- generated a curve passing through all polygon vertices
- had some way of controlling tightness of fit to the polygon
- fell within the domain of my very limited mathematical grasp

Luckily for me, Maxim Shemanarev had not only worked out a Bezier curve algorithm that satisfied all of the above, but he also published a beautifully clear explanation of how it works

^{note 1}. Maxim also provided some illustrative C code, but my Java implementation below is an independent effort (ie. all mistakes are my fault). It uses geometry objects from the JTS library.

Below is an example of the algorithm in action. The original polygon is in black. The smoothed polygon composed of fitted cubic Bezier segments is in blue. The red points and lines indicate positions of the Bezier control points calculated with Maxim's algorithm. There are a pair of control points for each vertex of the input polygon.

The tightness of fit of the smoothed boundary to the original boundary is controlled by a single parameter which varies between 0 and 1. In the figure above the tightness was set to 0. Below is the same polygon smoothed with tightness set to 0.6.

Notice that the higher tightness parameter has shifted each pair of control points closer to their respective polygon vertices.

Finally, here's the Java code. Share and enjoy...

/*

* Copyright 2010 Michael Bedward

*

* This file is part of jai-tools.

*

* jai-tools is free software: you can redistribute it and/or modify

* it under the terms of the GNU Lesser General Public License as

* published by the Free Software Foundation, either version 3 of the

* License, or (at your option) any later version.

*

* jai-tools is distributed in the hope that it will be useful,

* but WITHOUT ANY WARRANTY; without even the implied warranty of

* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the

* GNU Lesser General Public License for more details.

*

* You should have received a copy of the GNU Lesser General Public

* License along with jai-tools. If not, see <http://www.gnu.org/licenses/>.

*/

package mxb.interpolation;

import com.vividsolutions.jts.geom.Coordinate;

import com.vividsolutions.jts.geom.GeometryFactory;

import com.vividsolutions.jts.geom.LinearRing;

import com.vividsolutions.jts.geom.Polygon;

/**

* Polygon smoothing by interpolation with cubic Bazier curves.

* This code implements an algorithm which was devised by Maxim Shemanarev

* and described by him at:

* <a href="http://www.antigrain.com/research/bezier_interpolation/index.html">

* http://www.antigrain.com/research/bezier_interpolation/index.html</a>.

*

* Note: the code here is <b>not</b> that written by Maxim to accompany his

* algorithm description. Rather, it is an original implementation and any

* errors are my fault.

*/

public class PolygonSmoother {

private GeometryFactory geomFactory;

/**

* Calculates a pair of Bezier control points for each vertex in an

* array of {@code Coordinates}.

*

* @param coords vertex coordinates

* @param N number of coordinates in {@coords} to use

* @param alpha tightness of fit

*

* @return array of {@code Coordinates} for positions of control points

*/

private Coordinate[][] getControlPoints(Coordinate[] coords, int N, double alpha) {

if (alpha < 0.0 || alpha > 1.0) {

throw new IllegalArgumentException("alpha must be a value between 0 and 1 inclusive");

}

Coordinate[][] ctrl = new Coordinate[N][2];

Coordinate[] v = new Coordinate[3];

Coordinate[] mid = new Coordinate[2];

mid[0] = new Coordinate();

mid[1] = new Coordinate();

Coordinate anchor = new Coordinate();

double[] vdist = new double[2];

double mdist;

v[1] = coords[N - 1];

v[2] = coords[0];

mid[1].x = (v[1].x + v[2].x) / 2.0;

mid[1].y = (v[1].y + v[2].y) / 2.0;

vdist[1] = v[1].distance(v[2]);

for (int i = 0; i < N; i++) {

v[0] = v[1];

v[1] = v[2];

v[2] = coords[(i + 1) % N];

mid[0].x = mid[1].x;

mid[0].y = mid[1].y;

mid[1].x = (v[1].x + v[2].x) / 2.0;

mid[1].y = (v[1].y + v[2].y) / 2.0;

