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Euler Angles and Quaternions


Rotation Conventions

There are several methods of achieving rotations within BabylonJS all of which use a particular convention.

Euler Angles

In 3D space Euler angles can produce any possible orientation by providing three angles to rotate about each of three axes in a given order.

For three axes X, Y and Z there are 12 different permutations for the order of the angles. Since X, Y and Z can be treated as World or as local axes this means there is a potential of 24 different possibilities. Most, if not all,of these are in use in different systems around the world. So you need to be very careful that you know very clearly the convention that the system you are working in uses.

Mesh.rotation(alpha, beta, gamma) uses the three Euler angles alpha, beta and gamma which are rotations about the X, Y and Z axes respectively. The convention that Babylon.js uses is based on the yaw, pitch and roll convention and so is carried out around X, Y and Z as local axes in the order Y, X, Z.

References to Euler angles within the Babylon.js community can usually be taken to mean the angles to use with the rotation method.

YXZ Local Axes Yaw, Pitch, Roll

Yaw pitch roll

A pitch is about X, yaw about Y and roll about Z applied in the order yaw, pitch roll using local axes.

Applying independent rotations to a newly created mesh (ie one that has zero rotations) in the order YXZ using local axes

mesh.rotate(BABYLON.Axis.Y, yaw, BABYLON.Space.LOCAL);
mesh.rotate(BABYLON.Axis.X, pitch, BABYLON.Space.LOCAL);
mesh.rotate(BABYLON.Axis.Z, roll, BABYLON.Space.LOCAL);

produces the same orientation as

mesh.rotation = new BABYLON.Vector3(pitch, yaw, roll);

which will produce this orientation whatever the orientation of the mesh prior to its application. The playground below demonstrates this by randomly generating angles and then applying these two methods to two different boxes which remain in alignment.

Playground Example - YXZ yaw, pitch, roll -


ZXY World Axes

The YXZ convention with local axes has produced a particular orientation and it turns out that taking the same angles (alpha = pitch, beta = yaw and gamma = roll) and applying them in the order ZXY using world axes will produce exactly the same orientation.

Applying independent rotations to a newly created mesh (ie one that has zero rotations) in the order ZXY

mesh.rotate(BABYLON.Axis.Z, gamma, BABYLON.Space.WORLD);
mesh.rotate(BABYLON.Axis.X, alpha, BABYLON.Space.WORLD);
mesh.rotate(BABYLON.Axis.Y, beta, BABYLON.Space.WORLD);

produces the same orientation as

mesh.rotation = new BABYLON.Vector(alpha, beta, gamma);

which will produce this orientation whatever the orientation of the mesh prior to its application. The playground below demonstrates this by randomly generating angles and then applying these two methods to two different boxes which remain in alignment.

Playground Example - ZXY -


Quaternions

Imagine a disc with an axle through its center. The disc is able to rotate about the axle. The diagram below shows the disc at several different rotation points around the axle.

disc rotate

For all rotations of the disc the axle can be tilted as seen in the diagram below.

disc rotate and axle tilt

Together a rotation of the disc and a tilt of the axle can produce all possible 3D orientations of the disc. The tilt, or direction, of the axle can be given by a vector along the axle. This means that another way of giving the orientation of a mesh is with a vector (axle direction) and a rotation (of the disc).

So one way of producing any possible orientation is to use four values, three for the axis and one for the angle of rotation. Such a four dimensional vector is a rotational quaternion.

In Babylon.js this is obtained by using

mesh.rotationQuaternion = new BABYLON.Quaternion.RotationAxis(axis, angle);

where axis is a Vector3 and the angle is the rotation in radians.

Euler Angles to Quaternions

YXZ Local Axes Yaw, Pitch, Roll

Yaw pitch roll

As a reminder this convention is directly used in the rotation method in Babylon.js in the form

mesh.rotation = new BABYLON.Vector3(pitch, yaw, roll);

In this convention to take the three angles, yaw, pitch and roll and rotate rotate yaw about Y, then pitch about X and roll about Z using the local axes.

mesh.rotate(BABYLON.Axis.Y, yaw, BABYLON.Space.LOCAL);
mesh.rotate(BABYLON.Axis.X, pitch, BABYLON.Space.LOCAL);
mesh.rotate(BABYLON.Axis.Z, roll, BABYLON.Space.LOCAL);

which with quaternions is the same as using RotationYawPitchRoll

var yprQuaternion = BABYLON.Quaternion.RotationYawPitchRoll(yaw, pitch, roll);

Applying the above rotate sequence to a newly created mesh (ie one that has zero rotations) in the order YXZ in local space and applying RotationAlphaBetaGamma to a mesh, with any orientation, using the same angles will produce the same orientation. The playground below demonstrates this by randomly generating angles and then applying these two methods to two different boxes which remain in alignment.

Playground Example - Yaw Pitch Roll to Quaternion -


ZXZ World Axes A Standard Convention

A standard Euler angle convention is to take three given angles alpha, beta and gamma and rotate alpha about Z, then beta about X, then gamma about Z using the world axes. In Babylon.js this can be achieved by using rotate

mesh.rotate(BABYLON.Axis.Z, alpha, BABYLON.Space.WORLD);
mesh.rotate(BABYLON.Axis.X, beta, BABYLON.Space.WORLD);
mesh.rotate(BABYLON.Axis.Z, gamma, BABYLON.Space.WORLD);

which with quaternions is the same as using RotationAlphaBetaGamma

var abcQuaternion = BABYLON.Quaternion.RotationAlphaBetaGamma(alpha, beta, gamma);

Applying the above rotate sequence to a newly created mesh (ie one that has zero rotations) in the order ZXZ in world space and applying RotationAlphaBetaGamma to a mesh, with any orientation, using the same angles will produce the same orientation. The playground below demonstrates this by randomly generating angles and then applying these two methods to two different boxes which remain in alignment.

Playground Example ZXZ to Quaternion -


Quaternions to Euler Angles

The Euler angles that can be used in mesh.rotation can be found from any rotation quaternion by the following method

var euler = quaternion.toEulerAngles();

To illustrate this the following playground generates three random angles, puts the axes XYZ into a random order and selects at random either to use world or local for all axes. This data is then used to randomise the orientation of a just created box using the rotate method. The rotate method achieves the rotation by generating and using a rotationQuaternion on the box. The rotationQuaternion generated is used to produce the Euler angles to rotate another box, box1, using box1.rotation to obtain the same orientation as the first box.

Playground Example Random Orientation to Euler Angles for mesh.rotation -