Distortion Minimization: A Framework for the Design of Plane Geometric Anamorphosis
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Abstract
Anamorphosis, as a drawing, represents shapes on a surface such that they appear in their natural form only under specific viewing conditions. Although anamorphoses are mainly studied in a historical context, they are currently experiencing a revival. Plane geometric anamorphoses are a specific subtype of anamorphic drawings. Some practical problems may arise during the design and realization of plane geometric anamorphosis causing the 3D illusionistic effect to be impaired. The aim of this paper is to identify and analyze these problems. In the paper we use parametric analysis to quantify the distortion that may appear because the point of view is offset from the preferred point of view, and to simulate the deviations that can appear because of the errors in onsite realization. The analysis leads to a framework for the design of plane geometric anamorphosis that minimizes the impairment of the anamorphic illusion.
Keywords
Anamorphosis Geometry Perspective Parametric analysisIntroduction
The human visual system relies on retinal 2D images to perceive a 3D shape. The relationship between the information in the images perceived by the retinas and their realworld sources may cause shape ambiguity (Pizlo 2008, pp. 21–27; Schröter 2014, pp. 38–51). In the perception of geometry, the spatial properties of 3D objects are projected onto a plane. Hence, people often do not experience a veridical representation of the space surrounding them. This phenomenon is often referred to as an ‘illusion’ (Howe and Purves 2005, pp. 4–24).
Anamorphosis is an illusion that is intentionally created in order to simulate a virtual spatial structure. It represents shapes such that they appear in their natural form only under specific viewing conditions (an unusual position of the observer, direction of viewing, or the reflection from a curved mirror are the most common scenarios). While there is no difference between anamorphosis and a perspective image in the geometric construction, the perception of a plane anamorphic picture differs from the perception of a perspective image. The crucial difference is that the anamorphic picture is distorted and to see the 3D illusion the direction of the observer’s view has to be oblique to the plane in which the picture is drawn.
Development of Plane Anamorphic Pictures
Anamorphic pictures appear in European art and architecture shortly after the development of linear perspective geometric constructions in the period of the Renaissance (Veltman 1986, pp. 2–21). Anamorphoses were used in many different forms and for different reasons, most often to simulate nonexisting spaces or to transmit secret messages (Hagi 2002, pp. 68–69; Castillo 2001, pp. 1–17). One of the most famous historical examples of anamorphic pictures, The Ambassadors by Hans Holbein, has been studied by many authors (Sharp 1998, pp. 157–165; Hagi 2002, pp. 61–68; Collins 1992, pp. 73–80).
Although anamorphoses are mainly studied in a historical context (PérezGómez and Pelletier 1977; Andersen 2007; Massey 2007, pp. 124–126), they are currently experiencing a revival (Topper 2000, pp. 115–124) and again finding favor, probably due to the ease of producing anamorphic images using computer graphics modeling (Di Paola et al. 2015, pp. 253–285; Hansford and Collins 2007, pp. 214–222; Solina and Batagelj 2007, pp. 1–4; De Comite 2011, pp. 33–38, Čučaković and Paunović 2015, p. 620), and the strong illusionistic effects which may be created. This study is different from other recent and relevant research on anamorphosis because it quantifies distortion and deviation for the specific type of anamorphic pictures using parametric analysis and simulations.
Anamorphic images appear in a variety of media such as architectural and urban design, industrial design, scenic design, and art for the purposes of revitalization, decor, advertising as well as other related areas. Its application varies in scale from smaller examples in fine art to the pictures that occupy large urban spaces. Anamorphoses can be drawn on complex surfaces or spread over several different planes. In the urban space, anamorphoses are most often drawn onto the horizontal plane, since in the urban space wide areas, such as streets and squares, are ideal for creating an illusionistic anamorphic effect for an observer standing at ground level. Such anamorphoses, created in an urban context became extremely attractive with the development of social networks since the urban locations of the anamorphoses allow people to interact with the image and the illusionary space and make creative photographs.
Plane Geometric Anamorphoses
In this paper we discuss plane geometric anamorphoses. The term geometric anamorphosis is used in this paper to refer to a particular style of picture, an anamorphic picture with geometrical motifs, which consists of solid color areas shaped like geometric figures.
