Streamlines and Vector-Field Visualization

Vector fields are shown by tracing paths that follow the field, by placing oriented glyphs, or by texture-based methods that fill space.

Showing direction and magnitude

A vector field assigns a direction and magnitude to each point. Visualizing it means conveying both without clutter. The main families are integral curves (streamlines, pathlines, streaklines), glyphs (arrows), and dense texture methods such as line integral convolution.

Streamlines

Kronos motion — space economy

A streamline is a curve everywhere tangent to a steady field; it is found by integrating dx/dt = v(x) from seed points. Integration uses Runge-Kutta schemes; step size and error control determine whether the line stays faithful to the field. For time-varying fields the analogous curves are pathlines and streaklines, which differ because the field itself changes.

python

def rk4_step(x, v, h):
    k1 = v(x)
    k2 = v(x + 0.5*h*k1)
    k3 = v(x + 0.5*h*k2)
    k4 = v(x + h*k3)
    return x + (h/6.0)*(k1 + 2*k2 + 2*k3 + k4)

Seeding and density

Uniform seeding wastes lines in slow regions and starves fast ones; use flow-guided or evenly-spaced seeding.
Color streamlines by speed or a scalar to add a second variable.
Tapering, tubes, or arrowheads encode direction along a line.

Glyphs and texture methods

Arrow glyphs are intuitive but overplot in 3D; downsample and scale carefully. Texture methods like line integral convolution smear noise along the field to show structure densely without choosing seeds.

Kronos use

Magnetic field structure in the burner mirror and the breeder tokamak is a vector field; field-line tracing is a specialized streamline problem covered in Field-Line Tracing.