Applied polar decomposition patch. Fixes Issue 621. Thanks to Christian for the report, Joshua for the fix, Dongsoo for checking the fix.

Applied picking cloth patch. Fixes Issue 646. Thanks to Dongsoo.
Applied patch Softbody updateConstraints. Fixes Issue 503. Thanks to Dave Bruce Phillips and Dongsoo.
Fix various warnigns under Mac OSX.

src/LinearMath/CMakeLists.txt

@@ -8,6 +8,7 @@ SET(LinearMath_SRCS
 	btConvexHull.cpp
 	btConvexHullComputer.cpp
 	btGeometryUtil.cpp
+	btPolarDecomposition.cpp
 	btQuickprof.cpp
 	btSerializer.cpp
 	btVector3.cpp
@@ -28,6 +29,7 @@ SET(LinearMath_HDRS
 	btMatrix3x3.h
 	btMinMax.h
 	btMotionState.h
+	btPolarDecomposition.h
 	btPoolAllocator.h
 	btQuadWord.h
 	btQuaternion.h

src/LinearMath/btIDebugDraw.h

@@ -280,6 +280,7 @@ class	btIDebugDraw
 		}
 	}
 	
+  
 	virtual void drawBox(const btVector3& bbMin, const btVector3& bbMax, const btVector3& color)
 	{
 		drawLine(btVector3(bbMin[0], bbMin[1], bbMin[2]), btVector3(bbMax[0], bbMin[1], bbMin[2]), color);

src/LinearMath/btPolarDecomposition.cpp

@@ -0,0 +1,99 @@
+#include "btPolarDecomposition.h"
+#include "btMinMax.h"
+
+namespace
+{
+  btScalar abs_column_sum(const btMatrix3x3& a, int i)
+  {
+    return btFabs(a[0][i]) + btFabs(a[1][i]) + btFabs(a[2][i]);
+  }
+
+  btScalar abs_row_sum(const btMatrix3x3& a, int i)
+  {
+    return btFabs(a[i][0]) + btFabs(a[i][1]) + btFabs(a[i][2]);
+  }
+
+  btScalar p1_norm(const btMatrix3x3& a)
+  {
+    const btScalar sum0 = abs_column_sum(a,0);
+    const btScalar sum1 = abs_column_sum(a,1);
+    const btScalar sum2 = abs_column_sum(a,2);
+    return btMax(btMax(sum0, sum1), sum2);
+  }
+
+  btScalar pinf_norm(const btMatrix3x3& a)
+  {
+    const btScalar sum0 = abs_row_sum(a,0);
+    const btScalar sum1 = abs_row_sum(a,1);
+    const btScalar sum2 = abs_row_sum(a,2);
+    return btMax(btMax(sum0, sum1), sum2);
+  }
+}
+
+const btScalar btPolarDecomposition::DEFAULT_TOLERANCE = btScalar(0.0001);
+const unsigned int btPolarDecomposition::DEFAULT_MAX_ITERATIONS = 16;
+
+btPolarDecomposition::btPolarDecomposition(btScalar tolerance, unsigned int maxIterations)
+: m_tolerance(tolerance)
+, m_maxIterations(maxIterations)
+{
+}
+
+unsigned int btPolarDecomposition::decompose(const btMatrix3x3& a, btMatrix3x3& u, btMatrix3x3& h) const
+{
+  // Use the 'u' and 'h' matrices for intermediate calculations
+  u = a;
+  h = a.inverse();
+
+  for (unsigned int i = 0; i < m_maxIterations; ++i)
+  {
+    const btScalar h_1 = p1_norm(h);
+    const btScalar h_inf = pinf_norm(h);
+    const btScalar u_1 = p1_norm(u);
+    const btScalar u_inf = pinf_norm(u);
+
+    const btScalar h_norm = h_1 * h_inf;
+    const btScalar u_norm = u_1 * u_inf;
+
+    // The matrix is effectively singular so we cannot invert it
+    if (btFuzzyZero(h_norm) || btFuzzyZero(u_norm))
+      break;
+
+    const btScalar gamma = btPow(h_norm / u_norm, 0.25f);
+    const btScalar inv_gamma = 1.0 / gamma;
+
+    // Determine the delta to 'u'
+    const btMatrix3x3 delta = (u * (gamma - 2.0) + h.transpose() * inv_gamma) * 0.5;
+
+    // Update the matrices
+    u += delta;
+    h = u.inverse();
+
+    // Check for convergence
+    if (p1_norm(delta) <= m_tolerance * u_1)
+    {
+      h = u.transpose() * a;
+      h = (h + h.transpose()) * 0.5;
+      return i;
+    }
+  }
+
+  // The algorithm has failed to converge to the specified tolerance, but we
+  // want to make sure that the matrices returned are in the right form.
+  h = u.transpose() * a;
+  h = (h + h.transpose()) * 0.5;
+
+  return m_maxIterations;
+}
+
+unsigned int btPolarDecomposition::maxIterations() const
+{
+  return m_maxIterations;
+}
+
+unsigned int polarDecompose(const btMatrix3x3& a, btMatrix3x3& u, btMatrix3x3& h)
+{
+  static btPolarDecomposition polar;
+  return polar.decompose(a, u, h);
+}
+

