kreistiefe instructions

By moving the mouse over the picture one aplies socalled Möbius tranformations to the pattern of circle packings (*), i.e. the pattern wiggles around as the mouse moves. Holding down the mouse button stops this action, i.e. the picture stands still.

The hue of a color is a variable which encodes the actual color of a screen color as there are red, green etc. The saturation value encodes how intensive this color appears, in red this means a range from grey (saturation value is 0) to shouting red (saturation value is 1). The brightness value encodes how bright the color appears, i.e. it mediates between black/dark and white/light. Via the three sliders ''Hue, Sat and Bright'' the applet Kreistiefe allows to vary the hue, saturation and brightness of the color of the filled circles.

The button ''reset view'' resets to the default position but keeps the adjusted color. The button ''reset colors'' resets the colors, but keeps the adjusted position.

With the sliders ''SatFade'' and ''BrightFade'' it is possible to ''fade'' saturation and brightness towards the values of the background color. In particular with these sliders one can specify how ''fast'' (with respect to the circle radius) the circles should fade away. Small values mean (nearly) no fading, big values mean strong fading (**). Saturation and Brightness of a circle color depend on the value of ''SatFade'' and ''BrightFade''.(***)

SatFade and BrightFade assume values between 0 and 1. The buttons ''toggle saturation'' and ''toggle brightness'' switch the values SatFade and BrightFade to 1-SatFade and 1-BrightFade, respectively, i.e. they invert the fading process - bigger circles are now lighter (the background color is white) then smaller ones.

For those who are interested:

A Möbius transformation is uniquely determined by specifying the image of three points in the complex plane C. In the applet two points are kept fixed. The position of the mouse pointer gives the third point and hence the corresponding Möbius transformation.

The radius r of the circles is normalized to assume a value between 0 and 1 via the ArcTan function: SatFadeConst = 2/Pi ArcTan(r/SatFade) and BrightFadeConst = 2/Pi ArcTan(r/BrightFade). Via the slider variable SatFade one finally obtains the fading variables SatFadeConst and BrightFadeConst.

Saturation and Brightness in the picture are ''weighted'' with the fading variables: PictureSaturation = Sat* SatFadeConst and PictureBrightness = (1-BrightFadeConst) + BrightFadeConst* Bright

Remark: For some positions there appears to be an unfilled circle, this is when a circle popps over infinity, i.e. it gets so big that the background should have been filled, instead of the inner of the circle. We had to leave it this way in order to keep processing time small. Its not a bug. Its a bad feature.

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