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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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image3D model vision graphics

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3D model image vision graphics ! ?

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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 14 th 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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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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Ascending Descending (1960) Penrose also devised the impossible staircase

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Why are Objects Impossible? (Huffman: Impossible Objects as Nonsense Sentences, 1971) Line Labeling Inconsistency +Convex edge. -Concave edge. ^Occluding edge (surface to the right). Apparent contour (surface to the right). ^ ^ Shigeo Fukuda

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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 Simanek’s Ambiguous Ring

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The Impossible Stall: The basis of Escher’s 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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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. Algorithm

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Challenges… Model by Shigeo Fukuda Non-even symmetry Line labeling inconsistency Impossible stereo/autostereograms. Impossible shading/lighting. Impossible motion.

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A computer model of Esher’s “High Low” by Sascha Ledinsky rendered in POV-ray.

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