LeapMath.h

/******************************************************************************\
* Copyright (C) 2012-2013 Leap Motion, Inc. All rights reserved.               *
* Leap Motion proprietary and confidential. Not for distribution.              *
* Use subject to the terms of the Leap Motion SDK Agreement available at       *
* https://developer.leapmotion.com/sdk_agreement, or another agreement         *
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#if !defined(__LeapMath_h__)
#define __LeapMath_h__

#include <cmath>
#include <iostream>
#include <sstream>
#include <float.h>

namespace Leap {

/// The constant pi as a single precision floating point number.
static const float PI          = 3.1415926536f;
/// The constant ratio to convert an angle measure from degrees to radians.
/// Multiply a value in degrees by this constant to convert to radians.
static const float DEG_TO_RAD  = 0.0174532925f;
/// The constant ratio to convert an angle measure from radians to degrees.
/// Multiply a value in radians by this constant to convert to degrees.
static const float RAD_TO_DEG  = 57.295779513f;

/// The Vector struct represents a three-component mathematical vector or point
/// such as a direction or position in three-dimensional space.
///
/// The Leap software employs a right-handed Cartesian coordinate system.
/// Values given are in units of real-world millimeters. The origin is centered
/// at the center of the Leap device. The x- and z-axes lie in the horizontal
/// plane, with the x-axis running parallel to the long edge of the device.
/// The y-axis is vertical, with positive values increasing upwards (in contrast
/// to the downward orientation of most computer graphics coordinate systems).
/// The z-axis has positive values increasing away from the computer screen.
///
/// \image html images/Leap_Axes.png
struct Vector {
  /// Creates a new Vector with all components set to zero.
  Vector() :
    x(0), y(0), z(0) {}

  /// Creates a new Vector with the specified component values.
  Vector(float _x, float _y, float _z) :
    x(_x), y(_y), z(_z) {}

  /// Copies the specified Vector.
  Vector(const Vector& vector) :
    x(vector.x), y(vector.y), z(vector.z) {}

  /// The zero vector: (0, 0, 0)
  static const Vector& zero() {
    static Vector s_zero(0, 0, 0);
    return s_zero;
  }

  /// The x-axis unit vector: (1, 0, 0)
  static const Vector& xAxis() {
    static Vector s_xAxis(1, 0, 0);
    return s_xAxis;
  }
  /// The y-axis unit vector: (0, 1, 0)
  static const Vector& yAxis() {
    static Vector s_yAxis(0, 1, 0);
    return s_yAxis;
  }
  /// The z-axis unit vector: (0, 0, 1)
  static const Vector& zAxis() {
    static Vector s_zAxis(0, 0, 1);
    return s_zAxis;
  }

  /// The unit vector pointing left along the negative x-axis: (-1, 0, 0)
  static const Vector& left() {
    static Vector s_left(-1, 0, 0);
    return s_left;
  }
  /// The unit vector pointing right along the positive x-axis: (1, 0, 0)
  static const Vector& right() {
    return xAxis();
  }
  /// The unit vector pointing down along the negative y-axis: (0, -1, 0)
  static const Vector& down() {
    static Vector s_down(0, -1, 0);
    return s_down;
  }
  /// The unit vector pointing up along the positive x-axis: (0, 1, 0)
  static const Vector& up() {
    return yAxis();
  }
  /// The unit vector pointing forward along the negative z-axis: (0, 0, -1)
  static const Vector& forward() {
    static Vector s_forward(0, 0, -1);
    return s_forward;
  }
  /// The unit vector pointing backward along the positive z-axis: (0, 0, 1)
  static const Vector& backward() {
    return zAxis();
  }

  /// The magnitude, or length, of this vector.
  ///
  /// The magnitude is the L2 norm, or Euclidean distance between the origin and
  /// the point represented by the (x, y, z) components of this Vector object.
  ///
  /// @returns The length of this vector.
  float magnitude() const {
    return std::sqrt(x*x + y*y + z*z);
  }

