling inconsistency
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A computer model of Eshers High Low by Sascha
Ledinsky rendered in POV-ray.
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