Mathematical utilities (skhep.math)¶

Module for mathematical functions and utilities.

This module contains in particular:

Vector classes.

Geometry classes.

Mathematical functions relevant to kinematics¶

skhep.math.kinematics.Kallen_function(x, y, z)¶

The Kallen function, aka triangle or lambda function, named after physicist Anders Olof Gunnar Kallen [Kallen].

\[\begin{split}\begin{eqnarray} \lambda(x,y,z) &=& x^2 + y^2 + z^2 - 2 x y - 2 y z - 2 z x \\ &=& (x-y-z)^2 - 4 y z \\ &=& [ x - (\sqrt{y}+\sqrt{z})^2 ] [ x - (\sqrt{y}-\sqrt{z})^2 ] \,\,\,\mathrm{if} \,\,\,y, z > 0 \end{eqnarray}\end{split}\]

Example

Calculate in the rest frame of a particle of mass M decaying to 2 particles labeled 1 and 2, \(P (M) \to p1 (m1) + p2 (m2)\), the momenta of 1 and 2 given by \(p = |\mathbf{p1}| = |\mathbf{p2}|\):

>>> from skhep.math  import Kallen_function
>>> from skhep.units import MeV, GeV
>>> from math import sqrt
>>> M = 5.279 * GeV; m1 = 493.7 * MeV; m2 = 139.6 * MeV
>>> p = sqrt( Kallen_function( M**2, m1**2, m2**2 ) ) / (2*M)
>>> print p / GeV   # print the CMS momentum in GeV
2.61453580221

[Kallen]

https://en.wikipedia.org/wiki/K%C3%A4ll%C3%A9n_function

skhep.math.kinematics.lifetime_to_width(tau)¶

Convert from a particle lifetime to a decay width.

Parameters:	tau (float > 0) – Particle lifetime, typically in picoseconds (any HEP time unit is OK).
Returns:
Return type:	Particle decay width, in the HEP standard energy unit MeV.

skhep.math.kinematics.width_to_lifetime(Gamma)¶

Convert from a particle decay width to a lifetime.

Parameters:	Gamma (float > 0) – Particle decay width, typically in MeV (any HEP energy unit is OK).
Returns:
Return type:	Particle lifetime, in the HEP standard time unit ns.

Vector classes¶

Two vector classes are available:

Vector3D : a 3-dimensional vector.
LorentzVector: a Lorentz vector, i.e. a 4-dimensional Minkowski space-time vector

or a 4-momentum vector. The metric is (-1,-1,-1,+1).

class skhep.math.vectors.LorentzVector(x=0.0, y=0.0, z=0.0, t=0.0)¶

Class representing a Lorentz vector, either a 4-dimensional Minkowski space-time vector or a 4-momentum vector. The 4-vector components can be seen as (x,y,z,t) or (px,py,pz,E).

Constructors:: __init__(x=0., y=0., z=0., t=0.) from4vector(avector) from3vector(vector3d, t)

beta¶: Return \(\beta = v/c\).

boost(*args)¶: Apply a Lorentz boost on the Lorentz vector.

boostvector¶: Return the spatial component divided by the time component.

copy()¶

Get a copy of the LorentzVector.

Example

>>> v = ...
>>> v1 = v.copy()

costheta()¶: Return the cosinus of the spherical coordinate theta.

deltaeta(other)¶: Return the pseudorapidity difference, \(\Delta \eta\), with another Lorentz vector.

deltaphi(other)¶: Return the phi angle difference, \(\Delta \phi\), with another Lorentz vector.

deltar(other)¶: Return \(\Delta R\) the distance in (eta,phi) space with another Lorentz vector, defined as: \(\Delta R = \sqrt{(\Delta \eta)^2 + (\Delta \phi)^2}\)

dot(other)¶: Dot product with another Lorentz vector.

e¶: Return the energy/time component, aka momentum coordinate at position 3.

et¶: Return the transverse energy.

eta¶: Return the pseudorapidity.

classmethod from3vector(vector3d, t)¶: Constructor from a Vector3D and the time/energy component.

