Fractals/curves
< Fractals
Curves
 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 edit
Cyrve types^{[1]}
 defined by equation
 Implicit curves
 Explicit curves
 Parametric curves
 spline curves
 Bezier curves
 Bspline curves
Level curves edit
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 ( d3contour 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 edit
 parameter plane
 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) ^{[2]}