Fractals/Mathematics/Vector field

Vector field

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

Dictionary

• 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

Forms of 2D vector:

• [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

gradient

The numerical gradient of a function:

• "is a way to estimate the values of the partial derivatives in each dimension using the known values of the function at certain points."

Gradient function G of function f at point (x0,y0)

  G(f,x0,y0) = (x1,y1)

Input

• function f
• point (x0,y0) where gradient is computed

Output:

• vector from (x0,yo) to (x1,y1) = gradient

See:

• Numerical differentiation in wikipedia

Computation:

• "The gradient is calculated as: (f(x + h) - f(x - h)) / (2*h) where h is a small number, typically 1e-5 f(x) will be called for each input elements with +h and -h pertubation. For gradient checking, recommend using float64 types to assure numerical precision."
• in matlab
• in R
• python

Field types

• time independent (= stationary = Steady ) or time dependent ( unsteady flow)
• dimension: 2D, 3D, ...
• mesh ( grid) type
• scalar function
  • potential
  • electric, magnetic, ...
• vector function
  • equation ( for symbolic computations) - ODE
  • numerical values ( for numerical computations)
• field line integration method (scheme):

Force field

A gravitational force fields

• the field lines are the solution of $g'=F$
• the trajectory of a test mass is the solution of $mg''=F$

where

• g is the standard gravity
• m is a mass
• F is a force field

An electric field

• The field lines are the paths that a point positive charge would follow as it is forced to move within the field. Field lines due to stationary charges have several important properties, including always originating from positive charges and terminating at negative charges, they enter all good conductors at right angles, and they never cross or close in on themselves.  The field lines are a representative concept; the field actually permeates all the intervening space between the lines. More or fewer lines may be drawn depending on the precision to which it is desired to represent the field. The study of electric fields created by stationary charges is called electrostatics.

• 

  Diagram of electric charges induced in conductive objects (shapes) by the electrostatic field (lines with arrows) of a nearby charge (+), due to electrostatic induction

• 

  Electric field around two identical conducting spheres at opposite electric potential

Algorithm

• 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:
  • filed lines ( stream lines )
    • External Rays on the Parameter Plane
    • dynamic external rays
  • contour lines ( equipotential lines )
  • map whole field using
    • Line Integral Convolution (LIC)

gradient descent

• Gradient Descent by Aytan Hajiyeva Aytan Hajiyeva Oct 1, 2021
• Gradient Descent algorithm by Raghunath D Raghunath D Jan 28, 2019
• An overview of gradient descent optimization algorithms by SEBASTIAN RUDER 19 JAN 2016 and pdf from arxiv
• Escaping from Saddle Points by Rong Ge • Mar 22, 2016
• An Easy Guide to Gradient Descent in Machine Learning By Great Learning Team -May 24, 2020
• Gradient Descent in Python Sagar Mainkar Sagar Mainkar Aug 25, 2018

Field line computing

Problem Statement:

• Field line tracing ( not curve sketching}
• drawing contour maps ( in computer graphic) = Numerical continuation ( in math)
• compute an integral curve from a seed point through a vector field without any analysis of its structure on the uniform grid ( raster scan or pixels)

• 

  Natural parameter continuation, a very simple adaptation of the iterative solver to a parametrized problem

• 

  Simplicial linear continuation algorithm.

• 

  Pseudo-arclength continuation, based on the observation that the "ideal" parameterization of a curve is arclength, is an approximation of the arclength in the tangent space of the curve\

• 

Methods ( solvers) available for the field-lines

• Euler
• RK2
• RK3
• RK4 - the original authors of those sampling algorithms: Runge und Kutta.

  None of these 4 methods generate an exact answer, but they are (from left to right) increasingly more accurate. They also take (from left to right) more and more time to finish as they require more samples for each iteration.
  You won't be able to create reliably closed curves using iterative sampling methods as small errors at any step may be amplified in successive steps. There is also no guarantee that the field-line ends up in the exact coordinate where it started.
  The Grasshopper metaball solver on the other hand uses a marching squares algorithm which is capable of finding closed loops because it is a grid-cell approach and sampling inaccuracy in one area doesn't carry over to another. 
  However the solving of iso-curves is a very different process    from the solving of particle trajectories through fields. ... 
  Typically field lines shoot to infinity rather than form closed loops. That is one reason why I chose the RK methods here, because marching-cubes is very bad at dealing with things that tend to infinity.

