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203 lines
6.9 KiB
203 lines
6.9 KiB
// David Eberly, Geometric Tools, Redmond WA 98052
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// Copyright (c) 1998-2021
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// Distributed under the Boost Software License, Version 1.0.
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// https://www.boost.org/LICENSE_1_0.txt
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// https://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
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// Version: 4.0.2019.08.13
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#pragma once
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#include <Mathematics/Matrix.h>
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#include <Mathematics/Vector2.h>
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#include <Mathematics/Hypersphere.h>
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#include <Mathematics/SymmetricEigensolver.h>
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namespace gte
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{
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// The quadratic fit is
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// 0 = C[0] + C[1]*X + C[2]*Y + C[3]*X^2 + C[4]*Y^2 + C[5]*X*Y
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// subject to Length(C) = 1. Minimize E(C) = C^t M C with Length(C) = 1
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// and M = (sum_i V_i)(sum_i V_i)^t where
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// V = (1, X, Y, X^2, Y^2, X*Y)
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// The minimum value is the smallest eigenvalue of M and C is a
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// corresponding unit length eigenvector.
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//
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// Input:
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// n = number of points to fit
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// p[0..n-1] = array of points to fit
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//
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// Output:
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// c[0..5] = coefficients of quadratic fit (the eigenvector)
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// return value of function is nonnegative and a measure of the fit
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// (the minimum eigenvalue; 0 = exact fit, positive otherwise)
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//
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// Canonical forms. The quadratic equation can be factored into
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// P^T A P + B^T P + K = 0 where P = (X,Y,Z), K = C[0],
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// B = (C[1],C[2],C[3]), and A is a 3x3 symmetric matrix with
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// A00 = C[4], A11 = C[5], A22 = C[6], A01 = C[7]/2, A02 = C[8]/2,
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// and A12 = C[9]/2. Matrix A = R^T D R where R is orthogonal and
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// D is diagonal (using an eigendecomposition). Define
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// V = R P = (v0,v1,v2), E = R B = (e0,e1,e2), D = diag(d0,d1,d2)
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// and f = K to obtain
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// d0 v0^2 + d1 v1^2 + d2 v^2 + e0 v0 + e1 v1 + e2 v2 + f = 0
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// The characterization depends on the signs of the d_i.
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template <typename Real>
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class ApprQuadratic2
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{
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public:
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Real operator()(int numPoints, Vector2<Real> const* points, Real coefficients[6])
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{
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Matrix<6, 6, Real> A; // constructor sets A to zero
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for (int i = 0; i < numPoints; ++i)
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{
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Real x = points[i][0];
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Real y = points[i][1];
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Real x2 = x * x;
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Real y2 = y * y;
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Real xy = x * y;
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Real x3 = x * x2;
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Real xy2 = x * y2;
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Real x2y = x * xy;
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Real y3 = y * y2;
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Real x4 = x * x3;
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Real x2y2 = x * xy2;
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Real x3y = x * x2y;
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Real y4 = y * y3;
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Real xy3 = x * y3;
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A(0, 1) += x;
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A(0, 2) += y;
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A(0, 3) += x2;
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A(0, 4) += y2;
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A(0, 5) += xy;
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A(1, 3) += x3;
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A(1, 4) += xy2;
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A(1, 5) += x2y;
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A(2, 4) += y3;
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A(3, 3) += x4;
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A(3, 4) += x2y2;
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A(3, 5) += x3y;
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A(4, 4) += y4;
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A(4, 5) += xy3;
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}
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A(0, 0) = static_cast<Real>(numPoints);
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A(1, 1) = A(0, 3);
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A(1, 2) = A(0, 5);
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A(2, 2) = A(0, 4);
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A(2, 3) = A(1, 5);
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A(2, 5) = A(1, 4);
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A(5, 5) = A(3, 4);
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for (int row = 0; row < 6; ++row)
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{
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for (int col = 0; col < row; ++col)
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{
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A(row, col) = A(col, row);
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}
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}
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Real invNumPoints = (Real)1 / static_cast<Real>(numPoints);
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for (int row = 0; row < 6; ++row)
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{
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for (int col = 0; col < 6; ++col)
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{
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A(row, col) *= invNumPoints;
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}
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}
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SymmetricEigensolver<Real> es(6, 1024);
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es.Solve(&A[0], +1);
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es.GetEigenvector(0, &coefficients[0]);
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// For an exact fit, numeric round-off errors might make the
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// minimum eigenvalue just slightly negative. Return the
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// absolute value because the application might rely on the
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// return value being nonnegative.
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return std::fabs(es.GetEigenvalue(0));
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}
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};
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// If you think your points are nearly circular, use this. The circle is
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// of the form C'[0]+C'[1]*X+C'[2]*Y+C'[3]*(X^2+Y^2), where
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// Length(C') = 1. The function returns
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// C = (C'[0]/C'[3],C'[1]/C'[3],C'[2]/C'[3]), so the fitted circle is
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// C[0]+C[1]*X+C[2]*Y+X^2+Y^2. The center is (xc,yc) = -0.5*(C[1],C[2])
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// and the radius is r = sqrt(xc*xc+yc*yc-C[0]).
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template <typename Real>
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class ApprQuadraticCircle2
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{
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public:
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Real operator()(int numPoints, Vector2<Real> const* points, Circle2<Real>& circle)
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{
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Matrix<4, 4, Real> A; // constructor sets A to zero
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for (int i = 0; i < numPoints; ++i)
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{
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Real x = points[i][0];
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Real y = points[i][1];
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Real x2 = x * x;
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Real y2 = y * y;
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Real xy = x * y;
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Real r2 = x2 + y2;
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Real xr2 = x * r2;
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Real yr2 = y * r2;
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Real r4 = r2 * r2;
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A(0, 1) += x;
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A(0, 2) += y;
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A(0, 3) += r2;
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A(1, 1) += x2;
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A(1, 2) += xy;
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A(1, 3) += xr2;
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A(2, 2) += y2;
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A(2, 3) += yr2;
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A(3, 3) += r4;
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}
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A(0, 0) = static_cast<Real>(numPoints);
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for (int row = 0; row < 4; ++row)
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{
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for (int col = 0; col < row; ++col)
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{
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A(row, col) = A(col, row);
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}
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}
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Real invNumPoints = (Real)1 / static_cast<Real>(numPoints);
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for (int row = 0; row < 4; ++row)
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{
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for (int col = 0; col < 4; ++col)
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{
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A(row, col) *= invNumPoints;
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}
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}
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SymmetricEigensolver<Real> es(4, 1024);
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es.Solve(&A[0], +1);
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Vector<4, Real> evector;
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es.GetEigenvector(0, &evector[0]);
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// TODO: Guard against zero divide?
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Real inv = (Real)1 / evector[3];
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Real coefficients[3];
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for (int row = 0; row < 3; ++row)
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{
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coefficients[row] = inv * evector[row];
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}
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circle.center[0] = (Real)-0.5 * coefficients[1];
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circle.center[1] = (Real)-0.5 * coefficients[2];
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circle.radius = std::sqrt(std::fabs(Dot(circle.center, circle.center) - coefficients[0]));
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// For an exact fit, numeric round-off errors might make the
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// minimum eigenvalue just slightly negative. Return the
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// absolute value because the application might rely on the
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// return value being nonnegative.
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return std::fabs(es.GetEigenvalue(0));
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}
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};
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}
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