spatial/r3/mat.go

// Copyright ©2021 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package r3

import "gonum.org/v1/gonum/mat"

// Mat represents a 3×3 matrix. Useful for rotation matrices and such.
// The zero value is usable as the 3×3 zero matrix.
type Mat struct {
	data *array
}

var _ mat.Matrix = (*Mat)(nil)

// NewMat returns a new 3×3 matrix Mat type and populates its elements
// with values passed as argument in row-major form. If val argument
// is nil then NewMat returns a matrix filled with zeros.
func NewMat(val []float64) *Mat {
	if len(val) == 9 {
		return &Mat{arrayFrom(val)}
	}
	if val == nil {
		return &Mat{new(array)}
	}
	panic(mat.ErrShape)
}

// Dims returns the number of rows and columns of this matrix.
// This method will always return 3×3 for a Mat.
func (m *Mat) Dims() (r, c int) { return 3, 3 }

// T returns the transpose of Mat. Changes in the receiver will be reflected in the returned matrix.
func (m *Mat) T() mat.Matrix { return mat.Transpose{Matrix: m} }

// Scale multiplies the elements of a by f, placing the result in the receiver.
//
// See the mat.Scaler interface for more information.
func (m *Mat) Scale(f float64, a mat.Matrix) {
	r, c := a.Dims()
	if r != 3 || c != 3 {
		panic(mat.ErrShape)
	}
	if m.data == nil {
		m.data = new(array)
	}
	for i := 0; i < 3; i++ {
		for j := 0; j < 3; j++ {
			m.Set(i, j, f*a.At(i, j))
		}
	}
}

// MulVec returns the matrix-vector product M⋅v.
func (m *Mat) MulVec(v Vec) Vec {
	if m.data == nil {
		return Vec{}
	}
	return Vec{
		X: v.X*m.At(0, 0) + v.Y*m.At(0, 1) + v.Z*m.At(0, 2),
		Y: v.X*m.At(1, 0) + v.Y*m.At(1, 1) + v.Z*m.At(1, 2),
		Z: v.X*m.At(2, 0) + v.Y*m.At(2, 1) + v.Z*m.At(2, 2),
	}
}

// MulVecTrans returns the matrix-vector product Mᵀ⋅v.
func (m *Mat) MulVecTrans(v Vec) Vec {
	if m.data == nil {
		return Vec{}
	}
	return Vec{
		X: v.X*m.At(0, 0) + v.Y*m.At(1, 0) + v.Z*m.At(2, 0),
		Y: v.X*m.At(0, 1) + v.Y*m.At(1, 1) + v.Z*m.At(2, 1),
		Z: v.X*m.At(0, 2) + v.Y*m.At(1, 2) + v.Z*m.At(2, 2),
	}
}

// CloneFrom makes a copy of a into the receiver m.
// Mat expects a 3×3 input matrix.
func (m *Mat) CloneFrom(a mat.Matrix) {
	r, c := a.Dims()
	if r != 3 || c != 3 {
		panic(mat.ErrShape)
	}
	if m.data == nil {
		m.data = new(array)
	}
	for i := 0; i < 3; i++ {
		for j := 0; j < 3; j++ {
			m.Set(i, j, a.At(i, j))
		}
	}
}

// Sub subtracts the matrix b from a, placing the result in the receiver.
// Sub will panic if the two matrices do not have the same shape.
func (m *Mat) Sub(a, b mat.Matrix) {
	if r, c := a.Dims(); r != 3 || c != 3 {
		panic(mat.ErrShape)
	}
	if r, c := b.Dims(); r != 3 || c != 3 {
		panic(mat.ErrShape)
	}
	if m.data == nil {
		m.data = new(array)
	}

	m.Set(0, 0, a.At(0, 0)-b.At(0, 0))
	m.Set(0, 1, a.At(0, 1)-b.At(0, 1))
	m.Set(0, 2, a.At(0, 2)-b.At(0, 2))
	m.Set(1, 0, a.At(1, 0)-b.At(1, 0))
	m.Set(1, 1, a.At(1, 1)-b.At(1, 1))
	m.Set(1, 2, a.At(1, 2)-b.At(1, 2))
	m.Set(2, 0, a.At(2, 0)-b.At(2, 0))
	m.Set(2, 1, a.At(2, 1)-b.At(2, 1))
	m.Set(2, 2, a.At(2, 2)-b.At(2, 2))
}