Construction

Construction of a field line

Given a vector field $\mathbf {F} (\mathbf {x} )$ and a starting point $\mathbf {x} _{\text{0}}$ a field line can be constructed iteratively by finding the field vector at that point $\mathbf {F} (\mathbf {x} _{\text{0}})$. The unit tangent vector at that point is: $\mathbf {F} (\mathbf {x} _{\text{0}})/|\mathbf {F} (\mathbf {x} _{\text{0}})|$. By moving a short distance $ds$ along the field direction a new point on the line can be found

$\mathbf {x} _{\text{1}}=\mathbf {x} _{\text{0}}+{\mathbf {F} (\mathbf {x} _{\text{0}}) \over |\mathbf {F} (\mathbf {x} _{\text{0}})|}ds$

Then the field at that point $\mathbf {F} (\mathbf {x} _{\text{1}})$ is found and moving a further distance $ds$ in that direction the next point of the field line is found

$\mathbf {x} _{\text{2}}=\mathbf {x} _{\text{1}}+{\mathbf {F} (\mathbf {x} _{\text{1}}) \over |\mathbf {F} (\mathbf {x} _{\text{1}})|}ds$

By repeating this and connecting the points,the field line can be extended as far as desired. This is only an approximation to the actual field line, since each straight segment isn't actually tangent to the field along its length, just at its starting point. But by using a small enough value for $ds$, taking a greater number of shorter steps, the field line can be approximated as closely as desired. The field line can be extended in the opposite direction from $\mathbf {x} _{\text{0}}$ by taking each step in the opposite direction by using a negative step $-ds$.

rk4 numerical integration method

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

$F$ is a vector function that for each point p

p = (x, y)

in a domain assigns a vector v

$v=F(p)=F(x,y)=(F_{1}(x,y),F_{2}(x,y))$

where each of the functions $F_{i}$ is a scalar function:

$F_{i}:\mathbb {R} ^{2}\to \mathbb {R}$

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:

${\frac {dr}{ds}}=F(r(s))$

where:

• s representing the arc length along the field line, like for example continous iteration count
• $r(0)=r_{0}$ is a seed point

2 variables

Given a seed point $r_{0}$ on the field line, the update rule ( RK4) to find the next point $r_{i}$along the field line is

$r_{i+1}=r_{i}+{\frac {h(k_{1}+2k_{2}+2k_{3}+k_{4})}{6}}$

where:

• h is the step size along field line = ds
• k are the intermediate vectors:

${\begin{array}{lcl}k_{1}=F(r_{i})\\k_{2}=F(r_{i}+{\frac {h}{2}}k_{1})\\k_{3}=F(r_{i}+{\frac {h}{2}}k_{2})\\k_{4}=F(r_{i}+hk_{3})\end{array}}$

only x

Here

$y'={\frac {dy}{dx}}=f(x)$

Given a seed point $r_{0}$ on the field line, the update rule ( RK4) to find the next point $r_{i}$along the field line is

$x_{i+1}=x_{i}+dx=x_{i}+h$
$y_{i+1}=y_{i}+{\frac {h(k_{1}+2k_{2}+2k_{3}+k_{4})}{6}}$

where:

• h is the step size along field line = dx
• k are the intermediate vectors:

${\begin{array}{lcl}k_{1}=F(x_{i})\\k_{2}=F(x_{i}+{\frac {h}{2}}k_{1})\\k_{3}=F(x_{i}+{\frac {h}{2}}k_{2})\\k_{4}=F(x_{i}+hk_{3})\end{array}}$

Examples:

• ${\frac {dy}{dx}}=x$
• $y^{'}+x^{2}+x=0$
• ${\frac {dy}{dx}}=x^{2}-x-2$

• 

  Slope field (black), some solutions (red) and isoclines (blue) of y'=x

• 

  Three integral curves for the slope field corresponding to the differential equation dy / dx = x² − x − 2

