10.1.5.2 Finding tangents

As mentioned above, Limn moves the control points on the spline to optimally fit the bitmap. But it cannot just move them arbitrarily, because it must make sure that the spline fitting one part of the bitmap blends smoothly with those fit to adjacent parts.

Technically, this smooth blending is called continuity, and it comes in degrees. Limn is concerned with the first two degrees: zero- and first-order. Zero-order continuity between two curves simply means the curves are connected; first-order geometric (G1) continuity means the tangents to each curve at the point of connection have the same direction. (There are other kinds of continuity besides "geometric", but they are not important for our purposes.)

Informally, this means that the final shape will not abruptly turn at the point where two splines meet. (Any computer graphics textbook will discuss the properties of tangents, continuity, and splines, if you're unfamiliar with the subject.)

To achieve G1 continuity, Limn puts the first control point of a spline on a line going in the direction of the tangent to the start of the spline; and it puts the second control point on a line in the direction of the tangent to the end of the spline. (It would be going far afield to prove that this together with the properties of Bezier splines imply G1 continuity, but they do. See Schneider's thesis referenced in `limn/README' for a complete mathematical treatment.)

For the purposes of using Limn, the important thing is that Limn must compute the tangents to the spline at the beginning and end, and must do so accurately in order to achieve a good fit to the bitmap. Since Limn has available only samples (i.e., the pixel coordinates) of the curve being fit, it cannot compute the true tangent. Instead, it must approximate the tangent by looking at some number of coordinates on either side of a point. By default, the number is 3, but you can specify a different number with the `-tangent-surround' option. If the resolution of the input font is different than 300dpi, or if the outline Limn fits to the bitmap seems off, you will want to scale it proportionately.