Explorations of multidimensional space and time with linear and angular motion.
This post follows others such as the one here and here. A background document is here. One-dimensional kinematics is like traveling in a vehicle, and on the dashboard are three instruments: (1) a clock, (2) an odometer, and (3) a speedometer. In principle the speedometer reading can be determined from the other instruments, so let’s […]
Let the vector r be a displacement vector of motion K that is a parametric function of its arc time t so that r = r(t). Then define s as the arc length of K such that s = s(t) = ∫ || r′(τ) || dτ, where the integral is from 0 to t. Let
This post follows the material on polar coordinates from MIT Open Courseware, here. Instead of the space position vector r, we’ll use the time position vector w, and replace (arc) time with arc length, s. In polar coordinates, the time position of a tempicle A is determined by the value of the radial duration to
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
