We will now look at how we can initialize new coordinate systems insympy.vector, transformed in user-definedways with respect to already-existing systems.
Locating new systems¶
We already know that the origin property of aCoordSys3D corresponds to the Point instancedenoting its origin reference point.
Consider a coordinate system \(N\). Suppose we want to definea new system \(M\), whose origin is located at\(\mathbf{3\hat{i} + 4\hat{j} + 5\hat{k}}\) from \(N\)’s origin.In other words, the coordinates of \(M\)’s origin from N’s perspectivehappen to be \((3, 4, 5)\). Moreover, this would also mean thatthe coordinates of \(N\)’s origin with respect to \(M\)would be \((-3, -4, -5)\).
This can be achieved programmatically as follows -
>>> from sympy.vector import CoordSys3D>>> N = CoordSys3D('N')>>> M = N.locate_new('M', 3*N.i + 4*N.j + 5*N.k)>>> M.position_wrt(N)3*N.i + 4*N.j + 5*N.k>>> N.origin.express_coordinates(M)(-3, -4, -5)
It is worth noting that \(M\)’s orientation is the same as that of\(N\). This means that the rotation matrix of :math: \(N\) with respectto \(M\), and also vice versa, is equal to the identity matrix ofdimensions 3x3.The locate_new method initializes a CoordSys3D thatis only translated in space, not re-oriented, relative to the ‘parent’system.
Orienting new systems¶
Similar to ‘locating’ new systems, sympy.vector also allows forinitialization of new CoordSys3D instances that are orientedin user-defined ways with respect to existing systems.
Suppose you have a coordinate system \(A\).
>>> from sympy.vector import CoordSys3D>>> A = CoordSys3D('A')
You want to initialize a new coordinate system \(B\), that is rotated withrespect to \(A\)’s Z-axis by an angle \(\theta\).
>>> from sympy import Symbol>>> theta = Symbol('theta')
The orientation is shown in the diagram below:
There are two ways to achieve this.
Using a method of CoordSys3D directly¶
This is the easiest, cleanest, and hence the recommended way of doingit.
>>> B = A.orient_new_axis('B', theta, A.k)
This initializes \(B\) with the required orientation information withrespect to \(A\).
CoordSys3D provides the following direct orientation methodsin its API-
orient_new_axisorient_new_bodyorient_new_spaceorient_new_quaternion
Please look at the CoordSys3D class API given in the docsof this module, to know their functionality and required argumentsin detail.
Using Orienter(s) and the orient_new method¶
You would first have to initialize an AxisOrienter instance forstoring the rotation information.
>>> from sympy.vector import AxisOrienter>>> axis_orienter = AxisOrienter(theta, A.k)
And then apply it using the orient_new method, to obtain \(B\).
>>> B = A.orient_new('B', axis_orienter)
orient_new also lets you orient new systems using multipleOrienter instances, provided in an iterable. The rotations/orientationsare applied to the new system in the order the Orienter instancesappear in the iterable.
>>> from sympy.vector import BodyOrienter>>> from sympy.abc import a, b, c>>> body_orienter = BodyOrienter(a, b, c, 'XYZ')>>> C = A.orient_new('C', (axis_orienter, body_orienter))
The sympy.vector API provides the following four Orienterclasses for orientation purposes:
AxisOrienterBodyOrienterSpaceOrienterQuaternionOrienter
Please refer to the API of the respective classes in the docs of thismodule to know more.
In each of the above examples, the origin of the new coordinate systemcoincides with the origin of the ‘parent’ system.
>>> B.position_wrt(A)0
To compute the rotation matrix of any coordinate system with respectto another one, use the rotation_matrix method.
>>> B = A.orient_new_axis('B', a, A.k)>>> B.rotation_matrix(A)Matrix([[ cos(a), sin(a), 0],[-sin(a), cos(a), 0],[ 0, 0, 1]])>>> B.rotation_matrix(B)Matrix([[1, 0, 0],[0, 1, 0],[0, 0, 1]])
Orienting AND Locating new systems¶
What if you want to initialize a new system that is not only orientedin a pre-defined way, but also translated with respect to the parent?
Each of the orient_new_<method of orientation> methods, as wellas the orient_new method, support a location keywordargument.
If a Vector is supplied as the value for this kwarg, thenew system’s origin is automatically defined to be located at thatposition vector with respect to the parent coordinate system.
Thus, the orientation methods also act as methods to support orientation+location of the new systems.
