Quat< FT > Class Template Reference

The Quaternion class represents 3D rotations and orientations. More...

#include <easy3d/core/quat.h>

Public Types
typedef Vec< 3, FT >	Vec3
typedef Quat< FT >	thisclass

Public Member Functions
	Quat ()
	Default constructor, builds an identity rotation.
	Quat (const Mat3< FT > &m)
	Constructor from rotation matrix. See also set_from_rotation_matrix().
	Quat (const Vec3 &axis, FT angle)
	Constructor from rotation axis (non null) and angle (in radians). See also set_axis_angle().
	Quat (const Vec3 &from, const Vec3 &to)
	Constructs a quaternion that will rotate from the from direction to the to direction. More...
	Quat (FT q0, FT q1, FT q2, FT q3)
	Constructor from the four values of a Quaternion. First three values are axis*std::sin(angle/2) and the last one is std::cos(angle/2). More...
	Quat (const thisclass &Q)
Quat &	operator= (const thisclass &Q)
void	set_axis_angle (const Vec3 &axis, FT angle)
void	set_value (FT q0, FT q1, FT q2, FT q3)
void	set_from_rotation_matrix (const Mat3< FT > &m)
	Set the Quaternion from a (supposedly correct) 3x3 rotation matrix. The matrix is expressed in European format: its three columns are the images by the rotation of the three vectors of an orthogonal basis. See http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion/index.htm.
void	set_from_rotated_basis (const Vec3 &X, const Vec3 &Y, const Vec3 &Z)
	Set a Quaternion from the three axis of a rotated frame. It actually fills the three columns of a matrix with these rotated basis vectors and calls this method.
Vec3	axis () const
	Returns the normalized axis direction of the rotation represented by the Quaternion. It is null for an identity Quaternion. See also angle() and get_axis_angle().
FT	angle () const
	Returns the angle (in radians) of the rotation represented by the Quaternion. This value is always in the range [0-pi]. Larger rotational angles are obtained by inverting the axis() direction. See also axis() and get_axis_angle().
void	get_axis_angle (Vec3 &axis, FT &angle) const
	Returns the axis vector and the angle (in radians) of the rotation represented by the Quaternion.
FT	operator[] (int i) const
	Bracket operator, with a constant return value. i must range in [0..3].
FT &	operator[] (int i)
	Bracket operator returning an l-value. i must range in [0..3].
Quat &	operator*= (const Quat &q)
	Quaternion rotation is composed with q. See operator*(), since this is equivalent to this = this * q. More...
Vec3	rotate (const Vec3 &v) const
	Returns the image of v by the Quaternion rotation. More...
Vec3	inverse_rotate (const Vec3 &v) const
	Returns the image of v by the Quaternion inverse() rotation. rotate() performs an inverse transformation. Same as inverse().rotate(v).
Quat	inverse () const
	Inversion. Returns the inverse Quaternion (inverse rotation). Result has a negated axis() direction and the same angle(). A composition (see operator*()) of a Quaternion and its inverse() results in an identity function. Use invert() to actually modify the Quaternion.
void	invert ()
	Inverses the Quaternion (same rotation angle(), but negated axis()). More...
void	negate ()
	Negates all the coefficients of the Quaternion. This results in an other representation of the same rotation (opposite rotation angle, but with a negated axis direction: the two cancel out). However, note that the results of axis() and angle() are unchanged after a call to this method since angle() always returns a value in [0,pi]. This method is mainly useful for Quaternion interpolation, so that the spherical interpolation takes the shortest path on the unit sphere. See slerp() for details.
FT	length () const
	Return the length of the quaternion.
FT	normalize ()
	Normalizes the Quaternion coefficients. This method should not need to be called since we only deal with unit Quaternions. This is however useful to prevent numerical drifts, especially with small rotational increments. More...
Quat	normalized () const
	Returns a normalized version of the Quaternion. More...
Mat4< FT >	matrix () const
	Returns the Quaternion associated 4x4 rotation matrix. Use glMultMatrixf(q.matrix()) to apply the rotation represented by Quaternion q to the current OpenGL matrix.
Mat4< FT >	inverse_matrix () const
	Returns the associated 4x4 inverse rotation matrix. This is simply the matrix() of the inverse().
Quat	log ()
	Returns the logarithm of the Quaternion. See also exp().
Quat	exp ()
	Returns the exponential of the Quaternion. See also log().

