Does the user have any control over the number of points used?
Not directly, but the size of the source image, the threshold value, and the filter setting all affect the result.
The source image of the lower case "a" was scanned at 300 dpi. A 600 dpi scan will most likely result in more points, but in this case I don't think it will benefit the result as that is already very good (if I may say so myself). A 600 dpi scan resolution is useful when your characters contain little details (e.g. thin strokes).
Scanahand needs to find out what pixels belong to the characters. The threshold value is used to make that decision. Usually the Auto value gives best results, but you can also choose between Low, Medium, and High. The Low threshold removes all light colored pixels, while the High threshold includes all light colored pixels. When you use the High threshold you get slightly fatter characters, but you risk inclusion of unwanted pixels.
You can also reduce the number of points by applying either the Smooth or Super filter. Those filters polish the outlines and that usually results in more rounded contours.