piecewise_geodesic {spheresmooth}

Piecewise Geodesic

Description

This function computes a piecewise geodesic path between control points.

Usage

piecewise_geodesic(t, control_points, knots)

Arguments

t

A numeric vector representing the time or location.

control_points

A matrix of control points where each row represents a control point.

knots

A numeric vector of knot values.

Details

This function calculates the piecewise geodesic curve between control points based on the provided knots. The geodesic curve is computed segment by segment between adjacent control points. It interpolates the path between control points in a geodesic manner, ensuring the shortest path along the surface.

Value

A matrix containing the piecewise geodesic path.

Examples

# `rgl` package and `sphereplot` pacakges are needed for the visualizaiton of the following example.
# Define control points and knots
library(rgl)
library(sphereplot)
control_points <- matrix(c(1, 0, 0,                         # Control point 1
                          1/sqrt(2), 1/sqrt(2), 0,          # Control point 2
                          -1/sqrt(3), 1/sqrt(3), 1/sqrt(3), # Control point 3
                          0, 0, 1),                         # Control point 4
                          nrow = 4, byrow = TRUE)
knots <- c(1, 2, 3, 3.5)  # Knots indicating transitions
# Example of generating piecewise geodesic curve
t_example <- seq(0, 4, by = 0.01)
gamma_example <- piecewise_geodesic(t_example, control_points, knots)
# Plotting the piecewise geodesic curve
rgl.sphgrid(deggap = 45, col.long = "skyblue", col.lat = "skyblue")
spheres3d(x = 0, y = 0, z = 0, radius = 1, col = "grey", alpha = 0.05)
pch3d(control_points, col = "blue", cex = 0.2, pch = 19)
lines3d(gamma_example, col = "red", lty = 1, lwd = 2)