  /// The square of the magnitude, or length, of this vector.
  ///
  /// @returns The square of the length of this vector.
  float magnitudeSquared() const {
    return x*x + y*y + z*z;
  }

  /// The distance between the point represented by this Vector
  /// object and a point represented by the specified Vector object.
  ///
  /// @param other A Vector object.
  /// @returns The distance from this point to the specified point.
  float distanceTo(const Vector& other) const {
    return std::sqrt((x - other.x)*(x - other.x) +
                     (y - other.y)*(y - other.y) +
                     (z - other.z)*(z - other.z));
  }

  /// The angle between this vector and the specified vector in radians.
  ///
  /// The angle is measured in the plane formed by the two vectors. The
  /// angle returned is always the smaller of the two conjugate angles.
  /// Thus <tt>A.angleTo(B) == B.angleTo(A)</tt> and is always a positive
  /// value less than or equal to pi radians (180 degrees).
  ///
  /// If either vector has zero length, then this function returns zero.
  ///
  /// \image html images/Math_AngleTo.png
  ///
  /// @param other A Vector object.
  /// @returns The angle between this vector and the specified vector in radians.
  float angleTo(const Vector& other) const {
    float denom = this->magnitudeSquared() * other.magnitudeSquared();
    if (denom <= 0.0f) {
      return 0.0f;
    }
    return std::acos(this->dot(other) / std::sqrt(denom));
  }

  /// The pitch angle in radians.
  ///
  /// Pitch is the angle between the negative z-axis and the projection of
  /// the vector onto the y-z plane. In other words, pitch represents rotation
  /// around the x-axis.
  /// If the vector points upward, the returned angle is between 0 and pi radians
  /// (180 degrees); if it points downward, the angle is between 0 and -pi radians.
  ///
  /// \image html images/Math_Pitch_Angle.png
  ///
  /// @returns The angle of this vector above or below the horizon (x-z plane).
  float pitch() const {
    return std::atan2(y, -z);
  }

  /// The yaw angle in radians.
  ///
  /// Yaw is the angle between the negative z-axis and the projection of
  /// the vector onto the x-z plane. In other words, yaw represents rotation
  /// around the y-axis. If the vector points to the right of the negative z-axis,
  /// then the returned angle is between 0 and pi radians (180 degrees);
  /// if it points to the left, the angle is between 0 and -pi radians.
  ///
  /// \image html images/Math_Yaw_Angle.png
  ///
  /// @returns The angle of this vector to the right or left of the negative z-axis.
  float yaw() const {
    return std::atan2(x, -z);
  }

  /// The roll angle in radians.
  ///
  /// Roll is the angle between the negative y-axis and the projection of
  /// the vector onto the x-y plane. In other words, roll represents rotation
  /// around the z-axis. If the vector points to the left of the negative y-axis,
  /// then the returned angle is between 0 and pi radians (180 degrees);
  /// if it points to the right, the angle is between 0 and -pi radians.
  ///
  /// \image html images/Math_Roll_Angle.png
  ///
  /// Use this function to get roll angle of the plane to which this vector is a
  /// normal. For example, if this vector represents the normal to the palm,
  /// then this function returns the tilt or roll of the palm plane compared
  /// to the horizontal (x-z) plane.
  ///
  /// @returns The angle of this vector to the right or left of the y-axis.
  float roll() const {
    return std::atan2(x, -y);
  }

  /// The dot product of this vector with another vector.
  ///
  /// The dot product is the magnitude of the projection of this vector
  /// onto the specified vector.
  ///
  /// \image html images/Math_Dot.png
  ///
  /// @param other A Vector object.
  /// @returns The dot product of this vector and the specified vector.
  float dot(const Vector& other) const {
    return (x * other.x) + (y * other.y) + (z * other.z);
  }

