Vector: arrays of 2D, 3D, and Lorentz vectors

Vector is a Python 3.8+ library (Python 3.6 and 3.7 supported till v0.9.0 and v1.0.0, respectively) for 2D, 3D, and Lorentz vectors, especially arrays of vectors, to solve common physics problems in a NumPy-like way.

Main features of Vector:

• Pure Python with NumPy as its only dependency. This makes it easier to install.
• Vectors may be represented in a variety of coordinate systems: Cartesian, cylindrical, pseudorapidity, and any combination of these with time or proper time for Lorentz vectors. In all, there are 12 coordinate systems: {x-y vs ρ-φ in the azimuthal plane} × {z vs θ vs η longitudinally} × {t vs τ temporally}.
• Uses names and conventions set by ROOT's TLorentzVector and Math::LorentzVector, as well as scikit-hep/math, uproot-methods TLorentzVector, henryiii/hepvector, and coffea.nanoevents.methods.vector.
• Implemented on a variety of backends:
  • pure Python objects
  • SymPy vectors
  • NumPy arrays of vectors (as a structured array subclass)
  • Awkward Arrays of vectors
  • potential for more: CuPy, TensorFlow, Torch...
• Awkward backend also implemented in Numba for JIT-compiled calculations on vectors.
• JAX and Dask support through Awkward Arrays.
• Distinction between geometrical vectors, which have a minimum of attribute and method names, and vectors representing momentum, which have synonyms like pt = rho, energy = t, mass = tau.

Installation

To install, use pip install vector or your favorite way to install in an environment.

Overview

This overview is based on the documentation here.

import vector
import numpy as np
import awkward as ak  # at least v2.0.0 (vector v1.4.* supports awkward v1 and v2 both)
import numba as nb

Constructing a vector or an array of vectors

The easiest way to create one or many vectors is with a helper function:

• vector.obj to make a pure Python vector object,
• vector.arr to make a NumPy array of vectors (or array, lowercase, like np.array),
• vector.awk to make an Awkward Array of vectors (or Array, uppercase, like ak.Array).
• vector.zip to make an Awkward Array of vectors (similar to ak.zip)

Pure Python vectors

You can directly use the VectorObject and MomentumObject classes to construct object type vectors:

vector.VectorObject2D(x=1.1, y=2.2)
vector.MomentumObject3D(px=1.1, py=2.2, pz=3.3)
vector.VectorObject4D(x=1.1, y=2.2, eta=3.3, tau=4.4)

and so on for every class.

Or, you can use a single wrapper function to construct all possible combinations of object type vectors:

# Cartesian 2D vector
vector.obj(x=3, y=4)
# same in polar coordinates
vector.obj(rho=5, phi=0.9273)
# use "isclose" unless they are exactly equal
vector.obj(x=3, y=4).isclose(vector.obj(rho=5, phi=0.9273))
# Cartesian 3D vector
vector.obj(x=3, y=4, z=-2)
# Cartesian 4D vector
vector.obj(x=3, y=4, z=-2, t=10)
# in rho-phi-eta-t cylindrical coordinates
vector.obj(rho=5, phi=0.9273, eta=-0.39, t=10)
# use momentum-synonyms to get a momentum vector
vector.obj(pt=5, phi=0.9273, eta=-0.39, E=10)
vector.obj(rho=5, phi=0.9273, eta=-0.39, t=10) == vector.obj(
    pt=5, phi=0.9273, eta=-0.390035, E=10
)
# geometrical vectors have to use geometrical names ("tau", not "mass")
vector.obj(rho=5, phi=0.9273, eta=-0.39, t=10).tau
# momentum vectors can use momentum names (as well as geometrical ones)
vector.obj(pt=5, phi=0.9273, eta=-0.39, E=10).mass
# any combination of azimuthal, longitudinal, and temporal coordinates is allowed
vector.obj(pt=5, phi=0.9273, theta=1.9513, mass=8.4262)
vector.obj(x=3, y=4, z=-2, t=10).isclose(
    vector.obj(pt=5, phi=0.9273, theta=1.9513, mass=8.4262)
)

