## 1 Introduction
### Research Motivations: Invariants and Their Applications in Motion Perception and VR  
The first kind of invariant can be demonstrated by showing that whenever a body is moving, there is a unique axis at every instant, designated the instantaneous screw axis (ISA), about which the body twists with a specific pitch \(h\). Note that every point on the body has a velocity vector tangential to the helix that passes through it (see Figure 1a). The pattern of these velocity vectors is described as a _helicoidal vector field_. The ISA is often referred to as the _motion screw_[9]. It can, thus, be argued that the body in Figure 1b twists _at the instant about its motion screw_. To describe this field, we may imagine that all helices of the same pitch as those of the ISA are drawn concentrically with the ISA. Theoretically, there must be a helix at every radius from zero to infinity, and there must be an infinite number of helices at each radius, each displaced from its neighbor at an infinitesimal distance in the axial direction [9]. The motion of objects produces a helicoidal vector field \(V_{1}\), \(V_{2}\), \(\ldots\), \(V_{5}\), which manifests perspective changes in a part of the optic array (Figure 1b). The second kind of invariant, the _motion perspective_, may be derived from the movement of an object through the use of the moving reference frame. The motion perspective denotes the optical flow pattern in the visual field resulting from the observer's own movement, in fact, the observers' own locomotion, and its direction [6]. While working on visually guided airplane landings, [PERSON] identified a perceptual invariant that could be observed in animals. An animal moving within its environment will create an optic flow with what [PERSON] termed a \"perspective transformation of perspective flow structure\". This structure can be conveniently described in a vector field \(V_{1}\), \(V_{2}\), \(\ldots\), \(V_{5}\), [5] (see Figure 2). The point toward which the animal is heading is the source of all optical motion vectors, and the point away from which the animal is heading is the sink for those vectors. This condition results in a pure translation parallel to the animal's heading, with the pitch being infinite (Figure 3). The screw theory has numerous points of connection with certain geometrical researchers on the linear complex by [PERSON] and [PERSON] [7]. In the analytical treatment by [PERSON] of the geometry of space, the point is the fundamental element of space. In contrast to [PERSON] and [PERSON], it is the line that serves as a space element. If only one condition is imposed, the lines that fulfill it are said to form the field lines [10]. Consequently, the field lines in the optic array flow are the tangents to a series of helices, with each helix being on a cylinder of radius r and an axis of the field lines (see Figure 1a). The axis of the field is conditioned as the ISA. Figure 1: **(a) Illustration of the helicoidal velocity field: A twist of pitch \(h=\tau/\omega\) about an instantaneous screw axis provides a complete description of the velocity of a rigid body. Every point of the body has its velocity vector tangential to the helix that passes through it. This velocity vector pattern is a helicoidal velocity field. (b) The helicoidal field of linear velocity for any point on a rigid body results from general motion. At any moment, the net velocity \(V_{i}\) (\(i=1,\ldots,5\)) consists of two parts: (1) translational velocity parallel to the screw axis; and (2) tangential velocity representing the rotational body motion about the screw axis. **  
Figure 2: [PERSON]’s representation of the optic flow at moving points of observation: The center of the flow pattern during the animal’s forward movement is the field’s axis. Each vector \(V_{i}\) (\(i=1,\ldots,5\)) represents the optical motion of the corresponding environmental element. All vectors radiate outward from a focus of expansion in relation to the animal’s virtual heading.