Description Usage Arguments Details Value Note Author(s) Examples
Functions for conversion of coordinates; rotation matrices for post(pre)-multiplication of row(column) 3-vectors; Vector product(right handed), length of vector, angle between vectors.
1 2 3 4 5 6 7 8 9 10 11 12 13 |
x,y,z,r,theta,phi |
Real, rectangular, spherical coordinates; x, y, z may be combined as c(x,y,z), and r, theta, phi as c(r,theta,phi) |
P |
c(x,y,z), coordinates of point or projection direction |
v, w |
3-vectors (x, y, z). |
N |
Order of the square rotation matrix >= 2. |
k,m |
Integers (m != k) describing the plane of rotation. m==k gives unit matrix. |
M |
3x3 rotation matrix. |
toPol,toRec
: Convert plane rectangular c(x,y) <-> polar c(r,phi); phi = angle(x-axis,point).
toSph, toXyz:
Rectangular c(x,y,z) <-> spherical coordinates c(r,theta,phi); theta = angle(z-axis,P-'O'), phi= angle[plane(P,z-axis), plane(x-z)].
toPol:
c( r, phi ), r=Mod(z), phi= Arg(z); Re(z)=x, Im(z)=y
toRec:
c( x , y ), x=Re(z) , y=Im(z); Mod(z)=r, Arg(z)=phi
toSph:
c(r, theta, phi), r=sqrt(x^2+y^2+z^2), theta=atan2(z,v),
phi=atan2(y,x) ; v=sqrt(x^2+y^2)
toXyz:
c(x, y, z), x=r*sin(phi)*sin(theta), y=r*cos(phi)*sin(theta), z=r*cos(theta)
rotZ:
c(x', y') = rotated (x, y) by angle phi, counter clockwise,
– Rotation matrices:
rotA
: Ratation matrix to rotate around axis P - 'O'
.
rotV
: Ratation matrix to rotate v
into w
.
rotL
: Matrix m
for multiplication m %*% vector
.
getAp
: List with rotation axis and rotation angle corresponding to input matrix.
– Other:
angle
angle between vectors
lV
Euclidean (spatial) length of vector
scprod
scalar product
vecprod
vector product = cross product
rotZ
: see toPol
angle
: uses acos
and asin
v %v% w
: same as vecprod(v, w)
v %s% w
: same as scprod(v, w)
Christian W. Hoffmann <christian@echoffmann.ch>
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | pkg <- TRUE # FALSE for direct use
(x <- toPol(1.0, 1.0) ) # $r 1.41421, $p 0.785398 = pi/4
(y <- toRec(2.0,pi) ) # $x -2, $y 2.44921e-16
toPol(y[1], y[2]) # 2, pi
toRec( x[1], x[2]) # 1, 1
rotZ( 1, 0, pi/2 ) # 6.123032e-17 1.000000e+00
x <- 1; y <- 2; z <- 3
(R <- toSph(c(x,y,z)) ) # r= 3.7416574, theta= 0.64052231, phi= 1.1071487
c(R[1],180/pi*(R[2:3])) # 3.741657 36.6992252 63.434949
(w <- toXyz(R[1], R[2], R[3])) # = x,y,z
rotZ(1,2,pi/2) # -2, 1
opar <- par(mfrow=c(2,4))
x <- seq(0,1,0.05)
phi <- c(pi/6,pi/4,-pi/6)
Data <- matrix(c(x^2*10,(x^2-10*x)*4,(x+10)*1.5),ncol=3)
## Data <- matrix(c(rnorm(99)*10,rnorm(99)*4,rnorm(99)*1.5),ncol=3)
lim <- range(c(Data,-Data))*1.5
RD <- Data %*% rotL(phi[1],1,2) # !! # rotate around z-axis
RD2 <- RD %*% rotL(phi[2],2,3) # !! # rotate further around x
RD3 <- RD2 %*% rotL(phi[3],1,2) # !! # rotate back around z
## Not run:
plot(Data[,-3],xlim=lim,ylim=lim,xlab="x",ylab="y",pty="s")
plot(RD[,-3],xlim=lim,ylim=lim,xlab="RD x",ylab="y",pty="s",pch=5,col="red")
plot(RD2[,-3],xlim=lim,ylim=lim,xlab="RD2 x",ylab="y",pch=6,col="blue")
plot(RD3[,-3],xlim=lim,ylim=lim,xlab="RD3 x",ylab="RD3 y",col="magenta")
plot(Data[,1],RD3[,1])
plot(Data[,2],RD3[,2])
plot(Data[,3],RD3[,3])
## End(Not run)
m <- rotL(phi[1],1,2) %*% rotL(phi[2],2,3) %*% rotL(phi[3],1,2) # !! #
if (pkg) {
m <- rotL(phi[1],1,2) %*% rotL(phi[2],2,3) %*% rotL(phi[3],1,2) # !! #
round(m %*% t(m),2) #!! # composite rotation matrix and orthogonality,
# should be diag(3)
} else {
m <- rotL(phi[1],1,2) %*% rotL(phi[2],2,3) %*% rotL(phi[3],1,2) # !! #
round(m %*% t(m),2) #!! # composite rotation matrix and orthogonality,
# should be diag(3)
}
eye <- c(0.5,2.5,4)
re <- rotV(eye)
getAp(re) #$A [1] -9.805807e-01 1.961161e-01 -1.193931e-16
# $phi [1] 0.5674505
round(rotA(pi/1.5, c(1,1,1)),2) # 60 degrees around octant bisector
# [1,] 0 1 0 is permutation of axes 1 -> 2 -> 3 -> 1
# [2,] 0 0 1
# [3,] 1 0 0
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