# Test instance type for any level of granularity.
(
    # is a vector or array of vectors
    isinstance(vector.obj(x=1.1, y=2.2), vector.Vector),
    # is 2D (not 3D or 4D)
    isinstance(vector.obj(x=1.1, y=2.2), vector.Vector2D),
    # is a vector object (not an array)
    isinstance(vector.obj(x=1.1, y=2.2), vector.VectorObject),
    # has momentum synonyms
    isinstance(vector.obj(px=1.1, py=2.2), vector.Momentum),
    # has transverse plane (2D, 3D, or 4D)
    isinstance(vector.obj(x=1.1, y=2.2, z=3.3, t=4.4), vector.Planar),
    # has all spatial coordinates (3D or 4D)
    isinstance(vector.obj(x=1.1, y=2.2, z=3.3, t=4.4), vector.Spatial),
    # has temporal coordinates (4D)
    isinstance(vector.obj(x=1.1, y=2.2, z=3.3, t=4.4), vector.Lorentz),
    # azimuthal coordinate type
    isinstance(vector.obj(x=1.1, y=2.2, z=3.3, t=4.4).azimuthal, vector.AzimuthalXY),
    # longitudinal coordinate type
    isinstance(
        vector.obj(x=1.1, y=2.2, z=3.3, t=4.4).longitudinal, vector.LongitudinalZ
    ),
    # temporal coordinate type
    isinstance(vector.obj(x=1.1, y=2.2, z=3.3, t=4.4).temporal, vector.TemporalT),
)

The allowed keyword arguments for 2D vectors are:

• x and y for Cartesian azimuthal coordinates,
• px and py for momentum,
• rho and phi for polar azimuthal coordinates,
• pt and phi for momentum.

For 3D vectors, you need the above and:

• z for the Cartesian longitudinal coordinate,
• pz for momentum,
• theta for the spherical polar angle (from $0$ to $\pi$, inclusive),
• eta for pseudorapidity, which is a kind of spherical polar angle.

For 4D vectors, you need the above and:

• t for the Cartesian temporal coordinate,
• E or energy to get four-momentum,
• tau for the "proper time" (temporal coordinate in the vector's rest coordinate system),
• M or mass to get four-momentum.

Since momentum vectors have momentum-synonyms in addition to the geometrical names, any momentum-synonym will make the whole vector a momentum vector.

If you want to bypass the dimension and coordinate system inference through keyword arguments (e.g. for static typing), you can use specialized constructors:

vector.VectorObject2D.from_xy(1.1, 2.2)
vector.MomentumObject3D.from_rhophiz(1.1, 2.2, 3.3)
vector.VectorObject4D.from_xyetatau(1.1, 2.2, 3.3, 4.4)

and so on, for all combinations of azimuthal, longitudinal, and temporal coordinates, geometric and momentum-flavored.

SymPy vectors

Note: Operations on SymPy vectors are only 100% compatible with numeric vectors (Python, NumPy, and Awkward backends) if the vectors are positive time-like, that is, if t**2 > x**2 + y**2 + z**2. The space-like and negative time-like cases have different sign conventions.

You can directly use the VectorSympy and MomentumSympy classes to construct object type vectors:

import sympy

x, y, z, t, px, py, pz, eta, tau = sympy.symbols(
    "x y z t px py pz eta tau",
    real=True,  # see sympy assumptions to add more restrictions on the symbols
)
vector.VectorSympy2D(x=x, y=y)
vector.MomentumSympy3D(px=px, py=py, pz=pz)
vector.VectorSympy4D(x=x, y=y, eta=eta, tau=tau)

and so on for every class.

# Test instance type for any level of granularity.
(
    # is a vector or array of vectors
    isinstance(vector.VectorSympy2D(x=x, y=y), vector.Vector),
    # is 2D (not 3D or 4D)
    isinstance(vector.VectorSympy2D(x=x, y=y), vector.Vector2D),
    # is a sympy vector (not an array)
    isinstance(vector.VectorSympy2D(x=x, y=y), vector.VectorSympy),
    # has momentum synonyms
    isinstance(vector.MomentumSympy2D(px=px, py=py), vector.Momentum),
    # has transverse plane (2D, 3D, or 4D)
    isinstance(vector.VectorSympy4D(x=x, y=y, z=z, t=t), vector.Planar),
    # has all spatial coordinates (3D or 4D)
    isinstance(vector.VectorSympy4D(x=x, y=y, z=z, t=t), vector.Spatial),
    # has temporal coordinates (4D)
    isinstance(vector.VectorSympy4D(x=x, y=y, z=z, t=t), vector.Lorentz),
    # azimuthal coordinate type
    isinstance(vector.VectorSympy4D(x=x, y=y, z=z, t=t).azimuthal, vector.AzimuthalXY),
    # longitudinal coordinate type
    isinstance(
        vector.VectorSympy4D(x=x, y=y, z=z, t=t).longitudinal, vector.LongitudinalZ
    ),
    # temporal coordinate type
    isinstance(vector.VectorSympy4D(x=x, y=y, z=z, t=t).temporal, vector.TemporalT),
)

Since VectorSympy2D, VectorSympy3D, VectorSympy4D, and their momentum equivalents operate on SymPy expressions, all of the normal SymPy methods and functions work on the results, coordinates, and the vectors.

