Type Definition nalgebra::geometry::UnitDualQuaternion[][src]

pub type UnitDualQuaternion<T> = Unit<DualQuaternion<T>>;
Expand description

A unit quaternions. May be used to represent a rotation followed by a translation.

Implementations

The underlying dual quaternion.

Same as self.as_ref().

Example
let id = UnitDualQuaternion::identity();
assert_eq!(*id.dual_quaternion(), DualQuaternion::from_real_and_dual(
    Quaternion::new(1.0, 0.0, 0.0, 0.0),
    Quaternion::new(0.0, 0.0, 0.0, 0.0)
));

Compute the conjugate of this unit quaternion.

Example
let qr = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let qd = Quaternion::new(5.0, 6.0, 7.0, 8.0);
let unit = UnitDualQuaternion::new_normalize(
    DualQuaternion::from_real_and_dual(qr, qd)
);
let conj = unit.conjugate();
assert_eq!(conj.real, unit.real.conjugate());
assert_eq!(conj.dual, unit.dual.conjugate());

Compute the conjugate of this unit quaternion in-place.

Example
let qr = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let qd = Quaternion::new(5.0, 6.0, 7.0, 8.0);
let unit = UnitDualQuaternion::new_normalize(
    DualQuaternion::from_real_and_dual(qr, qd)
);
let mut conj = unit.clone();
conj.conjugate_mut();
assert_eq!(conj.as_ref().real, unit.as_ref().real.conjugate());
assert_eq!(conj.as_ref().dual, unit.as_ref().dual.conjugate());

Inverts this dual quaternion if it is not zero.

Example
let qr = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let qd = Quaternion::new(5.0, 6.0, 7.0, 8.0);
let unit = UnitDualQuaternion::new_normalize(DualQuaternion::from_real_and_dual(qr, qd));
let inv = unit.inverse();
assert_relative_eq!(unit * inv, UnitDualQuaternion::identity(), epsilon = 1.0e-6);
assert_relative_eq!(inv * unit, UnitDualQuaternion::identity(), epsilon = 1.0e-6);

Inverts this dual quaternion in place if it is not zero.

Example
let qr = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let qd = Quaternion::new(5.0, 6.0, 7.0, 8.0);
let unit = UnitDualQuaternion::new_normalize(DualQuaternion::from_real_and_dual(qr, qd));
let mut inv = unit.clone();
inv.inverse_mut();
assert_relative_eq!(unit * inv, UnitDualQuaternion::identity(), epsilon = 1.0e-6);
assert_relative_eq!(inv * unit, UnitDualQuaternion::identity(), epsilon = 1.0e-6);

The unit dual quaternion needed to make self and other coincide.

The result is such that: self.isometry_to(other) * self == other.

Example
let qr = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let qd = Quaternion::new(5.0, 6.0, 7.0, 8.0);
let dq1 = UnitDualQuaternion::new_normalize(DualQuaternion::from_real_and_dual(qr, qd));
let dq2 = UnitDualQuaternion::new_normalize(DualQuaternion::from_real_and_dual(qd, qr));
let dq_to = dq1.isometry_to(&dq2);
assert_relative_eq!(dq_to * dq1, dq2, epsilon = 1.0e-6);

Linear interpolation between two unit dual quaternions.

The result is not normalized.

Example
let dq1 = UnitDualQuaternion::new_normalize(DualQuaternion::from_real_and_dual(
    Quaternion::new(0.5, 0.0, 0.5, 0.0),
    Quaternion::new(0.0, 0.5, 0.0, 0.5)
));
let dq2 = UnitDualQuaternion::new_normalize(DualQuaternion::from_real_and_dual(
    Quaternion::new(0.5, 0.0, 0.0, 0.5),
    Quaternion::new(0.5, 0.0, 0.5, 0.0)
));
assert_relative_eq!(
    UnitDualQuaternion::new_normalize(dq1.lerp(&dq2, 0.5)),
    UnitDualQuaternion::new_normalize(
        DualQuaternion::from_real_and_dual(
            Quaternion::new(0.5, 0.0, 0.25, 0.25),
            Quaternion::new(0.25, 0.25, 0.25, 0.25)
        )
    ),
    epsilon = 1.0e-6
);

Normalized linear interpolation between two unit quaternions.