Static Public Member Functions
static Quat	slerp (const Quat< FT > &a, const Quat< FT > &b, FT t, bool allowFlip=true)
	Slerp(Spherical Linear intERPolation) interpolation. Returns the slerp interpolation of Quaternions a and b, at time t. t should range in [0,1]. Result is a when t=0 and b when t=1. When allowFlip is true (default) the slerp interpolation will always use the "shortest path" between the Quaternions' orientations, by "flipping" the source Quaternion if needed (see negate()). More...
static Quat	squad (const Quat< FT > &a, const Quat< FT > &tgA, const Quat< FT > &tgB, const Quat< FT > &b, FT t)
	Returns the slerp interpolation of the two Quaternions a and b, at time t, using tangents tgA and tgB. The resulting Quaternion is "between" a and b (result is a when t=0 and b for t=1). Use squad_tangent() to define the Quaternion tangents tgA and tgB.
static FT	dot (const Quat< FT > &a, const Quat< FT > &b)
	Returns the "dot" product of a and b: a[0]*b[0] + a[1]*b[1] + a[2]*b[2] + a[3]*b[3].
static Quat	ln_dif (const Quat< FT > &a, const Quat< FT > &b)
	Returns log(a. inverse() * b). Useful for squad_tangent().
static Quat	squad_tangent (const Quat< FT > &before, const Quat< FT > &center, const Quat< FT > &after)
	Returns a tangent Quaternion for center, defined by before and after Quaternions. Useful for smooth spline interpolation of Quaternion with squad() and slerp().
static thisclass	random_quat ()
	Returns a random unit Quaternion. You can create a randomly directed unit vector using: More...

Public Attributes
union {
struct {
FT   x
FT   y
FT   z
FT   w
}
FT   _q [4]
};

Friends
Quat	operator* (const Quat &a, const Quat &b)
	Returns the composition of the a and b rotations. The order is important. When applied to a Vec v (see operator*(const Quaternion&, const Vec&) and rotate()) the resulting Quaternion acts as if b was applied first and then a was applied. This is obvious since the image v' of v by the composited rotation satisfies: v'= (a*b) * v = a * (b*v) Note that a*b usually differs from b*a. More...
Vec3	operator* (const Quat &q, const Vec3 &v)
	Returns the image of v by the rotation q. Same as q.rotate(v). More...

Detailed Description

template<typename FT>
class easy3d::Quat< FT >

The Quaternion class represents 3D rotations and orientations.

The Quaternion is an appropriate (although not very intuitive) representation for 3D rotations and orientations. Many tools are provided to ease the definition of a Quaternion: see constructors, set_axis_angle(), set_from_rotation_matrix(), set_from_rotated_basis().

You can apply the rotation represented by the Quaternion to 3D points using rotate() and inverse_rotate().

You can apply the Quaternion q rotation to the OpenGL matrices using glMultMatrixd(q.matrix()); which is equivalent to glRotate(q.angle()*180.0/M_PI, q.axis().x, q.axis().y, q.axis().z);

Internal representation The internal representation of a Quaternion is a set of 4 numbers, [x y z w], which represents rotations the following way: x = axis.x * std::sin(angle / 2) y = axis.y * std::sin(angle / 2) z = axis.z * std::sin(angle / 2) w = std::cos(angle / 2) \NOTE:

• the angle is in radians and the axis is a unit vector.
• certain implementations place the cosine term in the first position (instead of last).

A Quaternion is always normalized, so that its inverse() is actually its conjugate.

Euler angles VS Quaternion

Euler angles are the easiest way to represent rotation. This representation simply stores the three rotation angles around the X, Y and Z axes. These 3 rotations are then applied successively, usually in this order: first Y, then Z, then X (but not necessarily). Using a different order yields different results. Euler angles are usually used to set a character's orientation since game characters only rotate on the vertical axis. Therefore, it is easier to write, understand and maintain "float direction" than 3 different orientations. Another good use of Euler angles is an FPS camera: you have one angle for heading (Y), and one for up/down (X). However, when things get more complex, Euler angle will be hard to work with, eg.,:

• Interpolating smoothly between 2 orientations is hard. Naively interpolating the X, Y, and Z angles will be ugly.
• Applying several rotations is complicated and unprecise: you have to compute the final rotation matrix, and guess the Euler angles from this matrix.
• A well-known problem, the "Gimbal Lock", will sometimes block your rotations, and other singularities which will flip your model upside-down.
• Different angles make the same rotation (-180 and 180 degrees, for instance)
• It is a mess - as said above, usually the right order is YZX, but if you also use a library with a different order, you'll be in trouble.
• Some operations are complicated, e.g., rotation of N degrees around a specific axis. Quaternions are a tool to represent rotations, which solves these problems.

