More about Coordinate Systems

We will now look at how we can initialize new coordinate systems in sympy.vector, transformed in user-defined ways with respect to already-existing systems.

Locating new systems

We already know that the origin property of a CoordSys3D corresponds to the Point instance denoting its origin reference point.

Consider a coordinate system \(N\). Suppose we want to define a 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 perspective happen to be \((3, 4, 5)\). Moreover, this would also mean that the 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 respect to \(M\), and also vice versa, is equal to the identity matrix of dimensions 3x3. The locate_new method initializes a CoordSys3D that is only translated in space, not re-oriented, relative to the ‘parent’ system.

Orienting new systems

Similar to ‘locating’ new systems, sympy.vector also allows for initialization of new CoordSys3D instances that are oriented in 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 with respect 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 doing it.

>>> B = A.orient_new_axis('B', theta, A.k)

This initializes \(B\) with the required orientation information with respect to \(A\).

CoordSys3D provides the following direct orientation methods in its API-

orient_new_axis
orient_new_body
orient_new_space
orient_new_quaternion

Please look at the CoordSys3D class API given in the docs of this module, to know their functionality and required arguments in detail.

Using Orienter(s) and the orient_new method

You would first have to initialize an AxisOrienter instance for storing 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 multiple Orienter instances, provided in an iterable. The rotations/orientations are applied to the new system in the order the Orienter instances appear 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 Orienter classes for orientation purposes-

AxisOrienter
BodyOrienter
SpaceOrienter
QuaternionOrienter

Please refer to the API of the respective classes in the docs of this module to know more.

In each of the above examples, the origin of the new coordinate system coincides with the origin of the ‘parent’ system.

>>> B.position_wrt(A)
0

To compute the rotation matrix of any coordinate system with respect to 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 oriented in a pre-defined way, but also translated with respect to the parent?

Each of the orient_new_<method of orientation> methods, as well as the orient_new method, support a location keyword argument.

If a Vector is supplied as the value for this kwarg, the new system’s origin is automatically defined to be located at that position 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 use transformation parameter in CoordSys3D. Here we can define any transformation equations. If we are interested in some typical curvilinear coordinate system different that Cartesian, we can also use some predefined ones. It could be also possible to translate or rotate 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 works similarly 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 in different coordinate systems. In general, the same is true for scalar expressions and dyadic tensors.

sympy.vector supports the expression of vector/scalar quantities in 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 using express.

>>> 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 supports reexpression of scalars (general SymPy Expr) and Dyadic objects.

express also accepts a second coordinate system for 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 the re-expression of BaseVector instances. However, it does affect the way BaseScalar instances are expressed in different systems.

BaseScalar instances, are coordinate ‘symbols’ meant to denote the variables used in the definition of vector/scalar fields in sympy.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 the field would be \(a + b + c\). Now consider system \(R\), whose origin is located at \((1, 2, 3)\) with respect to \(N\) (no change 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 the variables keyword argument of express to True.

The above mentioned example, done programmatically, would look like this -

>>> 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 and express_coordinates method of Point also return different 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]])