  /// The cross product of this vector and the specified vector.
  ///
  /// The cross product is a vector orthogonal to both original vectors.
  /// It has a magnitude equal to the area of a parallelogram having the
  /// two vectors as sides. The direction of the returned vector is
  /// determined by the right-hand rule. Thus <tt>A.cross(B) == -B.cross(A).</tt>
  ///
  /// \image html images/Math_Cross.png
  ///
  /// @param other A Vector object.
  /// @returns The cross product of this vector and the specified vector.
  Vector cross(const Vector& other) const {
    return Vector((y * other.z) - (z * other.y),
                  (z * other.x) - (x * other.z),
                  (x * other.y) - (y * other.x));
  }

  /// A normalized copy of this vector.
  ///
  /// A normalized vector has the same direction as the original vector,
  /// but with a length of one.
  ///
  /// @returns A Vector object with a length of one, pointing in the same
  /// direction as this Vector object.
  Vector normalized() const {
    float denom = this->magnitudeSquared();
    if (denom <= 0.0f) {
      return Vector::zero();
    }
    denom = 1.0f / std::sqrt(denom);
    return Vector(x * denom, y * denom, z * denom);
  }

  /// A copy of this vector pointing in the opposite direction.
  ///
  /// @returns A Vector object with all components negated.
  Vector operator-() const {
    return Vector(-x, -y, -z);
  }

  /// Add vectors component-wise.
  Vector operator+(const Vector& other) const {
    return Vector(x + other.x, y + other.y, z + other.z);
  }

  /// Subtract vectors component-wise.
  Vector operator-(const Vector& other) const {
    return Vector(x - other.x, y - other.y, z - other.z);
  }

  /// Multiply vector by a scalar.
  Vector operator*(float scalar) const {
    return Vector(x * scalar, y * scalar, z * scalar);
  }

  /// Divide vector by a scalar.
  Vector operator/(float scalar) const {
    return Vector(x / scalar, y / scalar, z / scalar);
  }

  /// Multiply vector by a scalar on the left-hand side (C++ only).
#if !defined(SWIG)
  friend Vector operator*(float scalar, const Vector& vector) {
    return Vector(vector.x * scalar, vector.y * scalar, vector.z * scalar);
  }
#endif

  /// Add vectors component-wise and assign the sum.
  Vector& operator+=(const Vector& other) {
    x += other.x;
    y += other.y;
    z += other.z;
    return *this;
  }

  /// Subtract vectors component-wise and assign the difference.
  Vector& operator-=(const Vector& other) {
    x -= other.x;
    y -= other.y;
    z -= other.z;
    return *this;
  }

  /// Multiply vector by a scalar and assign the product.
  Vector& operator*=(float scalar) {
    x *= scalar;
    y *= scalar;
    z *= scalar;
    return *this;
  }

  /// Divide vector by a scalar and assign the quotient.
  Vector& operator/=(float scalar) {
    x /= scalar;
    y /= scalar;
    z /= scalar;
    return *this;
  }

  /// Returns a string containing this vector in a human readable format: (x, y, z).
  std::string toString() const {
    std::stringstream result;
    result << "(" << x << ", " << y << ", " << z << ")";
    return result.str();
  }
  /// Writes the vector to the output stream using a human readable format: (x, y, z).
  friend std::ostream& operator<<(std::ostream& out, const Vector& vector) {
    return out << vector.toString();
  }

  /// Compare Vector equality component-wise.
  bool operator==(const Vector& other) const {
    return x == other.x && y == other.y && z == other.z;
  }
  /// Compare Vector inequality component-wise.
  bool operator!=(const Vector& other) const {
    return x != other.x || y != other.y || z != other.z;
  }

  /// Returns true if all of the vector's components are finite.  If any
  /// component is NaN or infinite, then this returns false.
  bool isValid() const {
    return (x <= FLT_MAX && x >= -FLT_MAX) &&
           (y <= FLT_MAX && y >= -FLT_MAX) &&
           (z <= FLT_MAX && z >= -FLT_MAX);
  }

  /// Index vector components numerically.
  /// Index 0 is x, index 1 is y, and index 2 is z.
  /// @returns The x, y, or z component of this Vector, if the specified index
  /// value is at least 0 and at most 2; otherwise, returns zero.
  float operator[](unsigned int index) const {
    return index < 3 ? (&x)[index] : 0.0f;
  }