Constructor & Destructor Documentation

Quat() [1/3]

Quat	(	const Vec3 &	from,
		const Vec3 &	to
	)

Constructs a quaternion that will rotate from the from direction to the to direction.

Note

This rotation is not uniquely defined. The selected axis is usually orthogonal to from and to, minimizing the rotation angle. This method is robust and can handle small or almost identical vectors.

Quat() [2/3]

Quat ( FT  q0, FT  q1, FT  q2, FT  q3  )

inline

Constructor from the four values of a Quaternion. First three values are axis*std::sin(angle/2) and the last one is std::cos(angle/2).

Attention

The identity Quaternion is Quat(0,0,0,1) and not Quat(0,0,0,0) (which is not unitary). The default Quat() creates such identity Quaternion.

Quat() [3/3]

Quat ( const thisclass &  Q )

inline

brief Copy constructor.

Member Function Documentation

invert()

void invert ( )

inline

Inverses the Quaternion (same rotation angle(), but negated axis()).

See also

also inverse().

normalize()

FT normalize ( )

inline

Normalizes the Quaternion coefficients. This method should not need to be called since we only deal with unit Quaternions. This is however useful to prevent numerical drifts, especially with small rotational increments.

See also

normalized().

normalized()

Quat normalized ( ) const

inline

Returns a normalized version of the Quaternion.

See also

normalize().

operator*=()

Quat & operator*= ( const Quat< FT > &  q )

inline

Quaternion rotation is composed with q. See operator*(), since this is equivalent to this = this * q.

Note

For efficiency reasons, the resulting Quaternion is not normalized. You may normalize() it after each application in case of numerical drift.

operator=()

Quat & operator= ( const thisclass &  Q )

inline

brief Equal operator.

random_quat()

Quat< FT > random_quat

static

Returns a random unit Quaternion. You can create a randomly directed unit vector using:

Vec randomDir = Quat::random_quat() * Vec(1.0, 0.0, 0.0); // or any other Vec

Note

This function uses rand() to create pseudo-random numbers and the random number generator can be initialized using srand().

rotate()

Quat< FT >::Vec3 rotate

(

const Vec3 &

v

)

const

Returns the image of v by the Quaternion rotation.

See also

inverse_rotate() and operator*(const Quat&, const Vec&).

set_axis_angle()

void set_axis_angle	(	const Vec3 &	axis,
		FT	angle
	)

brief Sets the Quaternion as a rotation of axis and angle (in radians). axis does not need to be normalized. A null axis will result in an identity Quaternion.

set_value()

void set_value ( FT  q0, FT  q1, FT  q2, FT  q3  )

inline

brief Sets the Quaternion value.

slerp()

Quat< FT > slerp ( const Quat< FT > &  a, const Quat< FT > &  b, FT  t, bool  allowFlip = true  )

static

Slerp(Spherical Linear intERPolation) interpolation. Returns the slerp interpolation of Quaternions a and b, at time t. t should range in [0,1]. Result is a when t=0 and b when t=1. When allowFlip is true (default) the slerp interpolation will always use the "shortest path" between the Quaternions' orientations, by "flipping" the source Quaternion if needed (see negate()).

Interpolate between 2 quaternions

Friends And Related Function Documentation

operator* [1/2]

Quat operator* ( const Quat< FT > &  a, const Quat< FT > &  b  )

friend

Returns the composition of the a and b rotations. The order is important. When applied to a Vec v (see operator*(const Quaternion&, const Vec&) and rotate()) the resulting Quaternion acts as if b was applied first and then a was applied. This is obvious since the image v' of v by the composited rotation satisfies: v'= (a*b) * v = a * (b*v) Note that a*b usually differs from b*a.

Attention

For efficiency reasons, the resulting Quaternion is not normalized. Use normalize() in case of numerical drift with small rotation composition.

operator* [2/2]

Vec3 operator* ( const Quat< FT > &  q, const Vec3 &  v  )

friend

Returns the image of v by the rotation q. Same as q.rotate(v).

See also

rotate() and inverse_rotate().

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The documentation for this class was generated from the following files:

• /Users/lnan/Documents/Projects/Easy3D/easy3d/core/mat.h
• /Users/lnan/Documents/Projects/Easy3D/easy3d/core/quat.h