Kinetic space-time prisms

Abstract

The space-time path and prism demarcate the estimated and potential locations (respectively) of a moving object with respect to time. The path is typically formed through linear interpolation between sampled locations of a moving object, while the prism is the envelope of all possible paths between two locations given the maximum speed of travel. The classic path and prism, however, are not physically realistic since they imply the ability of the object to make instantaneous changes in direction and speed without acceleration and deceleration. This is not acceptable in applications where kinetics is vital for scientific understanding such as animal ecology, vehicles moving through media such as ships through water and planes through air, human-powered movement such as bicycling and walking and environmental applications of transportation such as energy consumption and emissions modeling. In this paper we demonstrate how imposing an upper bound on acceleration, as well as information such as the initial speed and heading, affects the geometry of the space-time prism. We discuss how to calculate kinetic paths and prisms in one-dimensional and two dimensional space, and provide examples comparing the kinetic prisms and classical prisms.

Toward a kinetic-based probabilistic time geography

“…However, the relationship between initial velocity (defined here using the 21 ratio of v t to v max ) and γ may be non-linear and alternative definitions of [4] may be 22 warranted provided they maintain γ on the range [-1, 1]. For instance, here we ignore the 23 effect of acceleration (Kuijpers et al 2011), an integral component of object kinetics, in 1 our model definition. A more complete model would include the effect of acceleration on 2 γ and on the variance component (ω).…”

“…As discussed by Kuijpers et al (2011), ignoring object kinetics may be reasonable 16 when estimating broad-scale patterns from finely sampled movement data; for example, 17 when looking at long-term transportation trends. Similarly with coarsely sampled 18 movement data the physical kinetics of movement will not be relevant.…”

“…11 < approximate location Figure 1 > 12 Object kinetics, defined by an objects current speed and direction of movement, 13 along with acceleration, can similarly influence movement opportunities defined by time 14 geography (Kuijpers et al 2011). For instance, in classical time geography, movement 15 boundaries are calculated with the unrealistic expectation that an object can make 16 instantaneous changes in velocity.…”

“…With object kinetics (and other physical constraints) 17 ignored, time geographic structures (i.e., the space-time cone and space-time prism) 18 substantially overestimate movement opportunities. Kuijpers et al (2011) have quantified 19 the influence of object kinetics (from velocity and acceleration) on time geographic 20 boundaries, termed kinetic-based time geography. Consideration of object kinetics 21 provides a more realistic representation of time geography's boundaries, as kinetic-based 22 time geography will exclude locations in space-time not accessible based on an 1 individual's kinetic movement abilities.…”

“…Assuming 19 random movement is a limitation, as most objects move non-randomly with directed, 20 linear movements and often revisit specific locations with regularity (Gonzalez et al 21 2008). 22 Kuijpers et al (2011) identify several lingering questions in terms of kinetic-1 based time geography, the first of which is quantifying unequal movement probabilities 2 in kinetic time geography structures, much like probabilistic time geography. The 3 objective of this research is to develop a model for quantifying movement probabilities 4 for kinetic-based time geography.…”