Animating Impossible Figure from Twisted Torus to Penrose Stairs

Kana Nakatsu*
Tokyo Denki University

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Tokiichiro Takahashi†Tokyo Denki University
UEI Research

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Tomoaki Moriya‡
Tokyo Denki University

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Figure 3. Rotation of Penrose Stairs

1. Introduction
When a start prism and an end prism’s vertices are in the same values
on the projection screen, the shape model is viewed as an impossible
figure. Therefore, if we can create the optical illusion object according
to an arbitrary viewpoint, it is possible to create impossible figure thatis viewed from the arbitrary viewpoint. We have already proposed anddeveloped a method which animates impossible rectangle twisted torus
figure from any viewpoint interactively and in real time[Nakatsu,et al.
2012]. Although the method of creating an optical illusion based on theoptical illusion objects is a simple and classic method, this is powerful
and robust all the more. Even when a viewpoint is moved to a wide
range of area, the animation of an impossible figure can be created. We
have extended the application of this principle to Penrose Stairs, andthis paper presents that it was successfully realized when the viewpoint
was moved freely.
2. 3D Shape Model for Optical Illusion Object
2.1 3D Shape Model
Three dimensional shape model as the optical illusion model to be
used in our method arranges 4 or more prisms parallel to X, Y, and Zaxis. There, each prism is connected vertically.
Two prisms connected with each other are fundamentally called 'startprism' and 'end prism' respectively. The start prism, which is the startof the object, is arranged along Y-axis, and the end prism, which is theend of the object, is arranged along X-axis. There is a gap between thestart prism and the end prism, and they are not connected (Figure 1(b)).
The viewpoint where the start prism and the end prism look connected
is called an optical illusion viewpoint. Here, the optical illusion can beseen as an impossible rectangle twisted torus figure (Figure 1(a))
We call the end prism and the prism connected to it as 'flexible
quadrangular prism' (‘flexible prism’ in short). We transform the shapemodel into the optical illusion object by stretching the flexible prisms
according to the given viewpoint.

viewpoint is expressed as, wherethat the projection method is parallel projection.

. Note3. Penrose Stairs
Penrose Stairs (Figure 2(a)) is a staircase shape impossible figure thathas a property to continue ascending and descending. This figure is amore complicated shaped impossible figure that is derived impossible
rectangle twisted torus figure (Figure 2(b)). We can model an optical
illusion object of Penrose Stairs (Figure 2 (c)) based on our method [1].
Thus, a rotating animation of Penrose Stairs can be created.
We have made a rotating animation of impossible figure of Penrose
Stairs that maintained the nature of impossible figure at all times,
which was made possible by applying our proposed method to theoptical illusion objects as its components. Figure 3 shows theapplication result at each
. We used the post effect for an
ideal line drawing result. We have also created an interactive animation
system for an impossible figure of Penrose Stairs. See the attachedmovies.
However, our proposed method as it stands is unsuitable for Penrose
Stairs figure because its optical illusion object has a depth gap thatmakes distortion when the range of θ is(the part shownwith slanting lines in Figure 2 (c)). To add to our proposed method, weneed to devise the way to show the bottom of the prism being incontact with the ground in a line, such as moving the bottom position
to Y- axis direction.

Figure 2. An Optical Illusion Object of Penrose Stairs Based onImpossible Rectangle Twisted Torus.

Figure 1. An Optical Illusion Object for An Impossible Rectangle
Twisted Torus Figure.
2.2 Illusion Viewpoint and Projection
The sphere containing an optical illusion object is called 'boundingsphere'. The illusion viewpoint moves on the upper half of thebounding sphere except the pole. The illusion viewpoint looks at thecenter of the sphere all the time. The illusion viewpoint is given by alatitude , and longitude on the bounding hemi-sphere. The illusion
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4. Conclusion
We have applied the method of creating an optical illusion based on
the optical illusion objects to Penrose Stairs, and realized theinteractive rotating animation of impossible figure. We have shownthat our proposed method can be generally used to the impossible
figures derived from impossible rectangle twisted torus figures. Thismethod can also be applied to stereoscopic vision.
References
K. Nakatsu, T. Takahashi, T. Moriya, “A stereoscopic representation ofimpossible rectangle twisted torus figure”, ACM SIGGRAPH 2012
Posters, Article 2 (2012).