Matters relating to length and duration in physics and transportation
This post follows the introduction to intrinsic coordinates given here, and makes it temporo-spatial (1D space + 3D time). So rather than the 3D space position vector r, we’ll use the 3D time position vector, w, and we’ll switch the arc time, t, with the arc length, s. We follow the motion of a point […]
The mathematical problem is this: given a curve with distance coordinates that are parametric functions of time (duration), find the reparametrization of the curve with duration (time) coordinates that are parametric functions of distance. Symbolically, given the regular curve α(t) = (a1(t), …, an(t)), find β(s) = (b1(s), …, bn(s)) such that bi(s) = t(ai)
The following is a continuation and revision of the previous post, here. Based on the differential geometry part of the book Shape Interrogation for Computer Aided Design and Manufacturing by Nicholas M. Patrikalakis and Takashi Maekawa of MIT. A pdf version in parallel is here. Let a three-dimensional curve be expressed in parametric form as
The following is slightly modified from the differential geometry part of the book Shape Interrogation for Computer Aided Design and Manufacturing by Nicholas M. Patrikalakis and Takashi Maekawa of MIT. A plane curve can be expressed in parametric form as x = x(t); y = y(t); where the coordinates of the point (x, y) of
This post continues the topic of the previous post here. This is a post about two kinds of people. First a warning: There may be said to be two classes of people in the world; those who constantly divide the people of the world into two classes, and those who do not. – Robert Benchley
Let us distinguish between observer-receivers and traveler-transmitters. Although observers can travel and travelers can observe, insofar as one is observing, one is not traveling, and insofar as one is traveling, one is not observing. The main difference is this: traveler-transmitters have a destination but observer-receivers do not (or at least not as observers). Compare the
See also the related post on the Center of vass. Relativity has been addressed before, such as here. Isaac Newton called mass “the quantity of matter”, which is still used sometimes, although Max Jammer points out how it has been criticized for centuries (see Concepts of Mass in Classical and Modern Physics, 1961). Other definitions
One question is how to translate from time rates to space rates and vice versa. Consider the scalars base and time, and designate the stantial position, s, initial stantial position, s0, temporal position, t, initial temporal position, t0, velocity, v, initial velocity, v0, acceleration a (assumed constant over time), lenticity, w, initial lenticity, w0, and
Among the instruments on a vehicle there may be a speedometer, an odometer, a clock, and a compass, which provide scalar (1D) readings of the vehicle’s location. But what is the location of the vehicle in a larger framework? The compass shows two dimensions must exist on a map of this framework, but of what
We need to distinguish between scalar (1D) and vector (3D) versions of both time and space. Motion in scalar (1D) time and scalar (1D) space is measured by clocks and linear references, respectively, and apply throughout the associated vector space or vector time (in Newtonian mechanics). Scalar time is what a clock measures, which is
