The fineness factor determines how high the resolution of the graph will be, and in effect, also the amount of time it takes to complete the graph. The fineness factor of 1 is quite adequate in most circumstances, but you can still manually control the fineness to fill the need for especially sharp graphs or for extra-quick renderings. As the fineness is increased, more of the curve will be smoothed out, but remember that if you increase the fineness factor to 5, the computer will be slowed down to 1/5 its normal graphing speed under the load of all the extra calculations.
Fineness is also linked to the rate at which the angle is allowed to change in polar graphs, and both Cartesian and Polar graphs will be of comparable quality at the same fineness factor. Fineness is also linked to the step rate of parametric graphs and differential equations; because they vary so much, it is harder to insure that all parametrics and ODEs will graph well at the default fineness, but those that I have tested look fine.
When graphing differential equations (slope fields), the fineness factor controls the interval at which the hatch marks are drawn. If you use too high of a fineness value on this type of graph, you may not only get an awful mess on the screen, but also exceed the capacity of the data structure that records the points that were graphed for later redrawing. [At this point Graphmatica simply stops recording, so no harm can be done, but the graph will not redraw properly when the screen is repainted but not recalculated.]
To change the fineness factor, choose Settings from the Options menu and enter a new value in the edit box. Any value greater than zero is valid; the default value is 1.0. Theoretically there are no other limits on the fineness, and you can type any value into the edit field you wish, but the scroll bar slider is limited to the reasonable range of about 0.2 to 6. I would recommend not going below 0.25, as the image quality suffers and the graphs begin to look like modern art. Also, a factor greater than 5 will likely exceed the resolution available on your screen, in addition to taking proportionately longer to calculate.
If you increase the fineness, the graphs on screen will be redrawn at the higher resolution. If you decrease the fineness, the change will take effect the next time you draw a graph.