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Geodesic Interpolation of Multiple Points

Source: R/basic_interps.R
riem.interps.Rd

Given 2 observations \(X_1, X_2 \in \mathcal{M}\), find the interpolated points of a geodesic \(\gamma(t)\) for \(t \in (0,1)\) which assumes two endpoints \(\gamma(0)=X_1\) and \(\gamma(1)=X_2\).

Usage

riem.interps(
  riemobj,
  vect = c(0.25, 0.5, 0.75),
  geometry = c("intrinsic", "extrinsic")
)

Arguments

riemobj

a S3 "riemdata" class for \(2\) manifold-valued data where the first object is the starting point.

vect

a length-\(T\) vector in \((0,1)\) for which the interpolations are taken.

geometry

(case-insensitive) name of geometry; either geodesic ("intrinsic") or embedded ("extrinsic") geometry.

Value

a 3d array where \(T\) slices along 3rd dimension are interpolated objects in matrix representation.

Validation status

This retained legacy interface is experimental. Its full numerical and statistical contract has not been independently verified across supported inputs. See riem-method-contracts and the installed contract table for method-specific assumptions, restrictions, and evidence scope.

Examples

#-------------------------------------------------------------------
#       Geodesic Interpolation between (1,0) and (0,1) in S^1
#-------------------------------------------------------------------
## PREPARE DATA
sp.start = c(1,0)
sp.end   = c(0,1)
sp.data  = wrap.sphere(rbind(sp.start, sp.end))

## FIND THE INTERPOLATED POINT AT FOR t=0.1, 0.2, ..., 0.9.
myvect  = seq(from=0.1, to=0.9, by=0.1)
geo.int = riem.interps(sp.data, vect=myvect, geometry="intrinsic")
geo.ext = riem.interps(sp.data, vect=myvect, geometry="extrinsic")

geo.int = matrix(geo.int, byrow=TRUE, ncol=2) # re-arrange for plotting
geo.ext = matrix(geo.ext, byrow=TRUE, ncol=2)

## VISUALIZE
#  Prepare Lines and Points
thetas  = seq(from=0, to=pi/2, length.out=100)
quarter = cbind(cos(thetas), sin(thetas))

pts.int = rbind(sp.start, geo.int, sp.end)
pts.ext = rbind(sp.start, geo.ext, sp.end)
col.int = c("black", rep("red",9),  "black")
col.ext = c("black", rep("blue",9), "black")

# Draw
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
plot(quarter, main="intrinsic interpolation", # intrinsic geodesic
     xlab="x", ylab="y", type="l")
points(pts.int, col=col.int, pch=19)
for (i in 1:9){
  text(geo.int[i,1]*0.9, geo.int[i,2]*0.9, 
       paste0(round(i/10,2)), col="red")
}
plot(quarter, main="extrinsic interpolation", # intrinsic geodesic
     xlab="x", ylab="y", type="l")
points(pts.ext, col=col.ext, pch=19)
for (i in 1:9){
  text(geo.ext[i,1]*0.9, geo.ext[i,2]*0.9, 
       paste0(round(i/10,2)), col="blue")
}

par(opar)