classmethod from4vector(other)¶: Copy constructor.

classmethod fromiterable(values)¶: Constructor from a suitable iterable object. Suitable means here that all entries are numbers and the length equals 4.

gamma¶: Return \(\gamma = 1/\sqrt{1-\beta^2}\).

islightlike()¶: Check if Lorentz Vector is light-like.

isspacelike()¶: Check if Lorentz Vector is space-like.

istimelike()¶: Check if Lorentz Vector is time-like.

m¶: Return the invariant mass.

m2¶: Return the square of the invariant mass.

mag¶: Magnitude, a.k.a. norm, of the Lorentz vector.

mag2¶: Square of the magnitude, a.k.a. norm, of the Lorentz vector.

mass¶: Return the invariant mass.

mass2¶: Return the square of the invariant mass.

mt¶: Return the transverse mass.

mt2¶: Return the square of the transverse mass.

p¶: Return the momentum, aka norm of the momentum vector.

perp¶: Transverse component of the spatial components.

perp2¶: Square of the transverse component, in the (x,y) plane, of the spatial components.

phi(deg=False)¶

Return the spherical or cylindrical coordinate phi.

deg : return the angle in degrees (default is radians)

pseudorapidity¶: “Return the pseudorapidity. Alternative to eta() method.

pt¶: Return the transverse momentum, aka transverse component of the momentum vector.

px¶: Return the Vector3D coordinate px, aka first momentum coordinate at position 0.

py¶: Return the Vector3D coordinate px, aka second momentum coordinate at position 1.

pz¶: Return the Vector3D coordinate pz, aka third momentum coordinate at position 2.

rapidity¶: Return the rapidity.

rotate(angle, *args)¶: Rotate vector by a given angle (in radians) around a given axis.

rotatex(angle)¶: Rotate vector by a given angle (in radians) around the x axis.

rotatey(angle)¶: Rotate vector by a given angle (in radians) around the y axis.

rotatez(angle)¶: Rotate vector by a given angle (in radians) around the z axis.

set(x, y, z, t)¶: Update/set all components at once.

setptetaphie(pt, eta, phi, e)¶: Set the transverse momentum, the pseudorapidity, the angle phi and the energy.

setptetaphim(pt, eta, phi, m)¶: Set the transverse momentum, the pseudorapidity, the angle phi and the mass.

setpxpypze(px, py, pz, e)¶: Set the px,py,pz components and the energy.

setpxpypzm(px, py, pz, m)¶: Set the px,py,pz components and the mass.

t¶: Return the time/energy component, aka coordinate at position 3.

theta(deg=False)¶

Return the spherical coordinate theta.

Options:: deg : return the angle in degrees (default is radians)

tolist()¶: Return the LorentzVector as a list.

transversemass¶: Return the transverse mass.

transversemass2¶: Return the square of the transverse mass.

vector¶: Return a copy of the vector of spatial components.

x¶: Return the coordinate x, aka first coordinate at position 0.

y¶: Return the coordinate x, aka second coordinate at position 1.

z¶: Return the coordinate z, aka third coordinate at position 2.

class skhep.math.vectors.Vector3D(x=0.0, y=0.0, z=0.0)¶

Vector class in 3 dimensions.

Constructors:: __init__(x=0., y=0., z=0.) origin() frompoint(x, y, z) fromvector(avector) fromsphericalcoords(r, theta, phi) fromcylindricalcoords(rho, phi, z) fromiterable(values)

angle(other, deg=False)¶

Angle with respect to another vector.

deg : return the angle in degrees (default is radians)

copy()¶

Get a copy of the vector.

Example

>>> v = ...
>>> v1 = v.copy()

cosdelta(other)¶: Get cos(angle) with respect to another vector

costheta()¶: Return the cosinus of the spherical coordinate theta.

cross(other)¶: Cross product with another vector.

dot(other)¶: Dot product with another vector.

classmethod fromcylindricalcoords(rho, phi, z)¶

Constructor from a space point specified in cylindrical coordinates.

