Curves

Polylines have abrupt, often sharp, corners as they trace out a circuit. These paths do not flow, they jerk. To smooth a polyline path to a flowing curve, Anton uses what mathematicians call spline interpolation. This is a bit like fitting a thin springy strip of steel around a set of pegs to form a curved path that touches each peg.

Cubic functions (the simplest is y = x3) have curvy S-shaped graphs. They have the remarkable property that given four points (not all on a line), there is a cubic function whose graph goes through those four points. If the three points are fairly close to each other, the piece of the cubic curve through them (called a spline) is a close approximation to line segments that connect the three points. Using splines, Anton can replace each sharp V corner of a polyline path with a U curve. The result is a smooth curvacous circuit that travels through all of the corners of the polyline path.

The curved loop that results from smoothing a polyline circuit in space is merely a skeleton doodle with no thickness, no body, and this must be provided by the artist. A simple thickening coats the curve so it has a uniform cross-section such as a circle (which produces a tube covering), a square, or triangle. For aesthetic reasons, it is more interesting to vary the width and thickness of the curve’s covering. This can suggest a change of speed and spread as the curve flows, like water flowing in a creek that meanders through changing terrain.