Animating Impossible Objects

Animating Impossible ObjectsPeter Kovesi and Chih Khoh

School of Computer Science & Software EngineeringThe 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.

image3D model

vision

graphics

3D model

image

vision

graphics

!

?

Impossibleshading...

Uccello: The Battle of San Romano ~1430

Uccello: The Hunt ~1460

UccelloDrawing 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 imageWaterfall (1961)

Ascending Descending (1960)

Penrose also devised the impossible staircase

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

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 simultaneouspresentation 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 byShigeo Fukuda

Model of Belvedere byShigeo Fukuda

Model of Waterfall byShigeo 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

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

Challenges…

Model by Shigeo Fukuda

Non-even symmetry

Line labeling inconsistency

Impossible stereo/autostereograms.

Impossible shading/lighting.

Impossible motion.

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