src/LinearMath/btPolarDecomposition.h

@@ -0,0 +1,73 @@
+#ifndef POLARDECOMPOSITION_H
+#define POLARDECOMPOSITION_H
+
+#include "btMatrix3x3.h"
+
+/**
+ * This class is used to compute the polar decomposition of a matrix. In
+ * general, the polar decomposition factorizes a matrix, A, into two parts: a
+ * unitary matrix (U) and a positive, semi-definite Hermitian matrix (H).
+ * However, in this particular implementation the original matrix, A, is
+ * required to be a square 3x3 matrix with real elements. This means that U will
+ * be an orthogonal matrix and H with be a positive-definite, symmetric matrix.
+ */
+class btPolarDecomposition
+{
+  public:
+    static const btScalar DEFAULT_TOLERANCE;
+    static const unsigned int DEFAULT_MAX_ITERATIONS;
+
+    /**
+     * Creates an instance with optional parameters.
+     *
+     * @param tolerance     - the tolerance used to determine convergence of the
+     *                        algorithm
+     * @param maxIterations - the maximum number of iterations used to achieve
+     *                        convergence
+     */
+    btPolarDecomposition(btScalar tolerance = DEFAULT_TOLERANCE, 
+      unsigned int maxIterations = DEFAULT_MAX_ITERATIONS);
+
+    /**
+     * Decomposes a matrix into orthogonal and symmetric, positive-definite
+     * parts. If the number of iterations returned by this function is equal to
+     * the maximum number of iterations, the algorithm has failed to converge.
+     *
+     * @param a - the original matrix
+     * @param u - the resulting orthogonal matrix
+     * @param h - the resulting symmetric matrix
+     *
+     * @return the number of iterations performed by the algorithm.
+     */
+    unsigned int decompose(const btMatrix3x3& a, btMatrix3x3& u, btMatrix3x3& h) const; 
+
+    /**
+     * Returns the maximum number of iterations that this algorithm will perform
+     * to achieve convergence.
+     *
+     * @return maximum number of iterations
+     */
+    unsigned int maxIterations() const;
+
+  private:
+    btScalar m_tolerance;
+    unsigned int m_maxIterations;
+};
+
+/**
+ * This functions decomposes the matrix 'a' into two parts: an orthogonal matrix
+ * 'u' and a symmetric, positive-definite matrix 'h'. If the number of
+ * iterations returned by this function is equal to
+ * btPolarDecomposition::DEFAULT_MAX_ITERATIONS, the algorithm has failed to
+ * converge.
+ *
+ * @param a - the original matrix
+ * @param u - the resulting orthogonal matrix
+ * @param h - the resulting symmetric matrix
+ *
+ * @return the number of iterations performed by the algorithm.
+ */
+unsigned int polarDecompose(const btMatrix3x3& a, btMatrix3x3& u, btMatrix3x3& h); 
+
+#endif // POLARDECOMPOSITION_H
+