Surface Roughness

crater_curvature_radius(r, h) -> R

Calculate the curvature radius of a concave spherical segment.

Arguments

• r::Real: Crater radius (same units as h)
• h::Real: Crater depth (same units as r)

Returns

• R::Real: Curvature radius of the spherical segment

Notes

The curvature radius is calculated using the formula: R = (r² + h²) / 2h This represents the radius of the sphere from which the crater segment is cut.

Example

# Small bowl-shaped crater: 100m radius, 10m deep
R = crater_curvature_radius(100.0, 10.0)  # Returns 505.0 m

# Deeper crater (smaller curvature radius): 100m radius, 50m deep
R = crater_curvature_radius(100.0, 50.0)  # Returns 125.0 m

See also: concave_spherical_segment

concave_spherical_segment(r, h, xc, yc, x, y) -> z

Calculate the z-coordinate (depth) of a concave spherical segment at a given (x,y) position.

Arguments

• r::Real : Crater radius
• h::Real : Crater depth (maximum depth at center)
• xc::Real : x-coordinate of crater center
• yc::Real : y-coordinate of crater center
• x::Real : x-coordinate where to calculate z
• y::Real : y-coordinate where to calculate z

Returns

• z::Real: Depth below the surface (negative value inside, 0 outside crater)

Notes

• Returns 0 for points outside the crater radius
• The crater profile follows a spherical cap geometry
• All spatial parameters should use consistent units

Example

# Crater at origin with 10m radius and 2m depth
z_center = concave_spherical_segment(10.0, 2.0, 0.0, 0.0, 0.0, 0.0)   # Returns -2.0
z_edge   = concave_spherical_segment(10.0, 2.0, 0.0, 0.0, 10.0, 0.0)  # Returns 0.0
z_mid    = concave_spherical_segment(10.0, 2.0, 0.0, 0.0, 5.0, 0.0)   # Returns ~-0.6

See also: crater_curvature_radius

concave_spherical_segment(r, h; Nx=2^5, Ny=2^5, xc=0.5, yc=0.5) -> xs, ys, zs

Generate a grid representation of a concave spherical segment (crater).

Arguments

• r::Real : Crater radius (in normalized units, typically 0-1)
• h::Real : Crater depth (in same units as radius)

Keyword Arguments

• Nx::Integer=32 : Number of grid points in x-direction (default: 2^5)
• Ny::Integer=32 : Number of grid points in y-direction (default: 2^5)
• xc::Real=0.5 : x-coordinate of crater center (normalized, 0-1)
• yc::Real=0.5 : y-coordinate of crater center (normalized, 0-1)

Returns

• xs::LinRange: x-coordinates of grid points (0 to 1)
• ys::LinRange: y-coordinates of grid points (0 to 1)
• zs::Matrix : z-coordinates (depths) at each grid point

Notes

• The grid spans a unit square [0,1] × [0,1]
• z-values are negative inside the crater, 0 outside
• Suitable for use with load_shape_grid to create crater shape models

Example

# Generate a crater covering 40% of the domain, 0.1 units deep
xs, ys, zs = concave_spherical_segment(0.4, 0.1; Nx=64, Ny=64)

# Convert to shape model
shape = load_shape_grid(xs, ys, zs)

# Off-center crater
xs, ys, zs = concave_spherical_segment(0.3, 0.05; xc=0.3, yc=0.7)

See also: load_shape_grid, grid_to_faces