Arithmetic Ops Group Methods for 3vel objects
Ops.3vec.RdArithmetic operations for three-velocities
Usage
# S3 method for class '3vel'
Ops(e1, e2)
# S3 method for class '4vel'
Ops(e1, e2)
massage3(u,v)
neg3(u)
prod3(u,v=u)
add3(u,v)
dot3(v,r)Details
The function Ops.3vel() passes unary and binary arithmetic
operators “+”, “-” and “*”
to the appropriate specialist function.
The most interesting operators are “+” and
“*”, which are passed to add3() and dot3()
respectively. These are defined, following Ungar, as:
$$ \mathbf{u}+\mathbf{v} = \frac{1}{1+\mathbf{u}\cdot\mathbf{b}/c^2} \left\{ \mathbf{u} + \frac{1}{\gamma_\mathbf{u}}\mathbf{v} + \frac{1}{c^2}\frac{\gamma_\mathbf{u}}{1+\gamma_\mathbf{u}} \left(\mathbf{u}\cdot\mathbf{v}\right)\mathbf{u} \right\} $$
and
$$ r\odot\mathbf{v} = c\tanh\left( r\tanh^{-1}\frac{\left|\left|\mathbf{v}\right|\right|}{c} \right)\frac{\mathbf{v}}{\left|\left|\mathbf{v}\right|\right|} $$
where \(\mathbf{u}\) and \(\mathbf{v}\) are three-vectors
and \(r\) a scalar. Function dot3() has special dispensation
for zero velocity and does not treat NA entries entirely
consistently.
Arithmetic operations, executed via Ops.4vel(), are not defined
on four-velocities.
The package is designed so that natural R idiom may be used for three velocity addition, see the examples section.
Examples
u <- as.3vel(c(-0.7, 0.1,-0.1))
v <- as.3vel(c( 0.1, 0.2, 0.3))
w <- as.3vel(c( 0.5, 0.2,-0.3))
x <- r3vel(10) # random three velocities
y <- r3vel(10) # random three velocities
u+v # add3(u,v)
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -0.648977 0.2557545 0.1246803
u-v # add3(u,neg3(v))
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -0.7434641 -0.03267974 -0.2913943
-v # neg3(v)
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -0.1 -0.2 -0.3
gyr(u,v,w)
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] 0.5134003 0.2390541 -0.2434611
## package is vectorized:
u+x
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -0.7841682 0.02734861 -0.50404915
#> [2,] -0.7036563 -0.30641050 -0.36843047
#> [3,] -0.8587346 -0.08670915 -0.21410624
#> [4,] -0.9480249 -0.12030137 0.02246383
#> [5,] -0.3781309 0.71645428 -0.14413320
#> [6,] -0.9691780 0.12138302 -0.13643205
#> [7,] -0.8994853 0.12338245 0.03639300
#> [8,] -0.7238049 0.38382395 0.47428504
#> [9,] -0.7111436 0.29068074 -0.26074149
#> [10,] -0.4477571 -0.31206541 -0.49728634
x+y
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -0.09600868 -0.7179919 -0.36065284
#> [2,] 0.28002674 -0.7598638 -0.51931521
#> [3,] -0.33840429 0.3279208 -0.15803202
#> [4,] -0.56337171 -0.7201825 0.09843981
#> [5,] 0.84478030 0.2635985 0.07562038
#> [6,] -0.85828999 0.3694929 -0.34551186
#> [7,] -0.34587079 -0.1200680 0.76226856
#> [8,] -0.14321273 0.1191482 0.88090820
#> [9,] -0.58013943 0.7018523 -0.17781538
#> [10,] 0.46907486 -0.5528338 -0.46070785
f <- gyrfun(u,v)
g <- gyrfun(v,u)
f(g(x)) - x # should be zero by eqn10
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -5.853023e-16 2.237921e-16 1.377182e-16
#> [2,] 4.847343e-17 -1.551150e-16 1.034100e-16
#> [3,] 4.440762e-16 3.633351e-16 1.412970e-16
#> [4,] -1.130274e-15 1.614678e-16 1.614678e-16
#> [5,] -6.654321e-16 -4.436214e-16 8.317901e-17
#> [6,] 1.449847e-15 2.718464e-16 1.812309e-16
#> [7,] 2.974155e-16 9.559783e-17 1.805737e-16
#> [8,] -8.776910e-16 1.101416e-15 8.260621e-16
#> [9,] -3.322553e-16 -1.200673e-16 4.140253e-18
#> [10,] -3.334777e-16 4.446369e-16 4.446369e-16
g(f(x)) - x
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -5.508728e-16 1.032886e-16 -4.131546e-16
#> [2,] -5.606760e-16 -2.585250e-16 -1.551150e-16
#> [3,] -8.275965e-16 8.074112e-17 -9.083376e-17
#> [4,] -1.130274e-15 0.000000e+00 4.036694e-16
#> [5,] -7.393690e-16 4.436214e-16 7.393690e-17
#> [6,] 5.074466e-15 -2.265387e-16 1.178001e-15
#> [7,] -8.922464e-16 2.443056e-16 5.098551e-16
#> [8,] -8.260621e-16 1.239093e-15 1.101416e-15
#> [9,] 1.749257e-16 -5.382329e-17 1.242076e-16
#> [10,] 0.000000e+00 1.111592e-16 -5.557961e-17
(u+v) - f(v+u) # zero by eqn 10
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -6.132309e-16 1.951189e-16 1.672448e-16
(u+(v+w)) - ((u+v)+f(w)) # zero by eqn 11
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -5.469679e-17 0 -5.469679e-18
((u+v)+w) - (u+(v+g(w))) # zero by eqn 11
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] 3.951349e-16 -2.99021e-16 1.975675e-16
## NB, R idiom is unambiguous. But always always ALWAYS use brackets.
## Ice report in lat 42.n to 41.25n Long 49w to long 50.30w saw much
## heavy pack ice and great number large icebergs also field
## ice. Weather good clear
## -u+v == (-u) + v == neg3(u) + v == add3(neg3(u),v)
## u+v+w == (u+v)+w == add3(add3(u,v),w)