Parameters:	rho (radial distance from the z-axis (rho > 0)) – phi (azimuthal angle in radians (phi in [-pi, pi) rad)) – z (height) –

classmethod fromiterable(values)¶: Constructor from a suitable iterable object. Suitable means here that all entries are numbers and the length equals 3.

classmethod frompoint(x, y, z)¶: Constructor from an explicit space point.

classmethod fromsphericalcoords(r, theta, phi)¶

Constructor from a space point specified in spherical coordinates.

Parameters:	r (radius, the radial distance from the origin (r > 0)) – theta (inclination in radians (theta in [0, pi] rad)) – phi (azimuthal angle in radians (phi in [0, 2pi) rad)) –

classmethod fromvector(other)¶: Copy constructor.

isantiparallel(other)¶: Check if another vector is antiparallel. Two vectors are antiparallel if they have opposite direction but not necessarily the same magnitude.

iscollinear(other)¶: Check if another vector is collinear Two vectors are collinear if they have parallel or antiparallel

isopposite(other)¶: Two vectors are opposite if they have the same magnitude but opposite direction.

isparallel(other)¶: Check if another vector is parallel. Two vectors are parallel if they have the same direction but not necessarily the same magnitude.

isperpendicular(other)¶: Check if another vector is perpendicular.

mag¶: Magnitude, a.k.a. norm, of the vector.

mag2¶: Square of the magnitude, a.k.a. norm, of the vector.

classmethod origin()¶: Shortcut constuctor for the origin (x=0.,y=0.,z=0.). Equivalent to the default constructor Vector3D().

phi(deg=False)¶

Return the spherical or cylindrical coordinate phi.

deg : return the angle in degrees (default is radians)

r¶: Return the spherical coordinate r.

rho¶: Return the cylindrical coordinate rho.

rotate(angle, *args)¶: Rotate vector by a given angle (in radians) around a given axis.

rotatex(angle)¶: Rotate vector by a given angle (in radians) around the x axis.

rotatey(angle)¶: Rotate vector by a given angle (in radians) around the y axis.

rotatez(angle)¶: Rotate vector by a given angle (in radians) around the z axis.

set(x, y, z)¶: Update the vector components all at once.

theta(deg=False)¶

Return the spherical coordinate theta.

deg : return the angle in degrees (default is radians)

tolist()¶: Return the vector as a list.

unit()¶: Return the normalized vector, i.e. the unit vector along the direction of itself.

x¶: Return the x, aka first coordinate at position 0.

y¶: Return the y, aka second coordinate at position 1.

z¶: Return the z, aka third coordinate at position 2.

Geometry classes¶

Three geometry classes are available: * Point3D : a point in 3-dimensional space * Line3D : a line in 3-dimensional space * Plane3D : a plane in 3-dimensional space

class skhep.math.geometry.Vector3D(x=0.0, y=0.0, z=0.0)¶

Vector class in 3 dimensions.

Constructors:: __init__(x=0., y=0., z=0.) origin() frompoint(x, y, z) fromvector(avector) fromsphericalcoords(r, theta, phi) fromcylindricalcoords(rho, phi, z) fromiterable(values)

angle(other, deg=False)¶

Angle with respect to another vector.

deg : return the angle in degrees (default is radians)

copy()¶

Get a copy of the vector.

Example

>>> v = ...
>>> v1 = v.copy()

cosdelta(other)¶: Get cos(angle) with respect to another vector

costheta()¶: Return the cosinus of the spherical coordinate theta.

cross(other)¶: Cross product with another vector.

dot(other)¶: Dot product with another vector.

classmethod fromcylindricalcoords(rho, phi, z)¶

Constructor from a space point specified in cylindrical coordinates.

Parameters:	rho (radial distance from the z-axis (rho > 0)) – phi (azimuthal angle in radians (phi in [-pi, pi) rad)) – z (height) –

classmethod fromiterable(values)¶: Constructor from a suitable iterable object. Suitable means here that all entries are numbers and the length equals 3.

classmethod frompoint(x, y, z)¶: Constructor from an explicit space point.

classmethod fromsphericalcoords(r, theta, phi)¶

Constructor from a space point specified in spherical coordinates.