  /// Cast the vector to a float array.
  const float* toFloatPointer() const {
    return &x; /// Note: Assumes x, y, z are aligned in memory.
  }

  /// Convert a Leap::Vector to another 3-component Vector type.
  ///
  /// The specified type must define a constructor that takes the x, y, and z
  /// components as separate parameters.
  template<typename Vector3Type>
  const Vector3Type toVector3() const {
    return Vector3Type(x, y, z);
  }

  /// Convert a Leap::Vector to another 4-component Vector type.
  ///
  /// The specified type must define a constructor that takes the x, y, z, and w
  /// components as separate parameters. (The homogeneous coordinate, w, is set
  /// to zero by default, but you should typically set it to one for vectors
  /// representing a position.)
  template<typename Vector4Type>
  const Vector4Type toVector4(float w=0.0f) const {
    return Vector4Type(x, y, z, w);
  }

  /// The horizontal component.
  float x;
  /// The vertical component.
  float y;
  /// The depth component.
  float z;
};


/// The FloatArray struct is used to allow the returning of native float arrays
/// without requiring dynamic memory allocation.  It represents a matrix
/// with a size up to 4x4.
struct FloatArray {
  /// Access the elements of the float array exactly like a native array
  float& operator[] (unsigned int index) {
    return m_array[index];
  }

  /// Use the Float Array anywhere a float pointer can be used
  operator float* () {
    return m_array;
  }

  /// Use the Float Array anywhere a const float pointer can be used
  operator const float* () const {
    return m_array;
  }

  /// An array containing up to 16 entries of the matrix
  float m_array[16];
};

/// The Matrix struct represents a transformation matrix.
///
/// To use this struct to transform a Vector, construct a matrix containing the
/// desired transformation and then use the Matrix::transformPoint() or
/// Matrix::transformDirection() functions to apply the transform.
///
/// Transforms can be combined by multiplying two or more transform matrices using
/// the * operator.
struct Matrix
{
  /// Constructs an identity transformation matrix.
  Matrix() :
    xBasis(1, 0, 0),
    yBasis(0, 1, 0),
    zBasis(0, 0, 1),
    origin(0, 0, 0) {
  }

  /// Constructs a copy of the specified Matrix object.
  Matrix(const Matrix& other) :
    xBasis(other.xBasis),
    yBasis(other.yBasis),
    zBasis(other.zBasis),
    origin(other.origin) {
  }

  /// Constructs a transformation matrix from the specified basis vectors.
  ///
  /// @param _xBasis A Vector specifying rotation and scale factors for the x-axis.
  /// @param _yBasis A Vector specifying rotation and scale factors for the y-axis.
  /// @param _zBasis A Vector specifying rotation and scale factors for the z-axis.
  Matrix(const Vector& _xBasis, const Vector& _yBasis, const Vector& _zBasis) :
    xBasis(_xBasis),
    yBasis(_yBasis),
    zBasis(_zBasis),
    origin(0, 0, 0) {
  }

  /// Constructs a transformation matrix from the specified basis and translation vectors.
  ///
  /// @param _xBasis A Vector specifying rotation and scale factors for the x-axis.
  /// @param _yBasis A Vector specifying rotation and scale factors for the y-axis.
  /// @param _zBasis A Vector specifying rotation and scale factors for the z-axis.
  /// @param _origin A Vector specifying translation factors on all three axes.
  Matrix(const Vector& _xBasis, const Vector& _yBasis, const Vector& _zBasis, const Vector& _origin) :
    xBasis(_xBasis),
    yBasis(_yBasis),
    zBasis(_zBasis),
    origin(_origin) {
  }

  /// Constructs a transformation matrix specifying a rotation around the specified vector.
  ///
  /// @param axis A Vector specifying the axis of rotation.
  /// @param angleRadians The amount of rotation in radians.
  Matrix(const Vector& axis, float angleRadians) :
    origin(0, 0, 0) {
    setRotation(axis, angleRadians);
  }