>>> C = A.orient_new_axis('C', a, A.k, location=2*A.j)>>> C.position_wrt(A)2*A.j>>> from sympy.vector import express>>> express(A.position_wrt(C), C)(-2*sin(a))*C.i + (-2*cos(a))*C.j
More on the express function in a bit.
Transforming new system¶
The most general way of creating user-defined system is to usetransformation parameter in CoordSys3D. Here we can defineany transformation equations. If we are interested in some typicalcurvilinear coordinate system different that Cartesian, we can alsouse some predefined ones. It could be also possible to translate orrotate system by setting appropriate transformation equations.
>>> from sympy.vector import CoordSys3D>>> from sympy import sin, cos>>> A = CoordSys3D('A', transformation='spherical')>>> B = CoordSys3D('A', transformation=lambda x,y,z: (x*sin(y), x*cos(y), z))
In CoordSys3D is also dedicated method, create_new which workssimilarly to methods like locate_new, orient_new_axis etc.
>>> from sympy.vector import CoordSys3D>>> A = CoordSys3D('A')>>> B = A.create_new('B', transformation='spherical')
Expression of quantities in different coordinate systems¶
Vectors and Dyadics¶
As mentioned earlier, the same vector attains different expressions indifferent coordinate systems. In general, the same is true for scalarexpressions and dyadic tensors.
sympy.vector supports the expression of vector/scalar quantitiesin different coordinate systems using the express function.
For purposes of this section, assume the following initializations:
>>> from sympy.vector import CoordSys3D, express>>> from sympy.abc import a, b, c>>> N = CoordSys3D('N')>>> M = N.orient_new_axis('M', a, N.k)
Vector instances can be expressed in user defined systems usingexpress.
>>> v1 = N.i + N.j + N.k>>> express(v1, M)(sin(a) + cos(a))*M.i + (-sin(a) + cos(a))*M.j + M.k>>> v2 = N.i + M.j>>> express(v2, N)(1 - sin(a))*N.i + (cos(a))*N.j
Apart from Vector instances, express also supportsreexpression of scalars (general SymPy Expr) andDyadic objects.
express also accepts a second coordinate systemfor re-expressing Dyadic instances.
>>> d = 2*(M.i | N.j) + 3* (M.j | N.k)>>> express(d, M)(2*sin(a))*(M.i|M.i) + (2*cos(a))*(M.i|M.j) + 3*(M.j|M.k)>>> express(d, M, N)2*(M.i|N.j) + 3*(M.j|N.k)
Coordinate Variables¶
The location of a coordinate system’s origin does not affect there-expression of BaseVector instances. However, it does affectthe way BaseScalar instances are expressed in different systems.
BaseScalar instances, are coordinate ‘symbols’ meant to denote thevariables used in the definition of vector/scalar fields insympy.vector.
For example, consider the scalar field\(\mathbf{{T}_{N}(x, y, z) = x + y + z}\) defined in system \(N\).Thus, at a point with coordinates \((a, b, c)\), the value of thefield would be \(a + b + c\). Now consider system \(R\), whoseorigin is located at \((1, 2, 3)\) with respect to \(N\) (nochange of orientation).A point with coordinates \((a, b, c)\) in \(R\) has coordinates\((a + 1, b + 2, c + 3)\) in \(N\).Therefore, the expression for \(\mathbf{{T}_{N}}\) in \(R\) becomes\(\mathbf{{T}_{R}}(x, y, z) = x + y + z + 6\).
Coordinate variables, if present in a vector/scalar/dyadic expression,can also be re-expressed in a given coordinate system, by setting thevariables keyword argument of express to True.
The above mentioned example, done programmatically, would look likethis -
>>> R = N.locate_new('R', N.i + 2*N.j + 3*N.k)>>> T_N = N.x + N.y + N.z>>> express(T_N, R, variables=True)R.x + R.y + R.z + 6
Other expression-dependent methods¶
The to_matrix method of Vector andexpress_coordinates method of Point also returndifferent results depending on the coordinate system being provided.
>>> P = R.origin.locate_new('P', a*R.i + b*R.j + c*R.k)>>> P.express_coordinates(N)(a + 1, b + 2, c + 3)>>> P.express_coordinates(R)(a, b, c)>>> v = N.i + N.j + N.k>>> v.to_matrix(M)Matrix([[ sin(a) + cos(a)],[-sin(a) + cos(a)],[ 1]])>>> v.to_matrix(N)Matrix([[1],[1],[1]])
