#
Inversion

Many years ago there was a brief craze for "anamorphic art": images
painted and viewed with the use of a cylindrical mirror, so that they
looked weirdly distorted on the canvas but correct in the distorted
reflection. (This byway of art history may be a useful defense when your
friends and family give you odd looks for staring at fractal images
color-cycling on a CRT.)

The Inversion option performs a related transformation on most of the
fractal types. You define the center point and radius of a circle;
Fractint maps each point inside the circle to a corresponding point
outside, and vice-versa. This is known to mathematicians as inverting
(or if you want to get precise, "everting") the plane, and is something
they can contemplate without getting a headache. John Milnor (also
mentioned in connection with the
Distance Estimator Method ),
made his name in the 1950s with a method for everting a seven-
dimensional sphere, so we have a lot of catching up to do.

For example, if a point inside the circle is 1/3 of the way from the
center to the radius, it is mapped to a point along the same radial
line, but at a distance of (3 * radius) from the origin. An outside
point at 4 times the radius is mapped inside at 1/4 the radius.

The inversion parameters on the **[Y]** options screen allow entry of the
radius and center coordinates of the inversion circle. A default choice
of -1 sets the radius at 1/6 the smaller dimension of the image
currently on the screen. The default values for Xcenter and Ycenter use
the coordinates currently mapped to the center of the screen.

Try this one out with a
Newton plot, so its radial "spokes"
will give you something to hang on to. Plot a Newton-method image, then
set the inversion radius to 1, with default center coordinates. The
center "explodes" to the periphery.

Inverting through a circle not centered on the origin produces bizarre
effects that we're not even going to try to describe. Aren't computers
wonderful?

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### This page maintained by

Noel Giffin,

noel@triumf.ca