stats = function(a) {
var r = {mean: 0, variance: 0, deviation: 0}, t = a.length;
for(var m, s = 0, l = t; l--; s += a[l]);
for(m = r.mean = s / t, l = t, s = 0; l--; s += Math.pow(a[l] - m, 2));
return r.deviation = Math.sqrt(r.variance = s / t), r;
}
// Matrices look like this:
// q = [[1,2,3],
// [4,5,6],
// [7,3,9]];
// M x B = C
// M-1 x M x B = M-1 x C
// I x B = M-1 x C
var generate141Matrix = function(N) {
result = [];
for (var row = 0; row < N; row++) {
var newRow = [];
for (var col = 0; col < N; col++) {
if (col == row) {
newRow.push(4);
} else if (Math.abs(row - col) == 1) {
newRow.push(1);
} else {
newRow.push(0);
};
};
result.push(newRow);
}
return result;
};
var generateConstantMatrix = function (sampledPoints) {
// given an vector of N points, this should return N-2 ordered pairs
result = [];
result.push(
numeric.sub(numeric.mul(sampledPoints[1], 6),
sampledPoints[0])
);
for (i = 2; i < sampledPoints.length - 2; i++) {
result.push( numeric.mul(sampledPoints[i], 6) );
}
result.push(
numeric.sub(numeric.mul(sampledPoints[sampledPoints.length - 2], 6),
sampledPoints[sampledPoints.length - 1])
);
return result;
}
var convertBSplineControlPointsToBezierControlPoints = function(BSplinePoints) {
// Each Bezier curve has P0 and P3 (endpoints) == sampled points, and the
// intermediate points P1 and P2 are 1/3 & 2/3 along the way between the BSpline
// control points
var beziers = [];
for (var i = 0; i < BSplinePoints.length - 1; i++) {
// bezier control points are:
if (i == 0) {
// First one is a special case
var p0 = BSplinePoints[0];
} else {
var p0 = p3;
}
var p1 = numeric.add( numeric.mul(2/3, BSplinePoints[i]),
numeric.mul(1/3, BSplinePoints[i + 1])
);
var p2 = numeric.add( numeric.mul(1/3, BSplinePoints[i]),
numeric.mul(2/3, BSplinePoints[i + 1])
);
if (i == BSplinePoints.length - 2) {
// Last one is a special case
var p3 = BSplinePoints[BSplinePoints.length - 1];
} else {
var p3 = numeric.add( numeric.mul(1/6, BSplinePoints[i]),
numeric.mul(2/3, BSplinePoints[i + 1]),
numeric.mul(1/6, BSplinePoints[i + 2])
);
}
var bezierSegmentControlPoints = [p0, p1, p2, p3];
beziers.push(bezierSegmentControlPoints);
}
return beziers;
};
var getBezierControlPoints = function(sampledPoints) {
if (sampledPoints.length < 4) {
// We need 4 sampled points to draw a *cubic* bezier through those points
// These 3 cases are for shorter lengths: single point, line, quadratic
if (sampledPoints.length === 3) {
beziers =[sampledPoints[0], sampledPoints[1], sampledPoints[1], sampledPoints[2]];
return [beziers]
} else if (sampledPoints.length === 2) {
beziers = [sampledPoints[0], sampledPoints[0], sampledPoints[1], sampledPoints[1]]
return [beziers]
} else if (sampledPoints.length === 1) {
beziers = [sampledPoints[0], sampledPoints[0], sampledPoints[0], sampledPoints[0]];
return [beziers]
};
}
// Below here we have enough points to generate a cubic bezier
var M = generate141Matrix(sampledPoints.length - 2);
var C = generateConstantMatrix(sampledPoints);
var B = numeric.dot(numeric.inv(M), C);
// B-spline control points also include the endpoints of the sampled data
// These are the B Splines3
B.splice(0, 0, sampledPoints[0]);
B.push(sampledPoints[sampledPoints.length - 1]);
var beziers = convertBSplineControlPointsToBezierControlPoints(B);
return beziers;
}
/*
How does this basically work? Like this:
var M = generate141Matrix(3);
var sampledPoints = [[1,-1],
[-1,2],
[1,4],
[4,3],
[7,5]]
var C = generateConstantMatrix(sampledPoints);
var B = numeric.dot(numeric.inv(M), C);
convertBSplineControlPointsToBezierControlPoints(B);
*/