  /// Constructs a transformation matrix specifying a rotation around the specified vector
  /// and a translation by the specified vector.
  ///
  /// @param axis A Vector specifying the axis of rotation.
  /// @param angleRadians The angle of rotation in radians.
  /// @param translation A Vector representing the translation part of the transform.
  Matrix(const Vector& axis, float angleRadians, const Vector& translation)
    : origin(translation) {
    setRotation(axis, angleRadians);
  }

  /// Returns the identity matrix specifying no translation, rotation, and scale.
  ///
  /// @returns The identity matrix.
  static const Matrix& identity() {
    static Matrix s_identity;
    return s_identity;
  }

  /// Sets this transformation matrix to represent a rotation around the specified vector.
  ///
  /// This function erases any previous rotation and scale transforms applied
  /// to this matrix, but does not affect translation.
  ///
  /// @param _axis A Vector specifying the axis of rotation.
  /// @param angleRadians The amount of rotation in radians.
  void setRotation(const Vector& _axis, float angleRadians) {
    const Vector axis = _axis.normalized();
    const float s = std::sin(angleRadians);
    const float c = std::cos(angleRadians);
    const float C = (1-c);

    xBasis = Vector(axis[0]*axis[0]*C + c, axis[0]*axis[1]*C - axis[2]*s, axis[0]*axis[2]*C + axis[1]*s);
    yBasis = Vector(axis[1]*axis[0]*C + axis[2]*s, axis[1]*axis[1]*C + c, axis[1]*axis[2]*C - axis[0]*s);
    zBasis = Vector(axis[2]*axis[0]*C - axis[1]*s, axis[2]*axis[1]*C + axis[0]*s, axis[2]*axis[2]*C + c);
  }

  /// Transforms a vector with this matrix by transforming its rotation,
  /// scale, and translation.
  ///
  /// Translation is applied after rotation and scale.
  ///
  /// @param in The Vector to transform.
  /// @returns A new Vector representing the transformed original.
  Vector transformPoint(const Vector& in) const {
    return xBasis*in.x + yBasis*in.y + zBasis*in.z + origin;
  }

  /// Transforms a vector with this matrix by transforming its rotation and
  /// scale only.
  ///
  /// @param in The Vector to transform.
  /// @returns A new Vector representing the transformed original.
  Vector transformDirection(const Vector& in) const {
    return xBasis*in.x + yBasis*in.y + zBasis*in.z;
  }

  /// Performs a matrix inverse if the matrix consists entirely of rigid
  /// transformations (translations and rotations).  If the matrix is not rigid,
  /// this operation will not represent an inverse.
  ///
  /// Note that all matricies that are directly returned by the API are rigid.
  ///
  /// @returns The rigid inverse of the matrix.
  Matrix rigidInverse() const {
    Matrix rotInverse = Matrix(Vector(xBasis[0], yBasis[0], zBasis[0]),
                               Vector(xBasis[1], yBasis[1], zBasis[1]),
                               Vector(xBasis[2], yBasis[2], zBasis[2]));
    rotInverse.origin = rotInverse.transformDirection( -origin );
    return rotInverse;
  }

  /// Multiply transform matrices.
  ///
  /// Combines two transformations into a single equivalent transformation.
  ///
  /// @param in A Matrix to multiply on the right hand side.
  /// @returns A new Matrix representing the transformation equivalent to
  /// applying the other transformation followed by this transformation.
  Matrix operator*(const Matrix& other) const {
    return Matrix(transformDirection(other.xBasis),
                  transformDirection(other.yBasis),
                  transformDirection(other.zBasis),
                  transformPoint(other.origin));
  }

  /// Multiply transform matrices and assign the product.
  Matrix& operator*=(const Matrix& other) {
    return (*this) = (*this) * other;
  }

  /// Compare Matrix equality component-wise.
  bool operator==(const Matrix& other) const {
    return xBasis == other.xBasis &&
           yBasis == other.yBasis &&
           zBasis == other.zBasis &&
           origin == other.origin;
  }
  /// Compare Matrix inequality component-wise.
  bool operator!=(const Matrix& other) const {
    return xBasis != other.xBasis ||
           yBasis != other.yBasis ||
           zBasis != other.zBasis ||
           origin != other.origin;
  }

