An accidental image feature that appears but not disappears

A cusp of a curve in a 2D image is an important feature of the curve for visual perception. It is intuitively obvious that the cusp of the 2D curve can be attributed to an angular feature contained in a 3D scene. It is accidental when a space curve with a cusp in a 3D scene is projected to a smooth curve without any cusp in a 2D image. Note that there is also an interesting case in which a smooth space curve without any cusp is accidentally projected to a 2D curve with a cusp. The angle of the cusp of the 2D curve is arbitrary but it is determined by the shape of the space curve. In this study, we will show the necessary and sufficient conditions that are needed to produce a space smooth curve that is projected to a 2D curve with a cusp under both perspective and orthographic projections. We will also show how the angle of the cusp is determined, and that these conditions are only satisfied accidentally.