ridge_curve: Fourier-fitted ridge curve and related utilities

Description

Given the angles theta in $[-\pi, \pi)$, ridge_curve computes the Fourier-fitted ridge curve $(\theta, r_1(\theta))$ or $(r_2(\theta), \theta)$, where $$r_j(\theta):=\mathrm{atan2}(S_m (\theta), C_m (\theta))$$ with $C_m(x) := a_0/2 + \sum_{k=1}^m a_k \cos(kx)$ and $S_m(x) := \sum_{k=1}^m b_k \sin(kx)$ for $j = 1,2$. der_ridge_curve and dist_ridge_curve compute the derivatives of and the distances along these curves, respectively. alpha_ridge_curve provides a uniform grid of the ridge curve using the arc-length parametrization. proj_ridge_curve gives the ridge's $\theta$ for which the curve is closer to any point on $[-\pi, \pi)^2$.

Usage

ridge_curve(
  theta,
  mu = c(0, 0),
  coefs = list(cos_a = c(0, 0), sin_b = 0),
  ind_var = 1,
  at2 = TRUE
)
der_ridge_curve(
  theta,
  mu = c(0, 0),
  coefs = list(cos_a = c(0, 0), sin_b = 0),
  ind_var = 1,
  norm = NULL,
  at2 = TRUE
)
dist_ridge_curve(
  alpha,
  mu = c(0, 0),
  coefs = list(cos_a = c(0, 0), sin_b = 0),
  ind_var = 1,
  N = 500,
  der = TRUE,
  shortest = TRUE,
  at2 = TRUE
)
arclength_ridge_curve(
  mu = c(0, 0),
  coefs = list(cos_a = c(0, 0), sin_b = 0),
  ind_var = 1,
  N = 500,
  L = 500,
  at2 = TRUE
)
proj_ridge_curve(
  x,
  mu = c(0, 0),
  coefs = list(cos_a = c(0, 0), sin_b = 0),
  ind_var = 1,
  N = 500,
  ridge_curve_grid = NULL,
  arclength = FALSE,
  at2 = TRUE
)

Value

ridge_curve: a matrix of size c(nth, 2) with the ridge curve evaluated at theta.
der_ridge_curve: a matrix of size c(nth, 2) with the derivatives of the ridge curve evaluated at theta.
dist_ridge_curve: the distance between two points along the ridge curve, a non-negative scalar.
proj_ridge_curve: a list with (1) the theta's that give the points in the ridge curve that are the closest (in the flat torus distance) to x (a matrix of size c(nx, 2)); (2) the indexes of ridge_curve_grid in which those theta's were obtained.
arclength_ridge_curve: a vector of size N giving the theta angles that yield a uniform-length grid of the ridge curve.

Arguments

theta: vector $\theta$ of size nth.
mu: a vector of size 2 giving $(\mu_1, \mu_2)$. Defaults to c(0, 0).
coefs: list of coefficients cos_a ($a_k$) and sin_b ($b_k$ giving the Fourier fit of the ridge curve. Defaults to list(cos_a = c(0, 0), sin_b = 0). See examples.
ind_var: index $j$ of the variable that parametrizes the ridge. Defaults to 1.
at2: do the atan2 fit instead of the sine fit (only using $S_m$)? Defaults to TRUE. at2 = FALSE is not recommended to use.
norm: normalize tangent vectors? If different from NULL (the default), the vectors are normalized to have the given norm.
alpha: a vector of size 2.
N: number of discretization points for approximating curve lengths. Defaults to 5e2.
der: use derivatives to approximate curve lengths? Defaults to TRUE.
shortest: return the shortest possible distance? Defaults to TRUE.
L: number of discretization points for computing the arc-length parametrization curve lengths. Defaults to 5e2.
x: a matrix of size c(nx, 2) with angular coordinates.
ridge_curve_grid: if provided, the ridge_curve evaluated at a grid of size N. If not provided, it is computed internally. Useful for saving computations.
arclength: use the arc-length parametrization to compute the projections? This yields a more uniform grid for searching the projections. Defaults to TRUE.

