This post continues the ones here and here. Realism considers what is perceived with full consciousness as reality. Apperception and reality correspond to each other. The role of theory is to clarify this correspondence, not to deny it. So realists understand observation to be correct, not to be altered by theory. Anti-realism considers what is […]
For some background, see here and here. Velocity is defined as: “The time rate of change of position of a body; it is a vector quantity having direction as well as magnitude.” Speed is defined as: “The time rate of change of position of a body without regard to direction; in other words, the magnitude
A dimension is informally regarded as the number of coordinates needed to specify the location of a body or point. That may suffice for a mathematical dimension, but a physical dimension is a dimension of something, that is, some unit. In that sense, the dimensions of force are different from the dimensions of velocity. However,
Relativity may be derived as an algebraic relation among differentials. Consider motion in the x spatial dimension, with a differential displacement, dx, differential velocity displacement, dv, and arc (elapsed) time t: dx² = (dx/dt)²dt² = dv²dt² = d(vt)². Let there be a constant, c: dx² = d(vt)² = d(cvt)²/c² = d(ct)² (v/c)² = d(ct)² (1 –
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