// Add adds a and b element-wise, placing the result in the receiver. Add will panic if the two matrices do not have the same shape.
func (m *Mat) Add(a, b mat.Matrix) {
	if r, c := a.Dims(); r != 3 || c != 3 {
		panic(mat.ErrShape)
	}
	if r, c := b.Dims(); r != 3 || c != 3 {
		panic(mat.ErrShape)
	}
	if m.data == nil {
		m.data = new(array)
	}

	m.Set(0, 0, a.At(0, 0)+b.At(0, 0))
	m.Set(0, 1, a.At(0, 1)+b.At(0, 1))
	m.Set(0, 2, a.At(0, 2)+b.At(0, 2))
	m.Set(1, 0, a.At(1, 0)+b.At(1, 0))
	m.Set(1, 1, a.At(1, 1)+b.At(1, 1))
	m.Set(1, 2, a.At(1, 2)+b.At(1, 2))
	m.Set(2, 0, a.At(2, 0)+b.At(2, 0))
	m.Set(2, 1, a.At(2, 1)+b.At(2, 1))
	m.Set(2, 2, a.At(2, 2)+b.At(2, 2))
}

// VecRow returns the elements in the ith row of the receiver.
func (m *Mat) VecRow(i int) Vec {
	if i > 2 {
		panic(mat.ErrRowAccess)
	}
	if m.data == nil {
		return Vec{}
	}
	return Vec{X: m.At(i, 0), Y: m.At(i, 1), Z: m.At(i, 2)}
}

// VecCol returns the elements in the jth column of the receiver.
func (m *Mat) VecCol(j int) Vec {
	if j > 2 {
		panic(mat.ErrColAccess)
	}
	if m.data == nil {
		return Vec{}
	}
	return Vec{X: m.At(0, j), Y: m.At(1, j), Z: m.At(2, j)}
}

// Outer calculates the outer product of the vectors x and y,
// where x and y are treated as column vectors, and stores the result in the receiver.
//
//	m = alpha * x * yᵀ
func (m *Mat) Outer(alpha float64, x, y Vec) {
	ax := alpha * x.X
	ay := alpha * x.Y
	az := alpha * x.Z
	m.Set(0, 0, ax*y.X)
	m.Set(0, 1, ax*y.Y)
	m.Set(0, 2, ax*y.Z)

	m.Set(1, 0, ay*y.X)
	m.Set(1, 1, ay*y.Y)
	m.Set(1, 2, ay*y.Z)

	m.Set(2, 0, az*y.X)
	m.Set(2, 1, az*y.Y)
	m.Set(2, 2, az*y.Z)
}

// Det calculates the determinant of the receiver using the following formula
//
//	    ⎡a b c⎤
//	m = ⎢d e f⎥
//	    ⎣g h i⎦
//	det(m) = a(ei − fh) − b(di − fg) + c(dh − eg)
func (m *Mat) Det() float64 {
	a := m.At(0, 0)
	b := m.At(0, 1)
	c := m.At(0, 2)

	deta := m.At(1, 1)*m.At(2, 2) - m.At(1, 2)*m.At(2, 1)
	detb := m.At(1, 0)*m.At(2, 2) - m.At(1, 2)*m.At(2, 0)
	detc := m.At(1, 0)*m.At(2, 1) - m.At(1, 1)*m.At(2, 0)
	return a*deta - b*detb + c*detc
}

// Skew sets the receiver to the 3×3 skew symmetric matrix
// (right hand system) of v.
//
//	                ⎡ 0 -z  y⎤
//	Skew({x,y,z}) = ⎢ z  0 -x⎥
//	                ⎣-y  x  0⎦
func (m *Mat) Skew(v Vec) {
	m.Set(0, 0, 0)
	m.Set(0, 1, -v.Z)
	m.Set(0, 2, v.Y)
	m.Set(1, 0, v.Z)
	m.Set(1, 1, 0)
	m.Set(1, 2, -v.X)
	m.Set(2, 0, -v.Y)
	m.Set(2, 1, v.X)
	m.Set(2, 2, 0)
}

