Generic 2D, 3D and 4D vectors classes and their transformations (rotations).
GenVector, is a new package intended to represent vectors and their operations and transformations, such as rotations and Lorentz transformations, in 2, 3 and 4 dimensions. The 2D and 3D space are used to describe the geometry vectors and points, while the 4D space-time is used for physics vectors representing relativistic particles. These 2D,3D and 4D vectors are different from vectors of the Linear Algebra package which describe generic N-dimensional vectors. Similar functionality is currently provided by the CLHEP Vector and Geometry packages and the ROOT Physics Vector classes (TVector2, TVector3 and TLorentzVector). It is also re-uses concepts and ideas from the CMS Common Vector package.
In contrast to CLHEP or the ROOT physics libraries, GenVector provides class templates for modelling the vectors. There is a user-controlled freedom on how the vector is internally represented. This is expressed by a choice of coordinate system which is supplied as a template parameter when the vector is constructed. Furthermore each coordinate system is itself a template, so that the user can specify the underlying scalar type. In more detail, the main characteristics of GenVector are:
Optimal runtime performances
We try to minimize any overhead in the run-time performances. We have deliberately avoided to have any virtual function and even virtual destructors in the classes and have inlined as much as possible all the functions. For this reason, we have chosen to use template classes to implement the GenVector concepts instead of abstract or base classes and virtual functions.
Points and Vector concept
Mathematically vectors and points are two distinct concepts. They have different transformations, like vectors only rotate while points rotate and translate. You can add two vectors but not two points and the difference between two points is a vector. We then distinguish for the 2 and 3 dimensional case, between points and vectors, modeling them with different classes:
Generic Coordinate System
The vector classes are based on a generic type of coordinate system, expressed as a template parameter of the class. Various classes exist to describe the various coordinates systems:
The angle theta is defined between [0, \(\pi\)] and phi between [- \(\pi\), \(\pi\)]. The angles are expressed in radians.
Users can define the Vectors according to the coordinate type which is most efficient for their use. Transformations between the various coordinate systems are available through copy constructors or the assignment (=) operator. For maximum flexibility and minimize in some use case memory allocation, the coordinate system classes are templated on the scalar type. To avoid exposing templated parameter to the users, typedefs are defined for all types of vectors based an double's. See the 3D Vector, 3D Point, 2D Vector and Point, and LorentzVector classes for all the possible types of vector classes which can be constructed by the user with the available coordinate system types.
Coordinate System Tag
The 2D and 3D points and vector classes can be associated to a tag defining the coordinate system. This can be used to distinguish between vectors of different coordinate systems like global or local vectors. The coordinate system tag is a template parameter of the ROOT::Math::DisplacementVector3D (and ROOT::Math::DisplacementVector2D) and ROOT::Math::PositionVector3D (and ROOT::Math::PositionVector2D) classes. A default tag, ROOT::Math::DefaultCoordinateSystemTag, exists for users who don't need this functionality.
Transformations
The transformations are modeled using simple (non-template) classes, using double as the scalar type to avoid too large numerical errors. The transformations are grouped in Rotations (in 3 dimensions), Lorentz transformations and Poincaré transformations, which are Translation/Rotation combinations. Each group has several members which may model physically equivalent trasformations but with different internal representations. Transformation classes can operate on all type of vectors using the operator() or the operator * and the transformations can also be combined via the operator *. In more detail the transformations available are:
3D Transformation
We describe the transformations defined as a composition between a rotation and a translation using the class ROOT::Math::Transform3D. It is important to note that transformations act differently on Vectors and Points. The Vectors only rotate, therefore when applying a transformation (rotation + translation) on a Vector, only the rotation operates while the translation has no effect. The interface for Transformations is similar to the one used in the CLHEP Geometry package (class Transform3D). A class, ROOT::Math::Translation3D. describe transformations consisting of only a translation. Translation can be applied only on Points, applying them on Vector objects has no effect. The Translation3D class can be combined with both ROOT::Math::Rotation3D and ROOT::Math::Transform3D using the operator * to obtain a new transformation as an instance of a Transform3D class.
Other main characteristics of the GenVector classes are:
Minimal Vector classes interface
We have tried to keep the interface to a minimal level:
Naming Convention
As part of ROOT, the GenVector package adheres to the prescribed ROOT naming convention, with some (approved) exceptions, as described here:
Connection to Linear Algebra package
In some use cases, like in track reconstruction, it is needed to use the content of the vector and rotation classes in conjunction with linear algebra operations. We prefer to avoid any direct dependency to any Linear algebra package. However, we provide some hooks to convert to and from Linear Algebra classes.
This GenVector package is part of the index and it can be built as an independent package. A tar file can be downloaded from here.
A more detailed description of all the GenVector classes is available in this document.
