Fractals/Mathematics/Vector field

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Vector field[1]

Here mainly numerical methods for time independent 2D vector fields are described.


Field types[edit]

Algorithm[edit]

  • start with
    • plane (parameter plane or dynamic plane)
    • scalar function
    • vector function
  • create scalar field using scalar function ( potential)
  • create vector field from scalar field using vector function ( gradient of the potential)
  • compute:

Field line integration method[edit]

rk4[edit]

Fourth-order Runge-Kutta (RK4) in case of 2D time independent vector field


is a vector function that for each point p

p = (x, y)

in a domain assigns a vector v

where each of the functions is a scalar function:

A field line is a line that is everywhere tangent to a given vector field.

Let r(s) be a field line given by a system of ordinary differential equations, which written on vector form is:

where:

  • s representing the arc length along the field line
  • is a seed point

Given a seed point on the field line, the update rule ( RK4) to find the next point along the field line is[3]

where:

  • h is the step size
  • k are the intermediate vectors:

Visualisation of vector field[edit]

Plot types (Visualization Techniques for Flow Data) : [4]

  • Glyphs = Icons or signs for visualizing vector fields
    • simplest glyph = Line segment (hedgehog plots)
    • arrow plot = quiver plot = Hedgehogs (global arrow plots)
  • Characteristic Lines [5]
    • stremlines = curve everywhere tangential to the instantaneous vector (velocity) field (time independent vector field). For time independent vector field streaklines = Path lines = streak lines [6]
  • texture (line integral convolution = LIC)[7]
  • Topological skeleton [8]
    • fixed point extraction ( Jacobian)

Examples[edit]

Potential of Mandelbrot set[edit]

Programs[edit]

Dictionary[edit]

  • vector function is a function that gives vector as an output
  • field : space (plane, sphere, ... )
  • field line is a line that is everywhere tangent to a given vector field
  • scalar/ vector / tensor:
    • Scalars are real numbers used in linear algebra. Scalar is a tensor of zero order
    • Vector is a tensor of first order. Vector is an extension of scalar
    • tensor is an extension of vector

Vector[edit]

Forms of 2D vector:[9]

  • [z1] ( only one complex number when first point is known , for example z0 is origin
  • [z0, z1] = two complex numbers
  • 4 scalars ( real numbers)
    • [x, y, dx , dy]
    • [x0, y0, x1, y1]
    • [x, y, angle, magnitude]
  • 2 scalars : [x1, y1] for second complex number when first point is known , for example z0 is origin

Examples[edit]

Shadertoy[edit]

Videos[edit]

See also[edit]

References[edit]

  1. Vector field in wikipedia
  2. Numerical Methods for Particle Tracing in Vector Fields by Kenneth I. Joy
  3. Classification and visualisation of critical points in 3d vector fields. Master thesis by Furuheim and Aasen
  4. Flow Visualisation from TUV
  5. Data visualisation by Tomáš Fabián
  6. A Streakline Representation of Flow in Crowded Scenes from UCF
  7. lic by Zhanping Liu
  8. Vector Field Topology in Flow Analysis and Visualization by Guoning Chen
  9. Euclidian vector in wikipedia