attitude

Attitude: orientation of an object in space.

A rotation of the sphere can be represented in various ways, such as:

• Euler Angles
• Axis-Angle
• Rotation Matrix
• Unit Quaternion (aka versor)
• Rotation Vector

The attitude module allows conversions and computations between all these representations.

See https://observablehq.com/@fil/attitude for details.

Installing

If you use NPM, npm install attitude. Otherwise, download the latest release. AMD, CommonJS, and vanilla environments are supported. In vanilla, an attitude global is exported:

<script src="https://unpkg.com/attitude"></script>
<script>

const attitude = attitude();

</script>

Try attitude in your browser.

Representations

Euler Angles

[lambda, phi, gamma], in degrees.

Axis-Angle

{ axis: [lon, lat], angle: alpha }, in degrees.

Rotation Matrix

[ [r11, r12, r13],
  [r21, r22, r23],
  [r31, r32, r33] ]

Unit Quaternion

q = [q0, q1, q2, q3, q4] is also called a versor when its norm is equal to 1.

Rotation Vector

[ x, y, z ] = f(a)B, where f(a) is a scalar encoding the angle, and B a unit vector in cartesian coordinates.

Note: there are many ways to encode the angle, we have to settle on a default. The useful functions f(a) are:

• tan(a/4): stereographic, ‘Modified Rodrigues Parameters’.
• tan(a/2): gnomonic, ‘Rodrigues Parameters’, ‘Gibbs vector’.
• a: equidistant, logarithm vector.
• (vector part of the) unit quaternion: Euler angles.

Defaults to the stereographic vector representation.

API Reference

# attitude([angles])

Returns an attitude object. Sets the rotation’s Euler angles if the angles argument is specified. attitude is equivalent to d3.geoRotation(angles), and can be used as a function to rotate a point [longitude, latitude].

Operations

# attitude.invert(point)

Returns the inverse rotation of the point.

# attitude.inverse()

Returns a new attitude, inverse of the original.

# attitude.compose(b)

Returns a new attitude, composition of the original with the argument. When c = a.compose(b) is applied to a point p, the result c(p) = a(b(p)): in other words, the rotation b will be applied first, then rotation a.

# attitude.power(power)

Returns a new partial attitude. a.power(2) is twice the rotation a, a.power(.5) is half the rotation a.

# attitude.arc(A, B)

Returns a new attitude that brings the point A to B by the shortest (geodesic) path.

# attitude.interpolateTo(b)

Returns an interpolator that continuously transitions the original attitude to the argument. The result is a function of t that is equivalent to attitude for t = 0, and equivalent to b for t = 1. Useful for spherical linear interpolation (SLERP).

Representations

# attitude.angles([angles])

Sets or reads the Euler angles of an attitude, as an array [φ, λ, γ] (in degrees).

# attitude.axis([axis])

Sets or reads the rotation axis of an attitude, as [lon, lat] coordinates.

# attitude.angle([angle])

Sets or reads the rotation angle of an attitude, in degrees.

# attitude.versor([versor])

Sets or reads the versor representation of an attitude, as a length-4 array.

# attitude.matrix([matrix])

Sets or reads the matrix representation of an attitude, as a matrix of size 3×3.

# attitude.vector([vector])

Sets or reads the vector representation of an attitude, as a length-3 array. That array can be written f(a)B, where f is a function of the rotation’s angle, and B a unit vector respresenting the axis in cartesian coordinates.

Defaults to the stereographic vector: f(a) = tan(a/4).

# attitude.vectorStereographic([vector])

Stereographic vector: f(a) = tan(a/4). Also called the ‘Modified Rodrigues Parameters’.

# attitude.vectorGnomonic([vector])

Gnomonic vector: f(a) = tan(a/2). Also called ‘Rodrigues Parameters’ or ‘Gibbs vector’.

# attitude.vectorEquidistant([vector])

Equidistant vector: f(a) = a. Also called the logarithm vector.

With thanks to Jacob Rus, Nadieh Bremer, Mike Bostock and Darcy Murphy.