convert coordinates, angles, simple vector operations
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.
toPol( x, y=0 ) toRec( r, phi=0 ) toSph( x, y, z ) toXyz( r, theta, phi ) rotZ( x, y, phi ) rotA( phi, P=c(0,0,1) ) rotV(v, w=c(0,0,1)) rotL(phi,k=1,m=2,N=3) getAp( M ) angle(v,w) scprod(v, w) vecprod(v, w) v %v% w
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)=ytoRec: c( x , y ), x=Re(z) , y=Im(z); Mod(z)=r, Arg(z)=phitoSph: 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 vectorslV Euclidean (spatial) length of vectorscprod scalar productvecprod vector product = cross product
Christian W. Hoffmann <christian@echoffmann.ch>
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 0Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.