Fractals/curves

Curves

• definition

2D scalar field

• level curves
• field lines
• boundary
  • component of Parameter plane (Mandelbrot set, exterior of M set)
  • of component of Mandebrot set
  • of dynamic plane (Fatou components , Julia set)
  • components of Julia set

types

Cyrve types

• defined by equation
  • Implicit curves
  • Explicit curves
  • Parametric curves
• spline curves
  • Bezier curves
  • B-spline curves

Level curves

Level cures:

• edge detection of level sets gives boundaries
• tracing equilevel curves
  • computes contour polygons by applying marching squares to a rectangular array of numeric values ( d3-contour library )
• inverse iteration of closed curve ( circle) around attractor

Level curves of 2D scalar field:

• equipotential lines ( isocurves , contour lines)
• boundary
• filed lines ( rays) = level curves of external angle

• 
• 
  preimages of circle : {z:abs(z)=ER} under fc(z)=z*z+c. Preimages of circle from the exterior gives level sets of the escaping time. Exterior of the circle is the trap for escaping points ( target set)
• 
  edge detection of boundaries of Level Sets of escape time
• 
  Level curves - method unknown, probably edge detection
• 
  equipotential lines and external rays as preimages under Riemann map
• 
  array of pixels for diminish number of points
• 
  preimages of the circle from the Fatou set's interior

Field lines

• parameter plane
  • external rays on the parameter plane = parameter rays
  • internal rays
• dynamic plane
  • external rays on dynamic plane = dynamic rays
  • internal rays

• 
  parameter external and internal rays
• 
• 
  Parameter external and internal rays
• 
  Dynamic internal and external rays
• 
• 

The field lines of a vector field ( the gradient of a scalar field ) can be drawn without solving a DE, because the gradient of a scalar field is orthogonal to the level curves of the scalar field. In general they can only be solved numerically using a computer.

the most convenient way to get a metric perpendicular to equipotentials is to take the arctan2 of the point at bailout. That gives field lines (or lines of longitude) that branch at each iteration boundary in exterior regions. In interior regions they do not branch, so it's easy to apply continuous textures. ( xenodreambuie) 

See also

• Category:External rays : images in Commons, some with src code
• Category:Level_set_method in commons
• 2D algorithms
• axismaps: go-with-the-flow
  • awoodruff: relief-flow
  • wind map and code