Examples

Run this code

mu <- c(-0.5, 1.65)
th <- seq(-pi, pi, l = 200)
K <- 5
coefs <- list(cos_a = 1 / (1:(K + 1))^3, sin_b = 1 / (1:K)^3)
rid1 <- ridge_curve(theta = th, mu = mu, coefs = coefs, ind_var = 1)
rid2 <- ridge_curve(theta = th, mu = mu, coefs = coefs, ind_var = 2)
plot(mu[1], mu[2], xlim = c(-pi, pi), ylim = c(-pi, pi), axes = FALSE,
     xlab = expression(theta[1]), ylab = expression(theta[2]),
     pch = "*", col = 5, cex = 3)
sdetorus::linesTorus(rid1[, 1], rid1[, 2], col = 1)
sdetorus::linesTorus(rid2[, 1], rid2[, 2], col = 2)
abline(v = mu[1], lty = 3, col = 5)
abline(h = mu[2], lty = 3, col = 5)
points(ridge_curve(theta = mu[1], mu = mu, coefs = coefs, ind_var = 1),
       col = 1)
points(ridge_curve(theta = mu[2], mu = mu, coefs = coefs, ind_var = 2),
       col = 2)
sdetorus::torusAxis()

## der_ridge_curve

th <- seq(-pi, pi, l = 10)
mu <- c(0.5, 1.5)
K <- 5
coefs <- list(cos_a = 1 / (1:(K + 1))^3, sin_b = 1 / (1:K)^3)
rid1 <- ridge_curve(theta = th, mu = mu, coefs = coefs, ind_var = 1)
rid2 <- ridge_curve(theta = th, mu = mu, coefs = coefs, ind_var = 2)
v1 <- der_ridge_curve(theta = th, mu = mu, coefs = coefs, ind_var = 1,
                      norm = 0.5)
v2 <- der_ridge_curve(theta = th, mu = mu, coefs = coefs, ind_var = 2,
                      norm = 0.5)
points(rid1, pch = 16, col = 1)
points(rid2, pch = 16, col = 2)
arrows(x0 = rid1[, 1], y0 = rid1[, 2],
       x1 = (rid1 + v1)[, 1], y1 = (rid1 + v1)[, 2],
       col = 3, angle = 5, length = 0.1)
arrows(x0 = rid2[, 1], y0 = rid2[, 2],
       x1 = (rid2 + v2)[, 1], y1 = (rid2 + v2)[, 2],
       col = 4, angle = 5, length = 0.1)

## dist_ridge_curve

# Distances accuracy
a <- c(-pi / 2, pi)
mu <- c(-pi / 2, pi / 2)
dist_ridge_curve(alpha = a, mu = mu, coefs = coefs, der = TRUE, N = 1e6)
dist_ridge_curve(alpha = a, mu = mu, coefs = coefs, der = FALSE, N = 1e6)
dist_ridge_curve(alpha = a, mu = mu, coefs = coefs, der = TRUE, N = 1e2)
dist_ridge_curve(alpha = a, mu = mu, coefs = coefs, der = FALSE, N = 1e2)

## arclength_ridge_curve

mu <- c(-pi / 2, pi / 2)
alpha <- arclength_ridge_curve(mu = mu, coefs = coefs, ind_var = 1, N = 25)
alpha <- sdetorus::toPiInt(c(alpha, alpha[1]))
rid <- ridge_curve(theta = alpha, mu = mu, coefs = coefs, ind_var = 1)
plot(mu[1], mu[2], pch = "*", col = 5, cex = 3, xlim = c(-pi, pi),
     ylim = c(-pi, pi), axes = FALSE, xlab = expression(theta[1]),
     ylab = expression(theta[2]))
sdetorus::linesTorus(rid[, 1], rid[, 2], col = 1, pch = 16)
points(rid[, 1], rid[, 2], pch = 16, col = 1)
abline(v = mu[1], lty = 3, col = 5)
abline(h = mu[2], lty = 3, col = 5)
sdetorus::torusAxis()

## proj_ridge_curve

mu <- c(0, 0)
n <- 25
x <- matrix(runif(2 * n, -pi, pi), nrow = n, ncol = 2)
col <- rainbow(n)
th <- seq(-pi, pi, l = 100)
old_par <- par(no.readonly = TRUE)
par(mfrow = c(1, 2))
for (j in 1:2) {

  plot(x, xlim = c(-pi, pi), ylim = c(-pi, pi), axes = FALSE,
       xlab = expression(theta[1]), ylab = expression(theta[2]), col = col)
  rid <- ridge_curve(theta = th, mu = mu, coefs = coefs, ind_var = j)
  sdetorus::linesTorus(x = rid[, 1], y = rid[, 2], lwd = 2)
  abline(v = mu[1], lty = 3)
  abline(h = mu[2], lty = 3)
  points(mu[1], mu[2], pch = "*", cex = 3)
  sdetorus::torusAxis()
  theta_projs <- proj_ridge_curve(x = x, mu = mu, coefs = coefs, ind_var = j,
                                  ridge_curve_grid = rid)$theta_proj
  projs <- ridge_curve(theta = theta_projs, mu = mu, coefs = coefs,
                       ind_var = j)
  points(projs, col = col, pch = 3)
  for (i in 1:n) {

    sdetorus::linesTorus(x = c(x[i, 1], projs[i, 1]),
                         y = c(x[i, 2], projs[i, 2]), col = col[i], lty = 3)

  }

}
par(old_par)

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