sympy.init_session()  # latex printing

v1 = vector.VectorSympy2D(x=x, y=y)
sympy.Eq(v1.rho, sympy.sqrt(x**2 + y**2))

v2 = vector.VectorSympy4D(x=x, y=y, z=z, t=t)
v2.to_rhophithetatau().tau

values = {x: 3, y: 2, z: 1, t: 10}  # t**2 > x**2 + y**2 + z**2
v2.is_timelike()
v2.is_timelike().subs(values)

v2.to_rhophithetatau().tau.subs(values).evalf()

v2.boost(v2.to_beta3())
v2.boost(v2.to_beta3()).t
v2.boost(v2.to_beta3()).t.simplify()
v2.boost(v2.to_beta3()).t.subs(values)
v2.boost(v2.to_beta3()).t.subs(values).evalf()

All of the keyword arguments and rules that apply to vector.obj construction apply to vector.VectorSympyND and vector.MomentumObjectND objects.

NumPy arrays of vectors

You can directly use the VectorNumpy classes to construct object type vectors:

# NumPy-like arguments (literally passed through to NumPy)
vector.VectorNumpy2D(
    [(1.1, 2.1), (1.2, 2.2), (1.3, 2.3), (1.4, 2.4), (1.5, 2.5)],
    dtype=[("x", float), ("y", float)],
)

# Pandas-like arguments (dict from names to column arrays)
vector.VectorNumpy2D({"x": [1.1, 1.2, 1.3, 1.4, 1.5], "y": [2.1, 2.2, 2.3, 2.4, 2.5]})

# As with objects, the coordinate system and dimension is taken from the names of the fields.
vector.VectorNumpy4D(
    {
        "x": [1.1, 1.2, 1.3, 1.4, 1.5],
        "y": [2.1, 2.2, 2.3, 2.4, 2.5],
        "z": [3.1, 3.2, 3.3, 3.4, 3.5],
        "t": [4.1, 4.2, 4.3, 4.4, 4.5],
    }
)

and so on for every class.

Or, you can use a single wrapper function to construct all possible combinations of NumPy type vectors:

# NumPy-like arguments (literally passed through to NumPy)
vector.array(
    [(1.1, 2.1), (1.2, 2.2), (1.3, 2.3), (1.4, 2.4), (1.5, 2.5)],
    dtype=[("x", float), ("y", float)],
)

# Pandas-like arguments (dict from names to column arrays)
vector.array({"x": [1.1, 1.2, 1.3, 1.4, 1.5], "y": [2.1, 2.2, 2.3, 2.4, 2.5]})

# As with objects, the coordinate system and dimension is taken from the names of the fields.
vector.array(
    {
        "x": [1.1, 1.2, 1.3, 1.4, 1.5],
        "y": [2.1, 2.2, 2.3, 2.4, 2.5],
        "z": [3.1, 3.2, 3.3, 3.4, 3.5],
        "t": [4.1, 4.2, 4.3, 4.4, 4.5],
    }
)

vector.array(
    {
        "pt": [1.1, 1.2, 1.3, 1.4, 1.5],
        "phi": [2.1, 2.2, 2.3, 2.4, 2.5],
        "eta": [3.1, 3.2, 3.3, 3.4, 3.5],
        "M": [4.1, 4.2, 4.3, 4.4, 4.5],
    }
)

Existing NumPy arrays can be viewed as arrays of vectors, but it needs to be a structured array with recognized field names.

np.arange(0, 24, 0.1).view(  # NumPy array
    [
        ("x", float),
        ("y", float),
        ("z", float),
        ("t", float),
    ]  # interpret groups of four values as named fields
).view(
    vector.VectorNumpy4D
)  # give it vector properties and methods

Since VectorNumpy2D, VectorNumpy3D, VectorNumpy4D, and their momentum equivalents are NumPy array subclasses, all of the normal NumPy methods and functions work on them.

np.arange(0, 24, 0.1).view(
    [("x", float), ("y", float), ("z", float), ("t", float)]
).view(vector.VectorNumpy4D).reshape(6, 5, 2)

All of the keyword arguments and rules that apply to vector.obj construction apply to vector.arr dtypes.

Geometrical names are used in the dtype, even if momentum-synonyms are used in construction.

vector.arr({"px": [1, 2, 3, 4], "py": [1.1, 2.2, 3.3, 4.4], "pz": [0.1, 0.2, 0.3, 0.4]})

Awkward Arrays of vectors

Awkward Arrays are arrays with more complex data structures than NumPy allows, such as variable-length lists, nested records, missing and even heterogeneous data (multiple data types: use sparingly).