Visualisation of vector field

Plot types (Visualization Techniques for Flow Data) :

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

• 

  LIC

• 

  Streamlines and streamtube

• 

  Image-based flow visualization

• Integral Curves

• 

  integral curves

  "path lines, streak lines, and stream lines are identical for stationary flows" Leif Kobbelt

quiver plot

Definition

• "A quiver plot displays velocity vectors as arrows with components (u,v) at the points (x,y)"

stream plot

A stream plot uses stream lines

Example fields

flowmaps

• FlowMap Painter by TECK LEE TAN
• Flowmaps, gradient maps, gas giants by Martin Donald
• godot game engine

fBM

fBM stands for Fractional Brownian Motion

• i quilezles : fbmsdf
• i quilezles: fbm

SDF

SDF = Signed Distance Function

• SDF for 2d primitives
• normals to SDF
• gradient
• i quilezles : interior distance ( SDF)
• Glyphs, shapes, fonts, signed distance fields by Martin Donald
• CSC2547 DeepSDF Learning Continuous Signed Distance Functions for Shape Representation
• 2D Signed Distance Field Basics by Ronja
• snelly is a WebGL SDF pathtracer by Jamie Portsmouth
• shadertoy : SDF Raymarch Quadtree by paniq

Potential of Mandelbrot set

• i quilezles: distance to fractals
• Parameter External Rays are field lines of potential field

• 

  ( Empty) field : points of rectangular mesh

• 

  Scalar field : potential of Mandelbrot set

• 

  Vector field: Array plot of the gradient

Programs

• CGAL - 2D_Placement_of_Streamlines by Abdelkrim Mebarki
• Python
  • Matplotlib (python library ):
    • quiver
    • stream plot
    • quiver-plot-with-matplotlib-and-jupyter-notebooks
    • short scripts for making 2D vector plots for visualizing an equation by Benjamin Badger
  • VectorFieldPlot , images created with VectorFieldPlot
• OpenProcessing
• G'MIC - display_quiver
• OpenCV
  • dsp.stackexchange: how-to-detect-gradients-in-images - Histogram of Oriented Gradients ( HOG )
  • arrowed line
• Maxima CAS
  • plotdf ( for ODE)
• C++
  • Visualizing 2D Vector Fields by Philip Rideout, 18 July 2018 and clumpy
• par_streamlines ( C, WebGl) by Philip Allan Rideout, github: prideout/par
• JavaScript
  • Streamlines calculator by anvaka (Andrei Kashcha)
• c
  • vfplot - program for plotting two-dimensional vector fields by J J Green

Examples

Images

• images from commons : Category:Field_lines
• images from commons : Category:Vector_fields
• fractalforums.org : cruising-through-fractal-flow-fields

Shadertoy

• arrows = quiver plot
  • 2D Vector Field Flow by morgan3d
  • static 2d by celeriac
• Line Integral Convolution (LIC)
  • Flow Field LIC Created by Thomas_Diewald in 2017-10-10
  • LIC 2D / flow 2D - precomp+lines by FabriceNeyret2 in 2017-03-04
• 3D vector field

Videos

• by Chris Thomasson
  • Test Animated Vector Field "Power Flux", try 0...
  • Test Animated Vector Field "Power Flux", try 1...
• Numerical Approximations of Gradients from Deeplearning.ai
• Sticking to the pointy ends: electric field lines in an evolving Julia set capacitor by Nils Berglund

See also

• Vector flow in wikipedia
• TYLER HOBBS : flow-fields

References

• How I built a wind map with WebGL by Vladimir Agafonkin
• math.stackexchange question : what-do-polynomials-look-like-in-the-complex-plane
• mathematica.stackexchange question : visualizing-a-complex-vector-field-near-poles
• fractalforums.com : smooth-external-angle-of-mandelbrot-set
• J.M. Hyman, M. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, Computers & Mathematics with Applications, Volume 33, Issue 4, 1997, Pages 81-104, ISSN 0898-1221