Oolite JavaScript Reference: Quaternion

The Quaternion class represents a quaternion, a four-dimensional number, which is used to express rotations. Explaining quaternion mathematics is way beyond the scope of this document, but a quick overview is provided below.

Quaternions for Rotations

This is a very quick, pragmatic discussion of quaternions as they apply to rotating things in Oolite. If you’re interested in the theory, see:

• Wikipedia: Quaternions and spatial rotation
• Wikipedia: Quaternion
• MathWorld: Quaternion

Consider a ship at point h oriented to face a station at point t. This can be expressed as the vector from the ship to the station, v = t − h. However, if the ship rolls, it is still heading along the same vector v, so additional information is required: a twist angle, α (FIXME: relative to what?). A rotation quaternion is a tuple Q = (w, x, y, z), such that

Q_w = cos α/2

Q_x = v_x sin α/2

Q_y = v_y sin α/2

Q_z = v_z sin α/2

Additionally, a rotation quaternion must be normalized; that is, it must fulfill the normal invariant Q_w² + Q_x² + Q_y² + Q_z² = 1. Unlike with property list scripting and specifications, quaternions will not be automatically normalized for you except where specified, but a normalize() method is provided.

An identity rotation – that is, one which, when applied, has no effect – is represented by the identity quaternion (1, 0, 0, 0).

The Quaternion class provides several methods to make construction of rotations easier: rotate(), rotateX(), rotateY(), rotateZ().

Rotations can be combined by quaternion multiplication (see the multiply() method). Note that quaternion multiplication is not commutative; that is, PQ is not the same as QP. If this seems strange, take a box or book and assign it x, y and z axes. Rotate it about the x axis and then the y axis. Then, rotate it about the y axis followed by the x axis. If the results of the two rotations are the same, you’re doing it wrong.

Quaternion Expressions

All Oolite-provided functions which take a quaternion as an argument may instead be passed an Entity instead, in which case the entity’s orientation is used. In specifications, this is represented by arguments named quaternionOrEntity.

Additionally, most Quaternion methods may be passed four numbers instead of a vector. In specifications, this is represented by arguments named quaternionExpression.

Properties

w

w [read-write double]

The w component of the quaternion.

x

x [read-write double]

The x component of the quaternion.

y

y [read-write double]

The y component of the quaternion.

z

z [read-write double]

The z component of the quaternion.

Methods

Constructor

new Quaternion([[[#Quaternion Expressions|quaternionExpression]]]);

Create a new quaternion with the specified value. If no value is provided, the vector is initialized to the identity quaternion (1, 0, 0, 0).

multiply

Quaternion multiply(quaternionExpression);

Returns the standard quaternion product (Grassman product) of the target and quaternionExpression. This is used to concatenate rotations together.

dot

Quaternion dot(quaternionExpression);

Returns the quaternion dot product (inner product) of the target and quaternionExpression. (This is not known to be useful – it’s not used anywhere in Oolite – but the functionality exists in Oolite’s maths library, so I’ve chosen to expose it. -- Ahruman)

rotate

Quaternion rotate(vectorExpression, angle);

Returns a quaternion rotated angle radians about the axis of vectorExpression. (FIXME: clockwise or anticlockwise?)

rotateX

Quaternion rotateX(angle);

Returns a quaternion rotated angle radians about the x axis. (FIXME: clockwise or anticlockwise?)

q.rotateX(a) is equivalent to q.rotate(1, 0, 0, a).

rotateY

Quaternion rotateY(angle);

Returns a quaternion rotated angle radians about the y axis. (FIXME: clockwise or anticlockwise?)

q.rotateY(a) is equivalent to q.rotate(0, 1, 0, a).

rotateZ

Quaternion rotateZ(angle);

Returns a quaternion rotated angle radians about the z axis. (FIXME: clockwise or anticlockwise?)

q.rotateZ(a) is equivalent to q.rotate(0, 0, 1, a).

normalize

Quaternion normalize(angle);

Returns the quaternion adjusted to fulfill the normal invariant. Specifically, this divides each component by the square root of (w² + x² + y² + z²).

vectorForward

Vector vectorForward();

Returns the forward vector from the quaternion.

To understand this, consider an entity which is aligned with the world co-ordinate system – that is, its orientation is the identity quaternion (1, 0, 0, 0), and thus its x axis is aligned with the world x axis, its y axis is aligned with the world y axis and its z axis is aligned with the world z axis. If it is rotated by a quaternion Q, Q.vectorForward() is the forward (z) axis after rotation. Similarly, Q.vectorUp() is the up (y) axis after rotation, and Q.vectorRight() is the right (x) axis after rotation.

vectorUp

Vector vectorUp();

Returns the up vector from the quaternion. See vectorForward() for a definition.

vectorRight

Vector vectorRight();

Returns the up vector from the quaternion. See vectorForward() for a definition.