In this paper we will consider the anamorphic pictures in a single plane (plane anamorphosis). The ranges of parameters are set to be representative of anamorphosis at an urban scale. We assume that the anamorphic image is in the same horizontal plane upon which the observer is standing. The observer can experience the anamorphosis via eyesight or a camera. Binocular vision is ignored in this paper, and we assume that the anamorphosis is viewed from a single point (one eye or camera). This point is the observer’s point of view.
The observer’s point of view needs to be at a specific position in relationship to the picture in order to perceive the illusionary spatial structure. This point of view is the center of the perspective projection and we will refer to it as the preferred point of view. The orthogonal projection of the preferred point of view onto the picture plane is the preferred standpoint. The line that contains the observer’s point of view and the point on the picture plane onto which the observer is looking represents the direction of the observer’s view and this line is the direction of view.
There are two main reasons that cause the observer to experience an inaccurate shape from an anamorphic projection: point of view offset and picture errors. To distinguish between these two causes of impairment in the illusionistic effect, we will use the terms distortion and deviation. If the observer is offset from the preferred point of view the anamorphic picture will appear distorted to them and we will refer to this mismatch as a distortion of the picture. On the other hand, if the picture is not accurately realized onsite, the mismatch will be referred to as deviation of the picture.
Since an anamorphic image has to simulate 3D space using a 2D image it is very important to emphasize lines that simulate the direction perpendicular to the picture plane. If such lines are not emphasized then the 3D effect is impaired and the picture looks flat. Assuming that the picture plane is horizontal, the main direction to be emphasized in an image is the perspective projection of verticals. Lines that simulate the vertical direction in an anamorphic picture, but actually lie in the picture plane, will be referred to as imaginary verticals. Lines that are perpendicular to the picture plane (lines on the surrounding objects) will be referred to as real world verticals.
Common Problems in Perceiving Geometric Anamorphosis
The effect of a threedimensional illusion of anamorphosis depends significantly on the parameters of the picture and on the accuracy of realization. If the design of an anamorphic picture is not planned carefully then the illusion of a 3D shape can be impaired. The aim of this paper is to identify and analyze problems that need to be considered during the design of a plane anamorphosis in order to successfully create the 3D illusion effect.
If the observer’s point of view is offset from the preferred point of view the distortion will be apparent and this is most apparent in the imaginary verticals. Parameters such as the width of the picture, the direction of view and the distance of observer from the picture all influence the intensity of the distortion.
Even if these picture parameters are carefully chosen, errors in realization can cause deviation of geometric anamorphosis. The position of the picture elements and the method of realization both influence apparent deviation in realized pictures.
In this paper a parametric analysis is used in order to provide a quantitative measure of the distortions and deviations of plane anamorphic pictures in relationship to the picture parameters. Such analysis is important because it reveals a methodology that decreases the potential distortion and deviation before realization and thus minimizes the impairment of the 3D illusionistic effect in the realized picture.
Method
All necessary analyses were performed in the Rhinoceros environment. We used the Grasshopper plugin for visual programming in order to perform a quantitative parametric analysis and to simulate the process of realization.
In order to quantify the distortion of the picture that appears because of the point of view offset we created an algorithm that generates automatic measurements of the mismatch between the imaginary verticals and the real world verticals for a given set of parameters. Data that influences the distortion (the offset of the observer’s point of view and parameters of the anamorphic picture) was used as input data for the algorithm. The output data was the quantified distortion of the picture as it appears to the observer.
Another algorithm was used to quantify the influence of the errors that may appear during the realization. For each point in the picture plane, the input data is the point offset in the picture plane and the output data is the ratio between the error in the picture plane and the error as seen by the observer.
Simulating the realization process was done using a third algorithm, which automatically creates random offsets of specified points within the defined range. The result is a deviated anamorphic picture.
Point of View and Geometric Anamorphosis
Anamorphoses may be used to create an impressive representation of a 3D shape under perfect observing conditions. The dual nature of the visual perception allows us to see a rigid projection and recognize the shape constancy in perspective pictures at the same time (Gibson 1979, pp. 151–160) even if the observer is not in a preferred point of view. In plane anamorphic pictures that are designed to have one preferred point of view the 3D effect is perceivable only from the point that is close to it.
