Keywords: Impossible Object, Animation, Complementary Half.
An impossible figure is commonly defined as being a two-dimensional image that is interpreted to give the impression of some three-dimensional object that cannot exist in a three-dimensional world. The keywords in this definition are ``impression'' and ``interpretation''. For a figure to be judged impossible (or not) it must first make the impression of some three-dimensional object. Furthermore, the property ``to be an impossible figure'' is not the property of the drawing alone, but the property of its spatial interpretation by a human observer [9].
However, in many cases it is possible to construct a three-dimensional model of the impossible figure. This model will not correspond to the normal interpretation of the figure by a human. Typically the 3D model must be viewed from a particular direction to create the illusion of impossibility. For example, 3D models of Escher's Belvédère [3,10] have been built by Shigeo Fukuda [2] and models of the impossible tri-bar and crazy crate have been built by Mathieu Hamaekers [2]. Illustrations of other models can be found at the IllusionWorks web site [5]. Sugihara [13] describes an interactive system that allows one to generate polyhedra from impossible figures, and for drawing the unfolded surfaces of them. Typically all these 3D models contain a number of gaps or twists. Once revealed, these gaps and twists destroy the illusion of impossibility and the model loses its ``magic'' (figure 1). At this point it is important that we make the distinction between the terms ``impossible figure'', ``impossible object'' and ``3D model''. The 3D model is a physically realizable object, which when projected appropriately into 2D, results in an impossible figure. This impossible figure then gives rise to the perception of an impossible object (different to the 3D model) which can only exist as an interpretation in one's mind.
Note that an interactive impossible object does not have to look like an impossible object from all viewpoints - it is perfectly acceptable for it to look like a possible object from certain angles, but it would be expected to satisfy the conditions above.
We can solve this problem by using a computer model of the object rather than a physical one. The computer provides a tool that allows us to continuously modify a 3D model so that as the viewpoint changes its 2D projection continues to satisfy the various properties that make the figure impossible. The ultimate goal might be to create an impossible world, like the ones depicted by Escher, that people can interact with in virtual reality.
Penrose and Penrose [11] describe impossible figures as follows: ``Each individual part is acceptable as a representation of an object normally situated in three-dimensional space; and yet, owing to false [connections] of the parts, acceptance of the whole figure on this basis leads to the illusory effect of an impossible structure.'' A similar view is taken by Huffman [4] who shows that in general impossible figures will have a locally consistent, but globally inconsistent, line labelling.
Consider the impossible rectangle shown in figure 2. This figure is also known as an impossible four cornered torus or impossible four-bar [1,2]. The key feature of this impossible figure is that it can be divided into two halves, which when viewed independently, correspond to a 3D object that is globally consistent and ``possible''.
We call the projections of the two (globally consistent) halves of the object, the complementary halves of the impossible figure [6]. These complementary halves are related by an inversion transformation. That is, one complementary half can be obtained from the other via two sequential reflections about orthogonal axes in the image plane.
Yet another approach to the construction of complementary halves is to use face visibility reversal of the 3D model of the complementary half. However this approach requires that the impossible object that we wish to construct has an obvious corresponding possible object. This process is best illustrated by example. Figure 4 shows the construction of a possible rectangle from two halves. If we reverse the visibility of the faces of the right half (that is, make the hidden faces visible, and the visible faces hidden) before joining the two halves, we will produce an impossible rectangle. The process of face visibility reversal corresponds to the interpretation switch that is experienced with the Necker Cube (figure 5).
The manner in which the inversion transformation that relates the complementary halves provides two very different aspects of an object can be illustrated via an aspect graph [7,8]. Figure 7 shows the aspect graph of a cube. Each node in the graph represents a generic view, or aspect, of the object. An aspect of the object is defined as a range of views of the object over which the topology of the projected image remains unchanged. The edges in the graph show the possible transitions between aspects. If we take an aspect of the cube that shows the bottom and two side surfaces, and apply an inversion transformation to it (or reverse the visibility of its faces) we will construct a different aspect of the object, some distance away in the graph, corresponding to a view showing the top and two side surfaces. Thus, the merging of two complementary halves can be seen in terms of a simultaneous presentation of two distant aspects of an object within the one figure. This creates the impression of impossibility.
