A 3D scene and 3D objects in the scene can be drawn with contours in a 2D image and people can perceive their 3D information from the contours (e.g. Cole et al., 2009; DeCarlo, 2012; Elder, 2018; Hertzmann, 2020, 2021; Pizlo et al., 2014; Sayim & Cavanagh, 2011; Farshchi et al., 2021; Sawada et al., 2022; Walther & Shen, 2014; Wilder et al., 2018). These contours represent image features in a photographic view of the scene. The contours also represent characteristic features in the 3D information of the scene and of the objects, such as self-occluding boundaries of the objects (DeCarlo et al., 2003; Koenderink & van Doorn, 1976, 1982; Koenderink, 1984) and the ridges and sharp edges on the surface (Judd et al., 2007; Todd, 2004).

A cusp of a curve is an important feature of the curve for the visual system. The visual system is sensitive to the cusp (Andrews et al., 1973; Dvoeglazova et al., 2021; Tyler, 1973; Uchikawa & Andrews, 1980; Watt, 1984) and makes use of the cusp for organizing contours (Koenderink & van Doorn, 1979; Kogo et al., 2010, see also Latecki & Rosenfeld, 1998), for determining figure-ground assignment (Kanizsa & Gerbino, 1976; Peterson & Salvagio, 2008), and for perceiving the shape of an object and for recognizing the object from a contour-drawing of the object (Attneave, 1954; Pizlo et al., 2005, 2010; Koenderink & van Doorn, 1979).

It is intuitively obvious to consider that a cusp of a curve in a 2D projection of a 3D scene can be attributed to an angular feature contained in the 3D information of the scene. But, there is an interesting case in which a smooth space curve in a 3D scene, which is C2 everywhere, is projected to a 2D curve with a cusp at which the curve is continuous but it has no derivative and its tangent vector turns for 180° like the Greek letter upsilon ‘ϒ’ (Fig. 1, Koenderink, 1990; 1984). Note that a 2D curve that is a projection of a space smooth curve can also have a cusp of an arbitrary angle if the space curve is C2 everywhere except for some isolated points at which the curve is only C1 (see Section Theorems). The 2D curve with the cusp can be a projection of a space smooth curve but it is rarely perceived in this way. This can be attributed to a fact that the space smooth curve can only be accidentally projected to the curve with the cusp (Nakayama & Shimojo, 1992, see Section Theorems). This accidental cusp is an interesting case because this feature in a 2D image accidentally “appears” while many other accidental cases are about features in 3D information that accidentally “disappear” (Freeman, 1994; Dickinson et al., 1999; Sawada, Li, & Pizlo, 2015; Albert, 2001). The accidental cusp appeared in some of Sugihara's visual illusions of 3D shapes (Sugihara, 2016, 2022; Sugihara & Pinna, 2022). Note that the 3D shapes of these illusions are generated by using an algorithm that makes use of the rotational-symmetry of the 3D shapes (Sugihara, 2016, 2022, see also Sawada et al., 2011, 2014; Sawada & Zaidi, 2018; Sawada, 2020). The accidental cusps of these shapes can be regarded as unintentional productions of the algorithm and this seems to have been overlooked (Sawada & Dvoeglazova, 2023). Understanding this accidental feature will help create new illusions by controlling this accidental feature (Fig. 2, see https://tadamasasawada.com/demos/accidental/ for an interaction demo).

In this study, we will discuss a case in which a smooth curve in 3D space is projected to a 2D curve with a cusp under both perspective and orthographic projections. The angle of the accidental cusp can be arbitrary. We will show that such a case can only happen accidentally (Theorem 2.3 and 2.7). We will also show the necessary and sufficient conditions for the case (Theorems 2.5, 2.6). This manuscript of the study is structured as follows. The Geometrical terms that are used in this study are defined in the “Definitions” section. In the “Proofs” section, a 2D orthographic projection of the smooth curve is discussed first (Theorems 2.3 and 2.5 in the “Orthographic projection” section) and this discussion is generalized to a 2D perspective projection of the curve (Theorems 2.6 and 2.7 in the “Perspective projection” section).