The vector.awk function behaves exactly like the ak.Array constructor, except that it makes arrays of vectors.

vector.awk(
    [
        [{"x": 1, "y": 1.1, "z": 0.1}, {"x": 2, "y": 2.2, "z": 0.2}],
        [],
        [{"x": 3, "y": 3.3, "z": 0.3}],
        [
            {"x": 4, "y": 4.4, "z": 0.4},
            {"x": 5, "y": 5.5, "z": 0.5},
            {"x": 6, "y": 6.6, "z": 0.6},
        ],
    ]
)

If you want any records named "Vector2D", "Vector3D", "Vector4D", "Momentum2D", "Momentum3D", or "Momentum4D" to be interpreted as vectors, register the behaviors globally.

vector.register_awkward()

ak.Array(
    [
        [{"x": 1, "y": 1.1, "z": 0.1}, {"x": 2, "y": 2.2, "z": 0.2}],
        [],
        [{"x": 3, "y": 3.3, "z": 0.3}],
        [
            {"x": 4, "y": 4.4, "z": 0.4},
            {"x": 5, "y": 5.5, "z": 0.5},
            {"x": 6, "y": 6.6, "z": 0.6},
        ],
    ],
    with_name="Vector3D",
)

All of the keyword arguments and rules that apply to vector.obj construction apply to vector.awk field names.

Finally, the VectorAwkward mixins can be subclassed to create custom vector classes. The awkward behavior classes and projections must be named as *Array. For example, coffea uses the following names - TwoVectorArray, ThreeVectorArray, PolarTwoVectorArray, SphericalThreeVectorArray, ...

Vector properties

Any geometrical coordinate can be computed from vectors in any coordinate system; they'll be provided or computed as needed.

vector.obj(x=3, y=4).rho
vector.obj(rho=5, phi=0.9273).x
vector.obj(rho=5, phi=0.9273).y
vector.obj(x=1, y=2, z=3).theta
vector.obj(x=1, y=2, z=3).eta

Some properties are not coordinates, but derived from them.

vector.obj(x=1, y=2, z=3).costheta
vector.obj(x=1, y=2, z=3).mag  # spatial magnitude
vector.obj(x=1, y=2, z=3).mag2  # spatial magnitude squared

These properties are provided because they can be computed faster or with more numerical stability in different coordinate systems. For instance, the magnitude ignores phi in polar coordinates.

vector.obj(rho=3, phi=0.123456789, z=4).mag2

Momentum vectors have geometrical properties as well as their momentum-synonyms.

vector.obj(px=3, py=4).rho
vector.obj(px=3, py=4).pt
vector.obj(x=1, y=2, z=3, E=4).tau
vector.obj(x=1, y=2, z=3, E=4).mass

Here's the key thing: arrays of vectors return arrays of coordinates.

vector.arr(
    {
        "x": [1.0, 2.0, 3.0, 4.0, 5.0],
        "y": [1.1, 2.2, 3.3, 4.4, 5.5],
        "z": [0.1, 0.2, 0.3, 0.4, 0.5],
    }
).theta

vector.awk(
    [
        [{"x": 1, "y": 1.1, "z": 0.1}, {"x": 2, "y": 2.2, "z": 0.2}],
        [],
        [{"x": 3, "y": 3.3, "z": 0.3}],
        [{"x": 4, "y": 4.4, "z": 0.4}, {"x": 5, "y": 5.5, "z": 0.5}],
    ]
).theta

# Make a large, random NumPy array of 3D momentum vectors.
array = (
    np.random.normal(0, 1, 150)
    .view([(x, float) for x in ("x", "y", "z")])
    .view(vector.MomentumNumpy3D)
    .reshape(5, 5, 2)
)

# Get the transverse momentum of each one.
array.pt

# The array and its components have the same shape.
array.shape
array.pt.shape

# Make a large, random Awkward Array of 3D momentum vectors.
array = vector.awk(
    [
        [
            {x: np.random.normal(0, 1) for x in ("px", "py", "pz")}
            for inner in range(np.random.poisson(1.5))
        ]
        for outer in range(50)
    ]
)

# Get the transverse momentum of each one, in the same nested structure.
array.pt

# The array and its components have the same list lengths (and can therefore be used together in subsequent calculations).
ak.num(array)
ak.num(array.pt)

Vector methods

Vector methods require arguments (in parentheses), which may be scalars or other vectors, depending on the calculation.

vector.obj(x=3, y=4).rotateZ(0.1)
vector.obj(rho=5, phi=0.4).rotateZ(0.1)

# Broadcasts a scalar rotation angle of 0.5 to all elements of the NumPy array.
print(
    vector.arr({"rho": [1, 2, 3, 4, 5], "phi": [0.1, 0.2, 0.3, 0.4, 0.5]}).rotateZ(0.5)
)

