Animating Impossible Objects

1. Animating Impossible Objects Peter Kovesi and ChihKhoh School of Computer Science & Software Engineering The University of Western Australia

2. An impossible figure is a two-dimensional image that is interpreted to give the impression of some three-dimensional object that cannot exist.

3. vision graphics 3D model image

4. ! vision 3D model ? graphics image

7. Impossible shading...

8. Uccello: The Battle of San Romano ~1430

9. Uccello: The Hunt ~1460

10. Uccello Drawing of a Chalice

11. False Perspective, William Hogarth (1753)

12. Giovanni Battista Piranesi 14th Prison (1760)

13. Swedish artist Oscar Reutesvard was the first to intentionally construct impossible figures. He devised this version of the impossible tri-bar in 1934

14. Oscar Reutesvard

15. Oscar Reutesvard

16. 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)

17. Penrose also devised the impossible staircase Ascending Descending (1960)

18. 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)

19. But some impossible objects can be labeled consistently…

20. 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

21. Aspect Graph of a Cube An impossible object can result from the simultaneous presentation of two distant aspects of an object.

22. “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…

25. Impossible triangle by Mathieu Hamaekers

26. 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 …

27. Constructing Impossible Figures via Complementary Halves An impossible rectangle and its two halves, each of which are globally consistent

28. One complementary half can be obtained from the other via reflections across two orthogonal axes

29. An impossible rectangle can also be created by reversing the visibility of the faces on one half of a possible rectangle

30. The Necker Cube and its two interpretations

31. Donald Simanek’s Ambiguous Ring

32. The Impossible Stall: The basis of Escher’s Belvedere

34. Model of Belvedere by Shigeo Fukuda

35. Model of Belvedere by Shigeo Fukuda

36. Model of Waterfall by Shigeo Fukuda

37. The Crazy Crate

38. Mathieu Hamaekers and his model of an impossible crate

39. Rotating the Impossible Rectangle

40. 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

41. Algorithm • 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.

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

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