Parameters:	r (radius, the radial distance from the origin (r > 0)) – theta (inclination in radians (theta in [0, pi] rad)) – phi (azimuthal angle in radians (phi in [0, 2pi) rad)) –

classmethod fromvector(other)¶: Copy constructor.

isantiparallel(other)¶: Check if another vector is antiparallel. Two vectors are antiparallel if they have opposite direction but not necessarily the same magnitude.

iscollinear(other)¶: Check if another vector is collinear Two vectors are collinear if they have parallel or antiparallel

isopposite(other)¶: Two vectors are opposite if they have the same magnitude but opposite direction.

isparallel(other)¶: Check if another vector is parallel. Two vectors are parallel if they have the same direction but not necessarily the same magnitude.

isperpendicular(other)¶: Check if another vector is perpendicular.

mag¶: Magnitude, a.k.a. norm, of the vector.

mag2¶: Square of the magnitude, a.k.a. norm, of the vector.

classmethod origin()¶: Shortcut constuctor for the origin (x=0.,y=0.,z=0.). Equivalent to the default constructor Vector3D().

phi(deg=False)¶

Return the spherical or cylindrical coordinate phi.

deg : return the angle in degrees (default is radians)

r¶: Return the spherical coordinate r.

rho¶: Return the cylindrical coordinate rho.

rotate(angle, *args)¶: Rotate vector by a given angle (in radians) around a given axis.

rotatex(angle)¶: Rotate vector by a given angle (in radians) around the x axis.

rotatey(angle)¶: Rotate vector by a given angle (in radians) around the y axis.

rotatez(angle)¶: Rotate vector by a given angle (in radians) around the z axis.

set(x, y, z)¶: Update the vector components all at once.

theta(deg=False)¶

Return the spherical coordinate theta.

deg : return the angle in degrees (default is radians)

tolist()¶: Return the vector as a list.

unit()¶: Return the normalized vector, i.e. the unit vector along the direction of itself.

x¶: Return the x, aka first coordinate at position 0.

y¶: Return the y, aka second coordinate at position 1.

z¶: Return the z, aka third coordinate at position 2.

class skhep.math.geometry.Point3D(*args, **kwargs)¶

Point in 3D space

copy()¶: Make a copy of this point.

distance(other)¶: Get ``distance’’ between point and other object.

classmethod fromcylindricalcoords(rho, phi, z)¶

Constructor from a space point specified in cylindrical coordinates.

Parameters:	rho (radial distance from the z-axis (rho > 0)) – phi (azimuthal angle in radians (phi in [-pi, pi) rad)) – z (height) –

classmethod fromiterable(values)¶: Constructor from a suitable iterable object. Suitable means here that all entries are numbers and the length equals 3.

classmethod frompoint(x, y, z)¶: Constructor from an explicit space point.

classmethod fromsphericalcoords(r, theta, phi)¶

Constructor from a space point specified in spherical coordinates.

Parameters:	r (radius, the radial distance from the origin (r > 0)) – theta (inclination in radians (theta in [0, pi] rad)) – phi (azimuthal angle in radians (phi in [0, 2pi) rad)) –

classmethod fromvector(other)¶: Copy constructor.

mag¶: Magnitude, a.k.a. norm, of the point.

mag2¶: Square of the magnitude, a.k.a. norm, of the point.

on_line(line)¶: In the point on the line?

on_plane(plane)¶: Is the point on the plane?

classmethod origin()¶: Shortcut constuctor for the origin (x=0.,y=0.,z=0.). Equivalent to the default constructor Point3D().

phi¶

Return the spherical or cylindrical coordinate phi.

deg : float. Return the angle in degrees (default is radians).

rho¶: Return the cylindrical coordinate rho.

theta¶

Return the spherical coordinate theta.

deg : float. Return the angle in degrees (default is radians).

tolist()¶: Return the vector as a list.

x¶: Return the x, aka first coordinate at position 0.

y¶: Return the y, aka second coordinate at position 1.

z¶: Return the z, aka third coordinate at position 2.

class skhep.math.geometry.Line3D(point=<Point3D(x=0, y=0, z=0)>, vector=<Vector3D (x=0, y=0, z=1)>)¶

Line in 3D space. Line is defined by a point on line and by the direction.

angle(other)¶: Get the ``angle’’ between a line and other object.

closest_point(point)¶

Get the point on line that is closest to the given point in space.