# Matches each rotation angle to an element of the NumPy array.
print(
    vector.arr({"rho": [1, 2, 3, 4, 5], "phi": [0.1, 0.2, 0.3, 0.4, 0.5]}).rotateZ(
        np.array([0.1, 0.2, 0.3, 0.4, 0.5])
    )
)

# Broadcasts a scalar rotation angle of 0.5 to all elements of the Awkward Array.
print(
    vector.awk(
        [[{"rho": 1, "phi": 0.1}, {"rho": 2, "phi": 0.2}], [], [{"rho": 3, "phi": 0.3}]]
    ).rotateZ(0.5)
)

# Broadcasts a rotation angle of 0.1 to both elements of the first list, 0.2 to the empty list, and 0.3 to the only element of the last list.
print(
    vector.awk(
        [[{"rho": 1, "phi": 0.1}, {"rho": 2, "phi": 0.2}], [], [{"rho": 3, "phi": 0.3}]]
    ).rotateZ([0.1, 0.2, 0.3])
)

# Matches each rotation angle to an element of the Awkward Array.
print(
    vector.awk(
        [[{"rho": 1, "phi": 0.1}, {"rho": 2, "phi": 0.2}], [], [{"rho": 3, "phi": 0.3}]]
    ).rotateZ([[0.1, 0.2], [], [0.3]])
)

Some methods are equivalent to binary operators.

vector.obj(x=3, y=4).scale(10)
vector.obj(x=3, y=4) * 10
10 * vector.obj(x=3, y=4)
vector.obj(rho=5, phi=0.5) * 10

Some methods involve more than one vector.

vector.obj(x=1, y=2).add(vector.obj(x=5, y=5))
vector.obj(x=1, y=2) + vector.obj(x=5, y=5)
vector.obj(x=1, y=2).dot(vector.obj(x=5, y=5))
vector.obj(x=1, y=2) @ vector.obj(x=5, y=5)

The vectors can use different coordinate systems. Conversions are necessary, but minimized for speed and numeric stability.

# both are Cartesian, dot product is exact
vector.obj(x=3, y=4) @ vector.obj(x=6, y=8)
# one is polar, dot product is approximate
vector.obj(rho=5, phi=0.9273) @ vector.obj(x=6, y=8)
# one is polar, dot product is approximate
vector.obj(x=3, y=4) @ vector.obj(rho=10, phi=0.9273)
# both are polar, a formula that depends on phi differences is used
vector.obj(rho=5, phi=0.9273) @ vector.obj(rho=10, phi=0.9273)

In Python, some "operators" are actually built-in functions, such as abs.

abs(vector.obj(x=3, y=4))

Note that abs returns

• rho for 2D vectors
• mag for 3D vectors
• tau (mass) for 4D vectors

Use the named properties when you want magnitude in a specific number of dimensions; use abs when you want the magnitude for any number of dimensions.

The vectors can be from different backends. Normal rules for broadcasting Python numbers, NumPy arrays, and Awkward Arrays apply.

vector.arr({"x": [1, 2, 3, 4, 5], "y": [0.1, 0.2, 0.3, 0.4, 0.5]}) + vector.obj(
    x=10, y=5
)

(
    vector.awk(
        [  # an Awkward Array of vectors
            [{"x": 1, "y": 1.1}, {"x": 2, "y": 2.2}],
            [],
            [{"x": 3, "y": 3.3}],
            [{"x": 4, "y": 4.4}, {"x": 5, "y": 5.5}],
        ]
    )
    + vector.obj(x=10, y=5)  # and a single vector object
)

(
    vector.awk(
        [  # an Awkward Array of vectors
            [{"x": 1, "y": 1.1}, {"x": 2, "y": 2.2}],
            [],
            [{"x": 3, "y": 3.3}],
            [{"x": 4, "y": 4.4}, {"x": 5, "y": 5.5}],
        ]
    )
    + vector.arr(
        {"x": [4, 3, 2, 1], "y": [0.1, 0.1, 0.1, 0.1]}
    )  # and a NumPy array of vectors
)

Some operations are defined for 2D or 3D vectors, but are usable on higher-dimensional vectors because the additional components can be ignored or are passed through unaffected.