vdist[0] = vdist[1];

vdist[1] = v[1].distance(v[2]);

double p = vdist[0] / (vdist[0] + vdist[1]);

anchor.x = mid[0].x + p * (mid[1].x - mid[0].x);

anchor.y = mid[0].y + p * (mid[1].y - mid[0].y);

double xdelta = anchor.x - v[1].x;

double ydelta = anchor.y - v[1].y;

ctrl[i][0] = new Coordinate(

alpha*(v[1].x - mid[0].x + xdelta) + mid[0].x - xdelta,

alpha*(v[1].y - mid[0].y + ydelta) + mid[0].y - ydelta);

ctrl[i][1] = new Coordinate(

alpha*(v[1].x - mid[1].x + xdelta) + mid[1].x - xdelta,

alpha*(v[1].y - mid[1].y + ydelta) + mid[1].y - ydelta);

}

return ctrl;

}

/**

* Calculates vertices along a cubic Bazier curve given start point, end point

* and two control points.

*

* @param start start position

* @param end end position

* @param ctrl1 first control point

* @param ctrl2 second control point

* @param nv number of vertices including the start and end points

*

* @return vertices along the Bezier curve

*/

private Coordinate[] cubicBezier(final Coordinate start, final Coordinate end,

final Coordinate ctrl1, final Coordinate ctrl2, final int nv) {

final Coordinate[] curve = new Coordinate[nv];

final Coordinate[] buf = new Coordinate[3];

for (int i = 0; i < buf.length; i++) {

buf[i] = new Coordinate();

}

curve[0] = new Coordinate(start);

curve[nv - 1] = new Coordinate(end);

for (int i = 1; i < nv-1; i++) {

double t = (double) i / (nv - 1);

double tc = 1.0 - t;

double t0 = tc*tc*tc;

double t1 = 3.0*tc*tc*t;

double t2 = 3.0*tc*t*t;

double t3 = t*t*t;

double tsum = t0 + t1 + t2 + t3;

Coordinate c = new Coordinate();

c.x = t0*start.x + t1*ctrl1.x + t2*ctrl2.x + t3*end.x;

c.x /= tsum;

c.y = t0*start.y + t1*ctrl1.y + t2*ctrl2.y + t3*end.y;

c.y /= tsum;

curve[i] = c;

}

return curve;

}

/**

* Creates a new {@code Polygon} whose exterior shell is a smoothed

* version of the input {@code Polygon}.

* <p>

* Note: this method presently ignores holes.

*

* @param p the input {@code Polygon}

*

* @param alpha a value between 0 and 1 (inclusive) specifying the tightness

* of fit of the smoothed boundary (0 is loose)

*

* @param pointsPerSegment the number of vertices to use to represent each

* Bezier curve derived from an edge of the input {@code Polygon}

*

* @return the smoothed {@code Polygon}

*/

public Polygon smooth(Polygon p, double alpha, int pointsPerSegment) {

if (geomFactory == null) {

geomFactory = new GeometryFactory();

}

Coordinate[] coords = p.getExteriorRing().getCoordinates();

int Nvertices = coords.length - 1; // first coord == last coord

Coordinate[][] controlPoints = getControlPoints(coords, Nvertices, alpha);

Coordinate[] smoothCoords = new Coordinate[Nvertices * pointsPerSegment];

for (int i = 0, k = 0; i < Nvertices; i++) {

int next = (i + 1) % Nvertices;

Coordinate[] segment = cubicBezier(

coords[i], coords[next],

controlPoints[i][1], controlPoints[next][0],

pointsPerSegment);

for (int j = 0; j < pointsPerSegment; j++, k++) {

smoothCoords[k] = segment[j];

}

}

LinearRing shell = geomFactory.createLinearRing(smoothCoords);

return geomFactory.createPolygon(shell, null);

}

}

^{1}Maxim Shemanarev's algorithm http://www.antigrain.com/research/bezier_interpolation/index.html