# Fractals/Computer graphic techniques/2D/transformation

Plane transformations

# Names

• transformations
• projections
• Map projection[1] ( in case of 3D to 2D graphic)
• mappings

# Theory

Mapping in 2D raster graphic is a complex function ${\displaystyle f}$ that maps from complex plane to complex plane:

${\displaystyle f:\mathbb {C} \to \mathbb {C} }$


It is implemented as point transformations. For instance, Mercator:[2]

function Spherical_mercator(x, y) {
return [x, Math.log(Math.tan(Math.PI / 4 + y / 2))];
}

function transverseMercatorRaw(lambda, phi) {
return [log(tan((halfPi + phi) / 2)), -lambda];
}

transverseMercatorRaw.invert = function(x, y) {
return [-y, 2 * atan(exp(x)) - halfPi];
};


If one work with discrete geometry ( polygons and polylines) then the projecting is much harder to imlement. One have to balance accuracy and performance.[3]

# classifications

Maps clasifications

• object-based mappings ( Image objects are sets of connected pixels having the same integer value )
• pixel based mappings

Transformations ( maps) in 2-D graphics

• basic types
• Composite Transformation [4]

# types

Examples:[5][6]

## Plane inversion

### Parabola inversion

Inversion of a parabola = cardioid

### Mandelbrot set inversion

Inversion of lambda Mandelbrot set with different translations

### Julia set inversion

• f(z) = z^2+c convert to f(z) = z^2 + 1/c
• inverting the polynomial formula Ax(1-x) in the circle of radius 1, resulting in the formula f(x)= x^2/A(1-x).[19]

#### basilica

• normal basilica f(z) = z^2 - 1
• inverted basilica
• rational with degree = 2 : reversed basilica[20] : f(z) = z^2/(z^2 -1)

## Conformal map

• "Conformal maps preserve angles and take infinitesimal circles to infinitesimal circles. Non-conformal maps take infinitesimal circles to infinitesimal ellipses (or worse)." Claude Heiland-Allen

Conformal maps

Dictionary of Conformal Mapping by John H. Mathews, Russell W. Howell ( 2008)

## cylindrical projection

The geometry for the normal (equatorial) tangent projections of the sphere to the cylinder.

Cylindrical projection ( or cylindric p. ) maps from sphere ( without poles) to cylinder [21][22]

Description :

## domain coloring or complex phase plots

Visualisation of complex functions