This is the same as self.lerp except that the result is normalized.

Example
let dq1 = UnitDualQuaternion::new_normalize(DualQuaternion::from_real_and_dual(
    Quaternion::new(0.5, 0.0, 0.5, 0.0),
    Quaternion::new(0.0, 0.5, 0.0, 0.5)
));
let dq2 = UnitDualQuaternion::new_normalize(DualQuaternion::from_real_and_dual(
    Quaternion::new(0.5, 0.0, 0.0, 0.5),
    Quaternion::new(0.5, 0.0, 0.5, 0.0)
));
assert_relative_eq!(dq1.nlerp(&dq2, 0.2), UnitDualQuaternion::new_normalize(
    DualQuaternion::from_real_and_dual(
        Quaternion::new(0.5, 0.0, 0.4, 0.1),
        Quaternion::new(0.1, 0.4, 0.1, 0.4)
    )
), epsilon = 1.0e-6);

Screw linear interpolation between two unit quaternions. This creates a smooth arc from one dual-quaternion to another.

Panics if the angle between both quaternion is 180 degrees (in which case the interpolation is not well-defined). Use .try_sclerp instead to avoid the panic.

Example

let dq1 = UnitDualQuaternion::from_parts(
    Vector3::new(0.0, 3.0, 0.0).into(),
    UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_4, 0.0, 0.0),
);

let dq2 = UnitDualQuaternion::from_parts(
    Vector3::new(0.0, 0.0, 3.0).into(),
    UnitQuaternion::from_euler_angles(-std::f32::consts::PI, 0.0, 0.0),
);

let dq = dq1.sclerp(&dq2, 1.0 / 3.0);

assert_relative_eq!(
    dq.rotation().euler_angles().0, std::f32::consts::FRAC_PI_2, epsilon = 1.0e-6
);
assert_relative_eq!(dq.translation().vector.y, 3.0, epsilon = 1.0e-6);

Computes the screw-linear interpolation between two unit quaternions or returns None if both quaternions are approximately 180 degrees apart (in which case the interpolation is not well-defined).

Arguments
  • self: the first quaternion to interpolate from.
  • other: the second quaternion to interpolate toward.
  • t: the interpolation parameter. Should be between 0 and 1.
  • epsilon: the value below which the sinus of the angle separating both quaternion must be to return None.

Return the rotation part of this unit dual quaternion.

let dq = UnitDualQuaternion::from_parts(
    Vector3::new(0.0, 3.0, 0.0).into(),
    UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_4, 0.0, 0.0)
);

assert_relative_eq!(
    dq.rotation().angle(), std::f32::consts::FRAC_PI_4, epsilon = 1.0e-6
);

Return the translation part of this unit dual quaternion.

let dq = UnitDualQuaternion::from_parts(
    Vector3::new(0.0, 3.0, 0.0).into(),
    UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_4, 0.0, 0.0)
);

assert_relative_eq!(
    dq.translation().vector, Vector3::new(0.0, 3.0, 0.0), epsilon = 1.0e-6
);

Builds an isometry from this unit dual quaternion.

let rotation = UnitQuaternion::from_euler_angles(std::f32::consts::PI, 0.0, 0.0);
let translation = Vector3::new(1.0, 3.0, 2.5);
let dq = UnitDualQuaternion::from_parts(
    translation.into(),
    rotation
);
let iso = dq.to_isometry();

assert_relative_eq!(iso.rotation.angle(), std::f32::consts::PI, epsilon = 1.0e-6);
assert_relative_eq!(iso.translation.vector, translation, epsilon = 1.0e-6);

Rotate and translate a point by this unit dual quaternion interpreted as an isometry.

