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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.

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