# deltaphi is a planar operation (defined on the transverse plane)
vector.obj(rho=1, phi=0.5).deltaphi(vector.obj(rho=2, phi=0.3))
# but we can use it on 3D vectors
vector.obj(rho=1, phi=0.5, z=10).deltaphi(vector.obj(rho=2, phi=0.3, theta=1.4))
# and 4D vectors
vector.obj(rho=1, phi=0.5, z=10, t=100).deltaphi(
    vector.obj(rho=2, phi=0.3, theta=1.4, tau=1000)
)
# and mixed dimensionality
vector.obj(rho=1, phi=0.5).deltaphi(vector.obj(rho=2, phi=0.3, theta=1.4, tau=1000))

This is especially useful for giving 4D vectors all the capabilities of 3D vectors.

vector.obj(x=1, y=2, z=3).rotateX(np.pi / 4)
vector.obj(x=1, y=2, z=3, tau=10).rotateX(np.pi / 4)
vector.obj(pt=1, phi=1.3, eta=2).deltaR(vector.obj(pt=2, phi=0.3, eta=1))
vector.obj(pt=1, phi=1.3, eta=2, mass=5).deltaR(
    vector.obj(pt=2, phi=0.3, eta=1, mass=10)
)

For a few operations - +, -, ==, !=, ... - the dimension of the vectors should be equal. This can be achieved by using the like method, to_{coordinate_name} methods, to_Vector*D methods. The to_Vector*D methods provide more flexibility to the users, that is, new coordinate values can be passed into the methods as named arguments.

v1 = vector.obj(x=1, y=2, z=3)
v2 = vector.obj(x=1, y=2)

v1 - v2.like(v1)  # transforms v2 to v1's coordinate system (imputes z=0)
v1.like(v2) - v2  # transforms v1 to v2's coordinate system (removes z)
v1 - v2.to_xyz()  # transforms v2 to xyz coordinates (imputes z=0)
v1.to_xy() - v2  # transforms v1 to xy coordinates (removes z)
v1 - v2.to_Vector3D(z=3)  # transforms v2 to 3D (imputes z=3)
v1.to_Vector2D() - v2  # transforms v1 to 2D (removes z)

Similarly, for a few vector methods, the dimension of the input vectors are type checked strictly.

For instance, a cross-product is only defined for 3D and 7D vectors; hence, running the method on a 4D vector will error out.

vector.obj(x=0.1, y=0.2, z=0.3).cross(vector.obj(x=0.4, y=0.5, z=0.6))

The (current) list of properties and methods is:

Planar (2D, 3D, 4D):

• x (px)
• y (py)
• rho (pt): two-dimensional magnitude
• rho2 (pt2): two-dimensional magnitude squared
• phi
• deltaphi(vector): difference in phi (signed and rectified to $-\pi$ through $\pi$)
• rotateZ(angle)
• transform2D(obj): the obj must supply components through obj["xx"], obj["xy"], obj["yx"], obj["yy"]
• is_parallel(vector, tolerance=1e-5): only true if they're pointing in the same direction
• is_antiparallel(vector, tolerance=1e-5): only true if they're pointing in opposite directions
• is_perpendicular(vector, tolerance=1e-5)

Spatial (3D, 4D):

• z (pz)
• theta
• eta
• costheta
• cottheta
• mag (p): three-dimensional magnitude, does not include temporal component
• mag2 (p2): three-dimensional magnitude squared
• cross: cross-product (strictly 3D)
• deltaangle(vector): difference in angle (always non-negative)
• deltaeta(vector): difference in eta (signed)
• deltaR(vector): $\Delta R = \sqrt{\Delta\phi^2 + \Delta\eta^2}$
• deltaR2(vector): the above, squared
• rotateX(angle)
• rotateY(angle)
• rotate_axis(axis, angle): the magnitude of axis is ignored, but it must be at least 3D
• rotate_euler(phi, theta, psi, order="zxz"): the arguments are in the same order as ROOT::Math::EulerAngles, and order="zxz" agrees with ROOT's choice of conventions
• rotate_nautical(yaw, pitch, roll)
• rotate_quaternion(u, i, j, k): again, the conventions match ROOT::Math::Quaternion.
• transform3D(obj): the obj must supply components through obj["xx"], obj["xy"], etc.
• is_parallel(vector, tolerance=1e-5): only true if they're pointing in the same direction
• is_antiparallel(vector, tolerance=1e-5): only true if they're pointing in opposite directions
• is_perpendicular(vector, tolerance=1e-5)

Lorentz (4D only):