This is the same as the multiplication self * pt.

let dq = UnitDualQuaternion::from_parts(
    Vector3::new(0.0, 3.0, 0.0).into(),
    UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_2, 0.0, 0.0)
);
let point = Point3::new(1.0, 2.0, 3.0);

assert_relative_eq!(
    dq.transform_point(&point), Point3::new(1.0, 0.0, 2.0), epsilon = 1.0e-6
);

Rotate a vector by this unit dual quaternion, ignoring the translational component.

This is the same as the multiplication self * v.

let dq = UnitDualQuaternion::from_parts(
    Vector3::new(0.0, 3.0, 0.0).into(),
    UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_2, 0.0, 0.0)
);
let vector = Vector3::new(1.0, 2.0, 3.0);

assert_relative_eq!(
    dq.transform_vector(&vector), Vector3::new(1.0, -3.0, 2.0), epsilon = 1.0e-6
);

Rotate and translate a point by the inverse of this unit quaternion.

This may be cheaper than inverting the unit dual quaternion and transforming the point.

let dq = UnitDualQuaternion::from_parts(
    Vector3::new(0.0, 3.0, 0.0).into(),
    UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_2, 0.0, 0.0)
);
let point = Point3::new(1.0, 2.0, 3.0);

assert_relative_eq!(
    dq.inverse_transform_point(&point), Point3::new(1.0, 3.0, 1.0), epsilon = 1.0e-6
);

Rotate a vector by the inverse of this unit quaternion, ignoring the translational component.

This may be cheaper than inverting the unit dual quaternion and transforming the vector.

let dq = UnitDualQuaternion::from_parts(
    Vector3::new(0.0, 3.0, 0.0).into(),
    UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_2, 0.0, 0.0)
);
let vector = Vector3::new(1.0, 2.0, 3.0);

assert_relative_eq!(
    dq.inverse_transform_vector(&vector), Vector3::new(1.0, 3.0, -2.0), epsilon = 1.0e-6
);

Rotate a unit vector by the inverse of this unit quaternion, ignoring the translational component. This may be cheaper than inverting the unit dual quaternion and transforming the vector.

let dq = UnitDualQuaternion::from_parts(
    Vector3::new(0.0, 3.0, 0.0).into(),
    UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_2, 0.0, 0.0)
);
let vector = Unit::new_unchecked(Vector3::new(0.0, 1.0, 0.0));

assert_relative_eq!(
    dq.inverse_transform_unit_vector(&vector),
    Unit::new_unchecked(Vector3::new(0.0, 0.0, -1.0)),
    epsilon = 1.0e-6
);

Converts this unit dual quaternion interpreted as an isometry into its equivalent homogeneous transformation matrix.

let dq = UnitDualQuaternion::from_parts(
    Vector3::new(1.0, 3.0, 2.0).into(),
    UnitQuaternion::from_axis_angle(&Vector3::z_axis(), std::f32::consts::FRAC_PI_6)
);
let expected = Matrix4::new(0.8660254, -0.5,      0.0, 1.0,
                            0.5,       0.8660254, 0.0, 3.0,
                            0.0,       0.0,       1.0, 2.0,
                            0.0,       0.0,       0.0, 1.0);

assert_relative_eq!(dq.to_homogeneous(), expected, epsilon = 1.0e-6);

The unit dual quaternion multiplicative identity, which also represents the identity transformation as an isometry.

let ident = UnitDualQuaternion::identity();
let point = Point3::new(1.0, -4.3, 3.33);

assert_eq!(ident * point, point);
assert_eq!(ident, ident.inverse());

Cast the components of self to another type.

Example
let q = UnitDualQuaternion::<f64>::identity();
let q2 = q.cast::<f32>();
assert_eq!(q2, UnitDualQuaternion::<f32>::identity());

Return a dual quaternion representing the translation and orientation given by the provided rotation quaternion and translation vector.

let dq = UnitDualQuaternion::from_parts(
    Vector3::new(0.0, 3.0, 0.0).into(),
    UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_2, 0.0, 0.0)
);
let point = Point3::new(1.0, 2.0, 3.0);

assert_relative_eq!(dq * point, Point3::new(1.0, 0.0, 2.0), epsilon = 1.0e-6);

Return a unit dual quaternion representing the translation and orientation given by the provided isometry.

let iso = Isometry3::from_parts(
    Vector3::new(0.0, 3.0, 0.0).into(),
    UnitQuaternion::from_euler_angles(std::f32::consts::FRAC_PI_2, 0.0, 0.0)
);
let dq = UnitDualQuaternion::from_isometry(&iso);
let point = Point3::new(1.0, 2.0, 3.0);

assert_relative_eq!(dq * point, iso * point, epsilon = 1.0e-6);

Creates a dual quaternion from a unit quaternion rotation.