  /// Convert a Leap::Matrix object to another 3x3 matrix type.
  ///
  /// The new type must define a constructor function that takes each matrix
  /// element as a parameter in row-major order.
  ///
  /// Translation factors are discarded.
  template<typename Matrix3x3Type>
  const Matrix3x3Type toMatrix3x3() const {
    return Matrix3x3Type(xBasis.x, xBasis.y, xBasis.z,
                         yBasis.x, yBasis.y, yBasis.z,
                         zBasis.x, zBasis.y, zBasis.z);
  }

  /// Convert a Leap::Matrix object to another 4x4 matrix type.
  ///
  /// The new type must define a constructor function that takes each matrix
  /// element as a parameter in row-major order.
  template<typename Matrix4x4Type>
  const Matrix4x4Type toMatrix4x4() const {
    return Matrix4x4Type(xBasis.x, xBasis.y, xBasis.z, 0.0f,
                         yBasis.x, yBasis.y, yBasis.z, 0.0f,
                         zBasis.x, zBasis.y, zBasis.z, 0.0f,
                         origin.x, origin.y, origin.z, 1.0f);
  }

  /// Writes the 3x3 Matrix object to a 9 element row-major float or
  /// double array.
  ///
  /// Translation factors are discarded.
  ///
  /// Returns a pointer to the same data.
  template<typename T>
  T* toArray3x3(T* output) const {
    output[0] = xBasis.x; output[1] = xBasis.y; output[2] = xBasis.z;
    output[3] = yBasis.x; output[4] = yBasis.y; output[5] = yBasis.z;
    output[6] = zBasis.x; output[7] = zBasis.y; output[8] = zBasis.z;
    return output;
  }

  /// Convert a 3x3 Matrix object to a 9 element row-major float array.
  ///
  /// Translation factors are discarded.
  ///
  /// Returns a FloatArray struct to avoid dynamic memory allocation.
  FloatArray toArray3x3() const {
    FloatArray output;
    toArray3x3((float*)output);
    return output;
  }

  /// Writes the 4x4 Matrix object to a 16 element row-major float
  /// or double array.
  ///
  /// Returns a pointer to the same data.
  template<typename T>
  T* toArray4x4(T* output) const {
    output[0]  = xBasis.x; output[1]  = xBasis.y; output[2]  = xBasis.z; output[3]  = 0.0f;
    output[4]  = yBasis.x; output[5]  = yBasis.y; output[6]  = yBasis.z; output[7]  = 0.0f;
    output[8]  = zBasis.x; output[9]  = zBasis.y; output[10] = zBasis.z; output[11] = 0.0f;
    output[12] = origin.x; output[13] = origin.y; output[14] = origin.z; output[15] = 1.0f;
    return output;
  }

  /// Convert a 4x4 Matrix object to a 16 element row-major float array.
  ///
  /// Returns a FloatArray struct to avoid dynamic memory allocation.
  FloatArray toArray4x4() const {
    FloatArray output;
    toArray4x4((float*)output);
    return output;
  }

  /// Write the matrix to a string in a human readable format.
  std::string toString() const {
    std::stringstream result;
    result << "xBasis:" << xBasis.toString() << " yBasis:" << yBasis.toString()
           << " zBasis:" << zBasis.toString() << " origin:" << origin.toString();
    return result.str();
  }

  /// Write the matrix to an output stream in a human readable format.
  friend std::ostream& operator<<(std::ostream& out, const Matrix& matrix) {
    return out << matrix.toString();
  }

  /// The rotation and scale factors for the x-axis.
  Vector xBasis;
  /// The rotation and scale factors for the y-axis.
  Vector yBasis;
  /// The rotation and scale factors for the z-axis.
  Vector zBasis;
  /// The translation factors for all three axes.
  Vector origin;
};

}; // namespace Leap

#endif // __LeapMath_h__