Construction

Construction of a field line

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

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

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

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

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 ${\displaystyle ds}$, 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 ${\displaystyle \mathbf {x} _{\text{0}}}$ by taking each step in the opposite direction by using a negative step ${\displaystyle -ds}$.

rk4 numerical integration method

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

${\displaystyle F}$ is a vector function that for each point p

p = (x, y)

in a domain assigns a vector v

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

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

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

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:

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

where:

• s representing the arc length along the field line
• ${\displaystyle r(0)=r_{0}}$ is a seed point

Given a seed point ${\displaystyle r_{0}}$ on the field line, the update rule ( RK4) to find the next point ${\displaystyle r_{i}}$along the field line is^[7]

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

where:

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

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

LIC

input:

• • white noise
  • original vector field

LIC examples:

• GPU-Based-Image-Processing-Tools by RaymondMcGuire in python
• LIC (Line Integral Convolution) principle and Python implementation
• github

quiver plot

Definition

• "A quiver plot displays velocity vectors as arrows with components (u,v) at the points (x,y)"^[13]

stream plot

A stream plot uses stream lines