# Fractals/curves

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

Cyrve types[1]

• 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:

# Field lines

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]
```