Animation requires the existence of a 3D model that can be translated and rotated into different views. Thus, to create an animation of an impossible figure we must first create a 3D model, which when projected onto the image plane, results in an impossible figure. This 3D model must then be continuously modified in some way, as the viewpoint changes, so that the impression of impossibility is maintained. One approach to this problem might be to employ Sugihara's method of 3D model construction [13]. However, Sugihara's approach requires that the starting point in the process is the impossible figure. Once a 3D model has been obtained it is not clear how one might modify it for a different viewpoint.
Being able to divide an impossible figure into complementary halves greatly simplifies this task of constructing an appropriate 3D model. We only have to construct a model that corresponds to one half of the figure, and typically each complementary half is readily modeled by a simple 3D object. This gives us the basis of an algorithm for animating impossible objects composed of complementary halves. The steps are as follows:
Figure 8 shows the results of rotating an impossible rectangle. The shaded half-rectangle is the 3D model that we are rotating and the white half-rectangle is its complementary half constructed in the image plane.
You may have noticed that the impossible rectangle appears thinner on its side view (the second and third image in the rotation shown in figure 8). This is a side effect of getting the lines of each complementary half to join up. In joining the two halves of the impossible rectangle the top surface of one complementary half has to be merged with the side surface of the prong of the other complementary half. That is, the projected widths of the top and side surfaces must be equal if we are to successfully merge the complementary halves (figure 9). As the object rotates the relative widths of the projections of these two surfaces will vary. To compensate for this we must adjust the thickness of the 3D model, and/or adjust the height of the sides of the prongs, so that the projected width of the top surface will match the side surface. Figure 10 shows this process in the construction of the crazy crate.
Projection is a linear operation. Thus, the rescaling required of the surfaces in the 2D projection to allow the surfaces to be joined will directly correspond to the rescaling required of the surfaces in the 3D model. Note that there will be some views in which an impossible object will be degenerate. These will correspond to views in which the projected width of one of the joining surfaces become zero. No amount of rescaling of the 3D model will make its projected width non-zero to permit a match. The only solution is to form a `degenerate match' by rescaling the 3D width of the surface that has a non-zero projected width, to zero. Another degeneracy arises when we attempt to construct an end view of the object. If one thinks of the complementary halves in terms of the two `U' shaped 3D objects as shown in figure 1 this situation corresponds to the two objects rotating into each other so that they merge and become identical. Examples of these situations can be seen in figure 11.
Overall, the use of complementary halves to describe an impossible figure greatly simplifies the construction of a corresponding impossible object. The constant adjustment of the 3D model that is required with viewpoint changes is reduced to a rescaling of the dimensions of the object being used to model one of the complementary halves. In the case of the impossible rectangle this geometry change manifests itself as a simple thickness and/or height adjustment of the joining prongs. This constant adjustment in the geometry of the 3D model contributes an extra aspect to the impossibility of the figure.
Impossible figures can be animated. To do this there are two problems that have to be solved. Firstly, one has to construct a 3D model that corresponds to the figure, and secondly, one has to identify how the 3D model must change as the viewpoint changes, so that an impossible figure continues to be produced.
This paper introduces the concept of a complementary half of an impossible figure. It is shown how complementary halves can be constructed either by inversion in the image plane or by face visibility reversal. The use of complementary halves offers a systematic way of constructing a particular class of impossible figures. It facilitates the animation of impossible figures by greatly simplifying the construction of impossible objects from their corresponding impossible figures. Only half of the figure has to be considered and each complementary half is readily modeled by a simple 3D object. The constant adjustment of the 3D model that is required to maintain the impossible figure as the viewpoint changes, is reduced to a simple rescaling of the dimensions of the object being used to model one of the complementary halves. This constant adjustment in the geometry of the 3D model contributes an extra air of impossibility to the figure.