Distance between Two Curves on Manifolds

Given two curves \(\gamma_1, \gamma_2 : I \rightarrow \mathcal{M}\), we are interested in measuring the discrepancy of two curves. Usually, data are given as discrete observations so we are offering several methods to perform the task. See the section below for detailed description.

Usage

riem.distlp(
  riemobj1,
  riemobj2,
  vect = NULL,
  geometry = c("intrinsic", "extrinsic"),
  ...
)

Arguments

riemobj1

a S3 "riemdata" class for \(N\) manifold-valued data along the curve.

riemobj2

a S3 "riemdata" class for \(N\) manifold-valued data along the curve.

vect

a vector of domain values. If given Null (default), sequence 1:N is set.

geometry

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

...

extra parameters including

p

an exponent (default: 2).

Value

the distance value.

Default Method

Trapezoidal Approximation Assume \(\gamma_1 (t_i) = X_i\) and \(\gamma_2 (t_i) = Y_i\) for \(i=1,2,\ldots,N\). In the Euclidean space, \(L_p\) distance between two scalar-valued functions is defined as $$L_p^p (\gamma_1 (x), \gamma_2 (x) = \int_{\mathcal{X}} |\gamma_1 (x) - \gamma_2 (x)|^p dx $$. We extend this approach to manifold-valued curves $$L_p^p (\gamma_1 (t), \gamma_2 (t)) = \int_{t\in I} d^p (\gamma_1 (t), \gamma_2 (t)) dt$$ where \(d(\cdot,\cdot)\) is an intrinsic/extrinsic distance on manifolds. With the given representations, the above integral is approximated using trapezoidal rule.

Examples

#-------------------------------------------------------------------
#                          Curves on Sphere
#
#  curve1 : y = 0.5*cos(x) on the tangent space at (0,0,1)
#  curve2 : y = 0.5*cos(x) on the tangent space at (0,0,1)
#  curve3 : y = 0.5*sin(x) on the tangent space at (0,0,1)
#
# * distance between curve1 & curve2 should be close to 0.
# * distance between curve1 & curve3 should be large.
#-------------------------------------------------------------------
## GENERATION
vecx  = seq(from=-0.9, to=0.9, length.out=50)
vecy1 = 0.5*cos(vecx) + rnorm(50, sd=0.05)
vecy2 = 0.5*cos(vecx) + rnorm(50, sd=0.05)
vecy3 = 0.5*sin(vecx) + rnorm(50, sd=0.05)

## WRAP AS RIEMOBJ
mat1 = cbind(vecx, vecy1, 1); mat1 = mat1/sqrt(rowSums(mat1^2))
mat2 = cbind(vecx, vecy2, 1); mat2 = mat2/sqrt(rowSums(mat2^2))
mat3 = cbind(vecx, vecy3, 1); mat3 = mat3/sqrt(rowSums(mat3^2))

rcurve1 = wrap.sphere(mat1)
rcurve2 = wrap.sphere(mat2)
rcurve3 = wrap.sphere(mat3)

## COMPUTE DISTANCES
riem.distlp(rcurve1, rcurve2, vect=vecx)
#> [1] 0.07121886
riem.distlp(rcurve1, rcurve3, vect=vecx)
#> [1] 0.5962591