Example

>>> line  = ...
>>> point = ...
>>> closest  = line.closest_point ( point )

closest_point_param(point)¶

Get the ``mu’’ parameter of the point on line that is closest to the given point in space.

Example

>>> line  = ...
>>> point = ...
>>> mu    = line.closest_point_param ( point )

closest_points(line)¶

Get two points on lines that is closest to each other.

Example

>>> line1 = ...
>>> line2 = ...
>>> p1 , p2  = line1.closest_points ( line2 )

closest_points_params(line)¶

Get the ``mu’’ parameters of two points on lines that is closest to each other.

Example

>>> line1 = ...
>>> line2 = ...
>>> mu1 , mu2  = line1.closest_points_param ( line2 )

copy()¶: Make a copy of this line.

distance(other)¶

Get ``distance’’ between line and other object.

Example

>>> line  = ...
>>> point = ...
>>> line2 = ...
>>> plane = ...
>>> print line.distance ( line2 )
>>> print line.distance ( point )
>>> print line.distance ( plane )

classmethod from_line(line)¶: Create the line from anoother line.

classmethod from_points(point1, point2)¶

Create the line from two points.

Parameters:	point1 (the first point on the line) – point2 (the second point on the line) –

intersect(other)¶: Get the ``intersection’’ between a line and other object.

iscollinear(other)¶: Is the line collinear to another line or vector?

on_plane(plane)¶: Is the line on the plane?

point_on_line(mu)¶

Get the point on line, that corresponds to parameter ``mu’‘.

Example

>>> line = ...
>>> p0 = line.point_on_line ( 0 ) ## should be equal to line.point
>>> p1 = line.point_on_line ( 1 ) ## shodul be equal to line.point+line.vector

class skhep.math.geometry.Plane3D(point=<Point3D(x=0, y=0, z=0)>, normal=<Vector3D (x=0, y=0, z=1)>)¶

Plane in 3D space Plane is defined by a point on plane and by the normal vector

angle(other)¶: Get the ``angle’’ between a plane and other object

copy()¶: Make a copy of this plane.

distance(other)¶

Get ``distance’’ between plane and other object.

Example

>>> line  = ...
>>> point = ...
>>> line2 = ...
>>> plane = ...
>>> print line.distance ( line2 )
>>> print line.distance ( point )
>>> print line.distance ( plane )

classmethod from_line_and_point(line, point)¶

Create the plane from the line and point.

Parameters:	line (the line in plane) – point (the point in plane) –

classmethod from_points(point1, point2, point3)¶

Create the plane from three points.

Parameters:	point1 (the first point on the plane) – point2 (the second point on the plane) – point3 (the third point on the plane) –

intersect(other)¶: Get the ``intersection’’ between a line and other object

isparallel(other)¶: Is the plane collinear to another line, vector or plane?

skhep.math.geometry.distance(obj1, obj2)¶

Calculate 3D-distance between two geometry objects (points, lines, planes).

Examples

>>> line1  = ...
>>> line2  = ...
>>> point1 = ...
>>> point2 = ...
>>> plane1 = ...
>>> plane2 = ...
>>> for o1 in ( line1, line2 , point1 , point2 , plane1 , plane2 ) :
...    for o2 in ( line1, line2 , point1 , point2 , plane1 , plane2 ) :
...        print distance ( o1 , o2 )

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Related Topics

Mathematical utilities (skhep.math)¶

Mathematical functions relevant to kinematics¶

Vector classes¶

Geometry classes¶