Map torus data to the tangent space at a base point.

Applies the logarithmic map to send torus data in \([0,1)^d\) into the tangent space \([-0.5, 0.5)^d\) at base point p: for each coordinate, (x - p + 0.5) %% 1 - 0.5, i.e. the shortest signed offset from p on that coordinate's circle. Points on the cut locus of p map to -0.5 in the affected coordinates; see is_valid_torus.

Usage

log_map_torus(X, p)

Arguments

X

A matrix or data frame of torus data in \([0,1)^d\); rows are samples, columns are coordinates.

p

Numeric vector of length ncol(X): the base point on the torus.

Value

A numeric matrix of the same shape as X, with values in \([-0.5, 0.5)^d\).

Examples

X <- matrix(runif(20), nrow = 10, ncol = 2)
log_map_torus(X, p = c(0.5, 0.5))
#>               [,1]       [,2]
#>  [1,] -0.240181756 -0.4959180
#>  [2,] -0.481093830  0.1568564
#>  [3,]  0.221405048 -0.3490711
#>  [4,]  0.452809649 -0.3008393
#>  [5,] -0.059776696 -0.2603617
#>  [6,]  0.004012379  0.1615740
#>  [7,] -0.456443002 -0.2582112
#>  [8,]  0.422474456 -0.4284503
#>  [9,]  0.423789556  0.2046719
#> [10,]  0.105647714 -0.1846031