• t (E, energy): follows the ROOT::Math::LorentzVector behavior of treating spacelike vectors as negative t and negative tau and truncating wrong-direction timelike vectors
• t2 (E2, energy2)
• tau (M, mass): see note above
• tau2 (M2, mass2)
• beta: scalar(s) between $0$ (inclusive) and $1$ (exclusive, unless the vector components are infinite)
• deltaRapidityPhi: $\Delta R_{\mbox{rapidity}} = \Delta\phi^2 + \Delta \mbox{rapidity}^2$
• deltaRapidityPhi2: the above, squared
• gamma: scalar(s) between $1$ (inclusive) and $\infty$
• rapidity: scalar(s) between $0$ (inclusive) and $\infty$
• boost_p4(four_vector): change coordinate system using another 4D vector as the difference
• boost_beta(three_vector): change coordinate system using a 3D beta vector (all components between $-1$ and $+1$)
• boost(vector): uses the dimension of the given vector to determine behavior
• boostX(beta=None, gamma=None): supply beta xor gamma, but not both
• boostY(beta=None, gamma=None): supply beta xor gamma, but not both
• boostZ(beta=None, gamma=None): supply beta xor gamma, but not both
• transform4D(obj): the obj must supply components through obj["xx"], obj["xy"], etc.
• to_beta3(): turns a four_vector (for boost_p4) into a three_vector (for boost_beta3)
• is_timelike(tolerance=0)
• is_spacelike(tolerance=0)
• is_lightlike(tolerance=1e-5): note the different tolerance

All numbers of dimensions:

• unit(): note the parentheses
• dot(vector): can also use the @ operator
• add(vector): can also use the + operator
• subtract(vector): can also use the - operator
• scale(factor): can also use the * operator
• equal(vector): can also use the == operator, but consider isclose instead
• not_equal(vector): can also use the != operator, but consider isclose instead
• sum(): can also use the numpy.sum or awkward.sum, only for NumPy and Awkward vectors
• count_nonzero(): can also use numpy.count_nonzero or awkward.count_nonzero, only for NumPy and Awkward vectors
• count(): can also use awkward.count, only for Awkward vectors
• isclose(vector, rtol=1e-5, atol=1e-8, equal_nan=False): works like np.isclose; arrays also have an allclose method
• to_VectorND(coordinates)/to_ND(coordinates): replace N with the required vector dimension
• to_{coordinate-names}: for example - to_rhophietatau
• like(other): projects the vector into the dimensions of other, for example - two_d_vector.like(three_d_vector)

Compiling your Python with Numba

Numba is a just-in-time (JIT) compiler for a mathematically relevant subset of NumPy and Python. It allows you to write fast code without leaving the Python environment. The drawback of Numba is that it can only compile code blocks involving objects and functions that it recognizes.

The Vector library includes extensions to inform Numba about vector objects, vector NumPy arrays, and vector Awkward Arrays. At the time of writing, the implementation of vector NumPy arrays is incomplete due to numba/numba#6148.

For instance, consider the following function:

@nb.njit
def compute_mass(v1, v2):
    return (v1 + v2).mass


compute_mass(vector.obj(px=1, py=2, pz=3, E=4), vector.obj(px=-1, py=-2, pz=-3, E=4))

When the two MomentumObject4D objects are passed as arguments, Numba recognizes them and replaces the Python objects with low-level structs. When it compiles the function, it recognizes + as the 4D add function and recognizes .mass as the tau component of the result.

Although this demonstrates that Numba can manipulate vector objects, there is no performance advantage (and a likely disadvantage) to compiling a calculation on just a few vectors. The advantage comes when many vectors are involved, in arrays.

# This is still not a large number. You want millions.
array = vector.awk(
    [
        [
            dict(
                {x: np.random.normal(0, 1) for x in ("px", "py", "pz")},
                E=np.random.normal(10, 1),
            )
            for inner in range(np.random.poisson(1.5))
        ]
        for outer in range(50)
    ]
)


@nb.njit
def compute_masses(array):
    out = np.empty(len(array), np.float64)
    for i, event in enumerate(array):
        total = vector.obj(px=0.0, py=0.0, pz=0.0, E=0.0)
        for vec in event:
            total = total + vec
        out[i] = total.mass
    return out


compute_masses(array)

Sub-classing Awkward mixins

At the moment, it is possible to sub-class vector awkward mixins to extend the vector functionalities. Although the mechanism in place works well, it is still being improved.

For instance, the MomentumAwkward classes can be extended in the following way:

behavior = vector.backends.awkward.behavior


@ak.mixin_class(behavior)
class TwoVector(vector.backends.awkward.MomentumAwkward2D):
    pass


@ak.mixin_class(behavior)
class ThreeVector(vector.backends.awkward.MomentumAwkward3D):
    pass


# required for transforming vectors
# the class names must always end with "Array"
TwoVectorArray.ProjectionClass2D = TwoVectorArray  # noqa: F821
TwoVectorArray.ProjectionClass3D = ThreeVectorArray  # noqa: F821
TwoVectorArray.MomentumClass = TwoVectorArray  # noqa: F821

ThreeVectorArray.ProjectionClass2D = TwoVectorArray  # noqa: F821
ThreeVectorArray.ProjectionClass3D = ThreeVectorArray  # noqa: F821
ThreeVectorArray.MomentumClass = ThreeVectorArray  # noqa: F821

vec = ak.zip(
    {
        "pt": [[1, 2], [], [3], [4]],
        "phi": [[1.2, 1.4], [], [1.6], [3.4]],
    },
    with_name="TwoVector",
    behavior=behavior,
)

vec

The binary operators are not automatically registered by awkward, but vector methods can be used to perform operations on sub-classed vectors.

vec.add(vec)

Similarly, other vector methods can be used by the new methods internally.

