BarryGoldman {qsplines}

Barry-Goldman quaternions spline

Description

Constructs a spline of unit quaternions by the Barry-Goldman method.

Usage

BarryGoldman(keyRotors, keyTimes = NULL, n_intertimes, times)

Arguments

keyRotors

a vector of unit quaternions (rotors) to be interpolated; it is automatically appended with the first one to have a closed spline

keyTimes

the times corresponding to the key rotors; must be an increasing vector of length length(keyRotors)+1; if NULL, it is set to c(1, 2, ..., length(keyRotors)+1)

n_intertimes

a positive integer used to linearly interpolate the times given in keyTimes in order that there are n_intertimes - 1 between two key times (so one gets the key times if n_intertimes = 1); if this argument is given, then it has precedence over times

times

the interpolating times, they must lie within the range of keyTimes; ignored if n_intertimes is given

Value

A vector of unit quaternions with the same length as times.

Note

The function does not check whether the quaternions given in keyRotors are unit quaternions.

Examples

library(qsplines)
# Using a Barry-Goldman quaternions spline to construct 
#   a spherical curve interpolating some key points on
#   the sphere of radius 5.

# helper function: spherical to Cartesian coordinates
sph2cart <- function(rho, theta, phi){
  return(c(
    rho * cos(theta) * sin(phi),
    rho * sin(theta) * sin(phi),
    rho * cos(phi)
  ))
}

# construction of the key points on the sphere
keyPoints <- matrix(nrow = 0L, ncol = 3L)
theta_ <- seq(0, 2*pi, length.out = 9L)[-1L]
phi <- 1
for(theta in theta_){
  keyPoints <- rbind(keyPoints, sph2cart(5, theta, phi))
  phi = pi - phi
}
n_keyPoints <- nrow(keyPoints)

# construction of the key rotors; the first key rotor is the 
#   identity quaternion and rotor i sends the first key point 
#   to the key point i
keyRotors <- quaternion(length.out = n_keyPoints)
rotor <- keyRotors[1L] <- H1
for(i in seq_len(n_keyPoints - 1L)){
  keyRotors[i+1L] <- rotor <-
    quaternionFromTo(
      keyPoints[i, ]/5, keyPoints[i+1L, ]/5
    ) * rotor
}

# Barry-Goldman quaternions spline
rotors <- BarryGoldman(keyRotors, n_intertimes = 10L)

# construction of the interpolating points on the sphere
points <- matrix(nrow = 0L, ncol = 3L)
keyPoint1 <- rbind(keyPoints[1L, ])
for(i in seq_along(rotors)){
  points <- rbind(points, rotate(keyPoint1, rotors[i]))
}

# visualize the result with the 'rgl' package
library(rgl)
spheres3d(0, 0, 0, radius = 5, color = "lightgreen")
spheres3d(points, radius = 0.2, color = "midnightblue")
spheres3d(keyPoints, radius = 0.25, color = "red")