Namespaces | |
namespace | ROOT::Math::VectorUtil |
Global Helper functions for generic Vector classes. | |
Classes | |
class | ROOT::Math::AxisAngle |
AxisAngle class describing rotation represented with direction axis (3D Vector) and an angle of rotation around that axis. More... | |
class | ROOT::Math::Boost |
Lorentz boost class with the (4D) transformation represented internally by a 4x4 orthosymplectic matrix. More... | |
class | ROOT::Math::BoostX |
Class representing a Lorentz Boost along the X axis, by beta. More... | |
class | ROOT::Math::BoostY |
Class representing a Lorentz Boost along the Y axis, by beta. More... | |
class | ROOT::Math::BoostZ |
Class representing a Lorentz Boost along the Z axis, by beta. More... | |
class | ROOT::Math::Cartesian2D< T > |
Class describing a 2D cartesian coordinate system (x, y coordinates) More... | |
class | ROOT::Math::Cartesian3D< T > |
Class describing a 3D cartesian coordinate system (x, y, z coordinates) More... | |
class | ROOT::Math::Cylindrical3D< T > |
Class describing a cylindrical coordinate system based on rho, z and phi. More... | |
class | ROOT::Math::CylindricalEta3D< T > |
Class describing a cylindrical coordinate system based on eta (pseudorapidity) instead of z. More... | |
class | ROOT::Math::DefaultCoordinateSystemTag |
DefaultCoordinateSystemTag Default tag for identifying any coordinate system. More... | |
class | ROOT::Math::DisplacementVector2D< CoordSystem, Tag > |
Class describing a generic displacement vector in 2 dimensions. More... | |
class | ROOT::Math::DisplacementVector3D< CoordSystem, Tag > |
Class describing a generic displacement vector in 3 dimensions. More... | |
class | ROOT::Math::EulerAngles |
EulerAngles class describing rotation as three angles (Euler Angles). More... | |
class | ROOT::Math::GlobalCoordinateSystemTag |
Tag for identifying vectors based on a global coordinate system. More... | |
class | ROOT::Math::LocalCoordinateSystemTag |
Tag for identifying vectors based on a local coordinate system. More... | |
class | ROOT::Math::LorentzRotation |
Lorentz transformation class with the (4D) transformation represented by a 4x4 orthosymplectic matrix. More... | |
class | ROOT::Math::LorentzVector< CoordSystem > |
Class describing a generic LorentzVector in the 4D space-time, using the specified coordinate system for the spatial vector part. More... | |
class | ROOT::Math::Impl::Plane3D< T > |
Class describing a geometrical plane in 3 dimensions. More... | |
class | ROOT::Math::Polar2D< T > |
Class describing a polar 2D coordinate system based on r and phi Phi is restricted to be in the range [-PI,PI) More... | |
class | ROOT::Math::Polar3D< T > |
Class describing a polar coordinate system based on r, theta and phi Phi is restricted to be in the range [-PI,PI) More... | |
class | ROOT::Math::PositionVector2D< CoordSystem, Tag > |
Class describing a generic position vector (point) in 2 dimensions. More... | |
class | ROOT::Math::PositionVector3D< CoordSystem, Tag > |
Class describing a generic position vector (point) in 3 dimensions. More... | |
class | ROOT::Math::PtEtaPhiE4D< ScalarType > |
Class describing a 4D cylindrical coordinate system using Pt , Phi, Eta and E (or rho, phi, eta , T) The metric used is (-,-,-,+). More... | |
class | ROOT::Math::PtEtaPhiM4D< ScalarType > |
Class describing a 4D cylindrical coordinate system using Pt , Phi, Eta and M (mass) The metric used is (-,-,-,+). More... | |
class | ROOT::Math::PxPyPzE4D< ScalarType > |
Class describing a 4D cartesian coordinate system (x, y, z, t coordinates) or momentum-energy vectors stored as (Px, Py, Pz, E). More... | |
class | ROOT::Math::PxPyPzM4D< ScalarType > |
Class describing a 4D coordinate system or momentum-energy vectors stored as (Px, Py, Pz, M). More... | |
class | ROOT::Math::Quaternion |
Rotation class with the (3D) rotation represented by a unit quaternion (u, i, j, k). More... | |
class | ROOT::Math::Rotation3D |
Rotation class with the (3D) rotation represented by a 3x3 orthogonal matrix. More... | |
class | ROOT::Math::RotationX |
Rotation class representing a 3D rotation about the X axis by the angle of rotation. More... | |
class | ROOT::Math::RotationY |
Rotation class representing a 3D rotation about the Y axis by the angle of rotation. More... | |
class | ROOT::Math::RotationZ |
Rotation class representing a 3D rotation about the Z axis by the angle of rotation. More... | |
class | ROOT::Math::RotationZYX |
Rotation class with the (3D) rotation represented by angles describing first a rotation of an angle phi (yaw) about the Z axis, followed by a rotation of an angle theta (pitch) about the Y axis, followed by a third rotation of an angle psi (roll) about the X axis. More... | |
class | ROOT::Math::Impl::Transform3D< T > |
Basic 3D Transformation class describing a rotation and then a translation The internal data are a 3D rotation data (represented as a 3x3 matrix) and a 3D vector data. More... | |
class | ROOT::Math::Impl::Translation3D< T > |
Class describing a 3 dimensional translation. More... | |