// Hessian sets the receiver to the Hessian matrix of the scalar field at the point p,
// approximated using finite differences with the given step sizes.
// The field is evaluated at points in the area surrounding p by adding
// at most 2 components of step to p. Hessian expects the field's second partial
// derivatives are all continuous for correct results.
func (m *Mat) Hessian(p, step Vec, field func(Vec) float64) {
	dx := Vec{X: step.X}
	dy := Vec{Y: step.Y}
	dz := Vec{Z: step.Z}
	fp := field(p)
	fxp := field(Add(p, dx))
	fxm := field(Sub(p, dx))
	fxx := (fxp - 2*fp + fxm) / (step.X * step.X)

	fyp := field(Add(p, dy))
	fym := field(Sub(p, dy))
	fyy := (fyp - 2*fp + fym) / (step.Y * step.Y)

	aux := Add(dx, dy)
	fxyp := field(Add(p, aux))
	fxym := field(Sub(p, aux))
	fxy := (fxyp - fxp - fyp + 2*fp - fxm - fym + fxym) / (2 * step.X * step.Y)

	fzp := field(Add(p, dz))
	fzm := field(Sub(p, dz))
	fzz := (fzp - 2*fp + fzm) / (step.Z * step.Z)

	aux = Add(dx, dz)
	fxzp := field(Add(p, aux))
	fxzm := field(Sub(p, aux))
	fxz := (fxzp - fxp - fzp + 2*fp - fxm - fzm + fxzm) / (2 * step.X * step.Z)

	aux = Add(dy, dz)
	fyzp := field(Add(p, aux))
	fyzm := field(Sub(p, aux))
	fyz := (fyzp - fyp - fzp + 2*fp - fym - fzm + fyzm) / (2 * step.Y * step.Z)

	m.Set(0, 0, fxx)
	m.Set(0, 1, fxy)
	m.Set(0, 2, fxz)

	m.Set(1, 0, fxy)
	m.Set(1, 1, fyy)
	m.Set(1, 2, fyz)

	m.Set(2, 0, fxz)
	m.Set(2, 1, fyz)
	m.Set(2, 2, fzz)
}

// Jacobian sets the receiver to the Jacobian matrix of the vector field at
// the point p, approximated using finite differences with the given step sizes.
//
// The Jacobian matrix J is the matrix of all first-order partial derivatives of f.
// If f maps an 3-dimensional vector x to an 3-dimensional vector y = f(x), J is
// a 3×3 matrix whose elements are given as
//
//	J_{i,j} = ∂f_i/∂x_j,
//
// or expanded out
//
//	    [ ∂f_1/∂x_1 ∂f_1/∂x_2 ∂f_1/∂x_3 ]
//	J = [ ∂f_2/∂x_1 ∂f_2/∂x_2 ∂f_2/∂x_3 ]
//	    [ ∂f_3/∂x_1 ∂f_3/∂x_2 ∂f_3/∂x_3 ]
//
// Jacobian expects the field's first order partial derivatives are all
// continuous for correct results.
func (m *Mat) Jacobian(p, step Vec, field func(Vec) Vec) {
	dx := Vec{X: step.X}
	dy := Vec{Y: step.Y}
	dz := Vec{Z: step.Z}

	dfdx := Scale(0.5/step.X, Sub(field(Add(p, dx)), field(Sub(p, dx))))
	dfdy := Scale(0.5/step.Y, Sub(field(Add(p, dy)), field(Sub(p, dy))))
	dfdz := Scale(0.5/step.Z, Sub(field(Add(p, dz)), field(Sub(p, dz))))
	m.Set(0, 0, dfdx.X)
	m.Set(0, 1, dfdy.X)
	m.Set(0, 2, dfdz.X)

	m.Set(1, 0, dfdx.Y)
	m.Set(1, 1, dfdy.Y)
	m.Set(1, 2, dfdz.Y)

	m.Set(2, 0, dfdx.Z)
	m.Set(2, 1, dfdy.Z)
	m.Set(2, 2, dfdz.Z)
}