One of the variations that have to be considered in anamorphoses is that the real point of view (which can be observer’s eye or camera) is often not exactly aligned with the preferred point of view. People vary in height and will often slightly move from the predetermined standing location. Offsetting of the viewpoint causes a distorted perception of an anamorphic picture, which impairs the illusion of 3D shape. These distortions can be easily noticeable in geometric anamorphoses because of the domination of regular shapes and straight lines. Therefore, awareness of the intensity of the distortion is very useful during the design stage.
The most critical distortion in viewing occurs when the observer is either in front of or behind the preferred point of view. The problem occurs because the imaginary verticals appear skewed to the observer, which causes the effect of a 3D shape illusion to be impaired. If the direction of view would be parallel to picture plane, then the vanishing point of real world verticals (\(V_{v1}\)) would be at infinity and verticals would appear as mutually parallel. Since the direction of view (d) is always pointing below the horizon plane, the vanishing point of real world verticals (\(V_{v1}\)) is below the observer, which is why all verticals appear to intersect below the scene.
If the observer moves forward from the preferred point of view, then the imaginary verticals will appear as if their upper parts are skewed towards the top of the picture. Since all verticals are skewed to the other side (they intersect below the scene) this distortion is not very apparent. A much less favorable scenario occurs when the observer moves back from the preferred point of view. Then the upper parts of the imaginary verticals are skewed towards the bottom of the picture and hence emphasize the skewing effect that appears because the vanishing point of the verticals is below the horizon. This distortion is the most noticeable one, and this is why it will be analyzed in more detail. It is important to plan the anamorphosis carefully during the design stage in order to minimize its effect.

direction of view,

width of the picture,

distance of the picture (or part of the picture).
The algorithm uses these input data (position of O _{1}, \(\varphi\), position of D and position of vertical \(v_{r}\)) and calculates the list of angles between the perspective projection of real verticals and imaginary verticals to plane \(\gamma\). These angles are used to quantify the distortion. For any particular setting of parameter values we can obtain the average angle of distortion in the verticals. We used the average angle instead of a list of angles, because we want to observe distortion for the picture as a whole, as the observer perceives it. By varying one parameter we obtain a range of average distortion values. For any particular parameter the changes made in the average distortion by varying the parameter will be considered an additional measure of distortion, with the difference between the smallest and the largest of these the range of distortion.
In order to analyze the influence of each parameter on the distortion, we varied each parameter while the other parameters were set to their default value (default values are: O _{1} is 0.3 m behind O, D is at a distance of 3.5 m, \(\varphi = 50^{ \circ }\), \(v_{r}\) is 3 m from preferred standpoint).
This parametric analysis is important because the distortion of the skewing verticals can be avoided during the design stage of the project if the designer takes into account the relationship between the picture parameters, such as width of the picture, distance of the picture to the observer and picture’s point of interest which guide the direction of view. The algorithm can be used for any plane anamorphic picture to check the distortion for an expected point of view offset. Nonoptimal parameters can be compensated by another parameter, in correspondence to the relative distortion range.
Project Realization
When the design stage is complete, even if the project is designed to avoid distortions, inaccuracy of the anamorphic picture may appear. Onsite errors of a few centimeters in the picture plane are very common for large scale anamorphoses. Deviations that emerge onsite during the project realization in this way can occur due to the method used and/or due to the precision of the realization.
Transformation of the concept from a small scale to a large scale (especially spaces such as an urban space) using the grid construction method may be inaccurate and may not provide satisfactory results for geometric anamorphosis, because geometrical relationships can easily be disrupted. Accurate construction onsite with the use of the vanishing points and other characteristic picture elements provides results in which the errors in geometric relationships, such as parallelism, are less apparent.
Another type of picture deviation is caused by imprecision during the realization. Such errors cause the offset of the points in the picture plane and the bending of the straight lines. In this case the distance of the point offset in the picture plane appears smaller to the observer. The factor of shortening (r) represents the ratio between the length of the line in the real world and the length as it appears to the observer. The factor of shortening depends on the direction of view, position of the point, and offset direction. The factor of shortening is maximal if the angle between the direction of view and the picture plane is the biggest, the point is close to the point of view, it is away from picture axes, and if the offset direction is perpendicular to the line that connects the point with the standpoint. The area of low offset shortening is not favorable for placing small details in an anamorphic picture, because the point offset error can be noticeable.
