Based on the results of the study of convex object motion1 (J. Hopcroft and G. Wilfong, “Motion of objects in contact,” Int. J. Robot. Res., 4(4), 32–46 (1986)), this paper addresses the problem of exact collision detection of a pair of scaled convex polyhedra in relative motion, and determines the contact conditions of tangential contact features, arbitrary relative motion involving translation and rotation, and uniform scaling of the objects about a fixed point. We propose a new concept of the decision curve based on analytical contact equations that characterize a continuum of scaling factors (or a single scaling factor), which ensures that a pair of objects undergoing a scaling transformation will maintain the same tangential contact feature pair (or make instantaneous tangential contact feature transitions). We propose a reliable simulation-based approach to construct the decision curve by hybridizing analytical contact equations and conventional collision detection method, called the Fast Collision Detection Method (FCDM). This method can determine whether two scaled objects will make contact at specific tangential contact features (vertices, edges, or faces) under particular uniform scaling factors and after distinctive relative motion with better accuracy and less computational time than the existing collision detection methods. Finally, we demonstrate our approach for solving motion design in simple assembly/disassembly problems.

Contextual influences shape our perception of local visual stimuli. Relative-motion stimuli represent an important contextual influence, yet the mechanism subserving relative-motion computation remains largely unknown. In the present work, we investigated the responses of an established model for simple and complex cells to relative-motion stimuli. A straightforward mathematical analysis showed that relative-motion computation is inherent in the nonlinear transformation of the complex-cell model. Tuning to relative velocity is achieved by applying a temporal filter to the complex-cell response. The mathematical inference is supported by simulations that quantitatively reproduce measured complex-cell responses in both cat and monkey to a variety of relative-motion stimuli. Importantly, the posited mechanism for cortical computation of relative motion does not require an intermediate neural representation of local velocities and does not require lateral or feedback interactions within a network.

Vernier onset/offset thresholds were measured both psychophysically and with the steady-state VEP by introducing a series of horizontal breaks in a vertical square-wave luminance grating. Several diagnostic tests indicated that the first harmonic component of the evoked response generated by periodic modulation of offset gratings taps mechanisms that encode the relative position of spatial features. In the first test, a first harmonic component was only found with targets that contained transitions between collinear and noncollinear states. VEP vernier onset/offset thresholds obtained with foveal viewing were in the range of 15–22 arc sec. Control experiments with transitions between symmetrical, noncollinear patterns (relative motion) did not produce first harmonic components, nor did full-field motion of a collinear grating. A second series of experiments showed that VEP thresholds based on the first harmonic component of the vernier onset/offset response had an eccentricity dependence that was very similar to that found in a psychophysical discrimination task that required a left/right position judgment (vernier acuity). Other recordings showed that the first harmonic of the vernier onset/offset VEP was degraded by the introduction of a gap between stimulus elements, as is the displacement threshold. The vernier onset/offset target also produced a second harmonic component that was virtually identical to the one produced by a relative motion stimulus. Displacement thresholds based on these second harmonic components showed a more gradual decline with retinal eccentricity than did the first harmonic component elicited by vernier offsets. The second harmonic of the vernier onset/offset VEP was relatively unaffected by the introduction of gaps between the stimulus elements. The first and second harmonic components of the vernier onset/offset VEP thus tap different mechanisms, both of which support displacement thresholds that are finer than the resolution limits set by the spacing of the photoreceptors (hyperacuity).