@ak.mixin_class(behavior)
class LorentzVector(vector.backends.awkward.MomentumAwkward4D):
    @ak.mixin_class_method(np.divide, {numbers.Number})
    def divide(self, factor):
        return self.scale(1 / factor)


# required for transforming vectors
# the class names must always end with "Array"
LorentzVectorArray.ProjectionClass2D = TwoVectorArray  # noqa: F821
LorentzVectorArray.ProjectionClass3D = ThreeVectorArray  # noqa: F821
LorentzVectorArray.ProjectionClass4D = LorentzVectorArray  # noqa: F821
LorentzVectorArray.MomentumClass = LorentzVectorArray  # noqa: F821

vec = ak.zip(
    {
        "pt": [[1, 2], [], [3], [4]],
        "eta": [[1.2, 1.4], [], [1.6], [3.4]],
        "phi": [[0.3, 0.4], [], [0.5], [0.6]],
        "energy": [[50, 51], [], [52], [60]],
    },
    with_name="LorentzVector",
    behavior=behavior,
)

vec / 2
vec.like(vector.obj(x=1, y=2))
vec.like(vector.obj(x=1, y=2, z=3))

It is also possible to manually add binary operations in vector's behavior dict to enable binary operations.

_binary_dispatch_cls = {
    "TwoVector": TwoVector,
    "ThreeVector": ThreeVector,
    "LorentzVector": LorentzVector,
}
_rank = [TwoVector, ThreeVector, LorentzVector]

for lhs, lhs_to in _binary_dispatch_cls.items():
    for rhs, rhs_to in _binary_dispatch_cls.items():
        out_to = min(lhs_to, rhs_to, key=_rank.index)
        behavior[(np.add, lhs, rhs)] = out_to.add
        behavior[(np.subtract, lhs, rhs)] = out_to.subtract

vec + vec
vec.to_2D() + vec.to_2D()

Finally, instead of manually registering the superclass ufuncs, one can use the utility copy_behaviors function to copy behavior items for a new subclass -

behavior.update(ak._util.copy_behaviors("Vector2D", "TwoVector", behavior))
behavior.update(ak._util.copy_behaviors("Vector3D", "ThreeVector", behavior))
behavior.update(ak._util.copy_behaviors("Momentum4D", "LorentzVector", behavior))

vec + vec
vec.to_2D() + vec.to_2D()

Talks about vector

• 3rd July 2024 - A new SymPy backend for vector: uniting experimental and theoretical physicists - PyHEP 2024 (virtual)
• 9th October 2023 - What’s new with Vector? First major release is out! - PyHEP 2023 (virtual) 🎥
• 13th September 2022 - Constructing HEP vectors and analyzing HEP data using Vector - PyHEP 2022 (virtual) 🎥
• 20th July 2022 - Analysis Grand Challenge / HEP Scientific Python Ecosystem - DANCE/CoDaS@Snowmass 2022 computational and data science software training
• 25th April 2022 - Foundation libraries (uproot, awkward, hist, mplhep) - IRIS-HEP AGC Tools 2022 Workshop 🎥
• 3rd November 2021 - Data handling: uproot, awkward & vector - IRIS-HEP AGC Tools 2021 Workshop 🎥

Status as of November 17, 2023

First major release of vector is out and the package has reached a stable position. The work is spearheaded by bug reports and feature requests created on GitHub. It can only be improved by your feedback!

Contributors ✨

Thanks goes to these wonderful people (emoji key):

Jim Pivarski 🚧 💻 📖	Henry Schreiner 🚧 💻 📖	Eduardo Rodrigues 🚧 💻 📖	N!no 📖	Peter Fackeldey 📖	Luke Kreczko 💻	Nicholas Smith 🤔
Jonas Eschle 🤔

This project follows the all-contributors specification. Contributions of any kind welcome! See CONTRIBUTING.md for information on setting up a development environment.

Acknowledgements

This library was primarily developed by Jim Pivarski, Henry Schreiner, Saransh Chopra, and Eduardo Rodrigues.

Support for this work was provided by the National Science Foundation cooperative agreement OAC-1836650 and PHY-2323298 (IRIS-HEP) and OAC-1450377 (DIANA/HEP). Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.