coords: convert coordinates, angles, simple vector operations

Description

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.

Usage

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)
  lV( v )
  scprod(v, w)
  vecprod(v, w)
  v %v% w

Arguments

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)

c(x,y,z), coordinates of point or projection direction P-'O', with 'O' =c(0,0,0).

v, w

3-vectors (x, y, z).

Order of the square rotation matrix >= 2.

k,m

Integers (m != k) describing the plane of rotation. m==k gives Unit matrix.

3x3 rotation matrix.

Value

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

Details

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)].

Examples

Run this code

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, $y 2.44921e-16
  toPol(y[1], y[2]) # 2, pi
  toRec( x[1], x[2]) # 1, 1
  rotZ( 1, 0, pi/2 )  # 6.123032e-17 1.000000e+00
  x <- 1;  y <- 2;  z <- 3
  (R <- toSph(x,y,z))  # r= 3.7416574, theta= 0.93027401, phi= 1.1071487
  c(R[1],180/pi*(R[2:3])) #  3.741657 53.300775 63.434949
  (w <- toXyz(R[1], R[2], R[3]))  #  = x,y,z
  rotZ(1,2,pi/2)  # -2, 1
  opar <- par(mfrow=c(2,4))
  x <- seq(0,1,0.05)
  phi <- c(pi/6,pi/4,-pi/6)
  Data <- matrix(c(x^2*10,(x^2-10*x)*4,(x+10)*1.5),ncol=3)
##  Data <- matrix(c(rnorm(99)*10,rnorm(99)*4,rnorm(99)*1.5),ncol=3)
  lim <- range(c(Data,-Data))*1.5
  RD <-  Data %*% rotL(phi[1],1,2)   # !! # rotate around z-axis
  RD2 <- RD %*% rotL(phi[2],2,3) # !! # rotate further around x
  RD3 <- RD2 %*% rotL(phi[3],1,2) # !! # rotate back around z
  plot(Data[,-3],xlim=lim,ylim=lim,xlab="x",ylab="y",pty="s")
  plot(RD[,-3],xlim=lim,ylim=lim,xlab="RD x",ylab="y",pty="s",pch=5,col="red")
  plot(RD2[,-3],xlim=lim,ylim=lim,xlab="RD2 x",ylab="y",pch=6,col="blue")
  plot(RD3[,-3],xlim=lim,ylim=lim,xlab="RD3 x",ylab="RD3 y",col="magenta")
  plot(Data[,1],RD3[,1])
  plot(Data[,2],RD3[,2])
  plot(Data[,3],RD3[,3])
  m <- rotL(phi[1],1,2) %*% rotL(phi[2],2,3) %*% rotL(phi[3],1,2) # !! #
  if (pkg) {
    m <- rotL(phi[1],1,2) %*% rotL(phi[2],2,3) %*% rotL(phi[3],1,2) # !! #
    round(m %*% t(m),2) #!! # composite rotation matrix and orthogonality,
       # should be diag(3)
  } else {
    m <- rotL(phi[1],1,2) %*% rotL(phi[2],2,3) %*% rotL(phi[3],1,2) # !! #
    round(m %*% t(m),2) #!! # composite rotation matrix and orthogonality,
       # should be diag(3)
  }
  eye <- c(0.5,2.5,4)
  re  <- rotV(eye)
  getAp(re) #$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