Example
let q = Quaternion::new(1.0, 2.0, 3.0, 4.0);
let rot = UnitQuaternion::new_normalize(q);

let dq = UnitDualQuaternion::from_rotation(rot);
assert_relative_eq!(dq.as_ref().real.norm(), 1.0, epsilon = 1.0e-6);
assert_eq!(dq.as_ref().dual.norm(), 0.0);

Trait Implementations

Used for specifying relative comparisons.

The default tolerance to use when testing values that are close together. Read more

A test for equality that uses the absolute difference to compute the approximate equality of two numbers. Read more

The inverse of AbsDiffEq::abs_diff_eq.

Returns the “default value” for a type. Read more

Formats the value using the given formatter. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

The resulting type after applying the / operator.

Performs the / operation. Read more

Performs the /= operation. Read more

Performs the /= operation. Read more

Performs the /= operation. Read more

Performs the /= operation. Read more

Performs the /= operation. Read more

Performs the /= operation. Read more

Performs the /= operation. Read more

Performs the /= operation. Read more

Performs the conversion.

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

The resulting type after applying the * operator.

Performs the * operation. Read more

Performs the *= operation. Read more

Performs the *= operation. Read more

Performs the *= operation. Read more

Performs the *= operation. Read more

Performs the *= operation. Read more

Performs the *= operation. Read more

Performs the *= operation. Read more

Performs the *= operation. Read more

The resulting type after applying the - operator.

Performs the unary - operation. Read more

The resulting type after applying the - operator.

Performs the unary - operation. Read more

Returns the multiplicative identity element of Self, 1. Read more

Sets self to the multiplicative identity element of Self, 1.

Returns true if self is equal to the multiplicative identity. Read more

This method tests for self and other values to be equal, and is used by ==. Read more

This method tests for !=.

The default relative tolerance for testing values that are far-apart. Read more

A test for equality that uses a relative comparison if the values are far apart.

The inverse of RelativeEq::relative_eq.

The inclusion map: converts self to the equivalent element of its superset.

Checks if element is actually part of the subset Self (and can be converted to it).

Use with care! Same as self.to_superset but without any property checks. Always succeeds.

The inverse inclusion map: attempts to construct self from the equivalent element of its superset. Read more

The inclusion map: converts self to the equivalent element of its superset.

Checks if element is actually part of the subset Self (and can be converted to it).

Use with care! Same as self.to_superset but without any property checks. Always succeeds.

The inverse inclusion map: attempts to construct self from the equivalent element of its superset. Read more

The inclusion map: converts self to the equivalent element of its superset.

Checks if element is actually part of the subset Self (and can be converted to it).

Use with care! Same as self.to_superset but without any property checks. Always succeeds.

The inverse inclusion map: attempts to construct self from the equivalent element of its superset. Read more

The inclusion map: converts self to the equivalent element of its superset.

Checks if element is actually part of the subset Self (and can be converted to it).

Use with care! Same as self.to_superset but without any property checks. Always succeeds.

The inverse inclusion map: attempts to construct self from the equivalent element of its superset. Read more

The inclusion map: converts self to the equivalent element of its superset.

Checks if element is actually part of the subset Self (and can be converted to it).

Use with care! Same as self.to_superset but without any property checks. Always succeeds.

The inverse inclusion map: attempts to construct self from the equivalent element of its superset. Read more

The default ULPs to tolerate when testing values that are far-apart. Read more

A test for equality that uses units in the last place (ULP) if the values are far apart.

The inverse of UlpsEq::ulps_eq.