Another consequence of the imprecision during the realization is line bending. During realization of a large anamorphosis, it is sometimes hard to construct a perfectly straight line onsite. The appearance of a bulge acts similarly to the point offset as described in the previous section. Besides the shortening factor of the bulge, the shortening factor of the line (the ratio between the length of the line in the picture plane and the line length as it is seen by an observer) influences the appearance of the bending deviation. This factor acts in a similar way to the previous case, which is why it will not be taken into further consideration. Lines have a potential for bending if the shortening factor of the line is large, and the shortening factor of the bulge is small. Therefore if the anamorphosis is to be realized using a low precision method then long and distant imaginary verticals should be avoided.
Case Studies
The detection of major problems described in the previous sections is based on the experiences in realization of urban space geometric anamorphoses during the period between 2010 and 2015. In this section, we present three case studies and use them to illustrate some of the problems and issues previously discussed in this paper as they were encountered in practice.
Amphitheater
In the realization the vanishing points method was used. Arches were constructed onsite using the coordinates of the center and radius. The realization of the project did not cause any deviations.
Cube
In the design of this project possible skewing of the verticals was taken into consideration and the parameters (\(\varphi = 17^{ \circ }\), distance of the closest point of the picture is 3 m and distance of D is around 4.5 m) were chosen to minimize the potential distortion.
In realization the vanishing points methods was used. This was significant for this project because at the end of the realization huge point offsets, up to 20 cm, appeared. In spite of this, the deviation is not obvious in the realized project (Fig. 16b).
City Vistas
In the design of this project the possible skewing of verticals was considered and the parameters (\(\varphi = 19^{ \circ }\), distance of the closest point of the picture is 5 m and distance of D is around 10 m) were chosen to minimize the potential for distortion. Using these parameters the distortions were not obvious even with the large point of view offsets.
During realization the vanishing points method was used, so the imaginary parallelism was not impacted. However, high deviations appeared during realization. The reasons for the deviations were small details in the area with a small shortening factor and imaginary verticals with high shortening factor. The point offset and bending are noticeable in the final project (Fig. 16c).
Conclusion
In the design and realization of plane geometric anamorphosis it is important to consider parameters that may cause distortions and/or deviations of the picture because this can lead to the impairment of the 3D effect. In this paper we have identified and analyzed several problems that cause distortion stemming from an offset point of view and the deviations that can appear during the realization of the project.
The distortion of an anamorphic picture appears when the point of view is offset from the preferred point of view. The situation in which the point of view is behind the preferred point of view is studied in detail because this offset impairs the 3D illusion the most. The parametric analysis revealed that the parameter with the strongest influence on the appearance of this kind of distortion is the width of the picture. The position of the vertical has less influence and the direction of view has the least influence on this kind of distortion. To avoid a mismatch between the imaginary verticals and the real world verticals picture parameters should be considered during the design stage of an anamorphic picture.
Deviations in an anamorphic picture appear because of imprecision during the realization. By simulating different methods of realizing geometric anamorphoses with emphasized imaginary parallelism with the same range of error in realization we have shown that the construction of vanishing points onsite is recommended in order to avoid impairment of parallelism.
The analysis of the parameters that influence the shortening factor can be used in preventing deviation caused by point offset and line bending. This analysis points out the areas of the picture that are least favorable for small details and the characteristics of the lines that are in a high risk of bending deviation.
The limitations of this study are that problems discussed in this paper are general issues and concerns. Since each design is different it is impossible to cover all the possible potential problems and provide adequate solutions. The parametric analysis and simulations described in this paper may need to be revised in order to apply them to other individual projects according to their specific concerns and constraints. This paper analyses the anamorphic picture as a perspective projection, and hence ignores binocular vision. Binocular vision can impair the 3D illusion, especially at small distances. We also only consider anamorphoses drawn on the horizontal plane, and similar studies can be applied to anamorphoses drawn on other projection planes and/or multiple planes, which are considered future directions of this work.
Notes
Acknowledgments
We would like to thank the reviewers for valuable comments and suggestions that improved our paper. This research was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (OI174012).
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