Complex shapes - The Triangle class

The field of a homogeneously charged magnet is, on the outside, the same as the field of a similarly shaped body with a magnetic surface charge. The surface charge is proportional to the projection of the magnetization vector onto the surface normal.

The Triangle class is set up so that it can easily be used for this purpose. Arbitrary surfaces are easily approximated by triangles, and given the magnetization vector, the surface charge density is automatically computed. One must be very careful to orient the triangles correctly, with surface normal vectors pointing outwards (right-hand-rule). The resulting H-field of such a collection is correct, but the B-field is only correct on the outside of the body. On the inside the magnetization must be added to the field.

Example: Triangular prisma magnet

Consider a Prisma with triangular base that is magnetized orthogonal to the base. All surface normals of the sides of the prisma are orthogonal to the magnetization vector. As a result the sides do not contribute to the magnetic field because their charge desity dissappears. Only top and bottom surfaces contribute. One must be very careful when defining those surfaces in such a way that the surface normals point outwards. For this purpose the surface normal of Triangle objects is graphically displayed by default. The following examples shows how the Triangle class can be used to create a prisma magnet object.

import magpylib as magpy

top = magpy.misc.Triangle(
    magnetization=(0,0,1000),
    vertices= ((-1,-1,1), (1,-1,1), (0,2,1)),
)
bott = magpy.misc.Triangle(
    magnetization=(0,0,1000),
    vertices= ((-1,-1,-1), (0,2,-1), (1,-1,-1)),
)

prisma = magpy.Collection(top, bott)

prisma.show(
    style_opacity=0.5,
    style_magnetization_size=0.2,
)

Example: Cuboctahedron magnet

More complex bodies are easy constructed from Triangle objects. The following code shows how a magnet with cuboctahedral shape can be constructed. Be aware that the B-field requires addition of the magnetization vector on the inside.

import magpylib as magpy

vertices = [
    ([0, 1, -1], [-1, 1, 0], [1, 1, 0]),
    ([0, 1, 1], [1, 1, 0], [-1, 1, 0]),
    ([0, 1, 1], [-1, 0, 1], [0, -1, 1]),
    ([0, 1, 1], [0, -1, 1], [1, 0, 1]),
    ([0, 1, -1], [1, 0, -1], [0, -1, -1]),
    ([0, 1, -1], [0, -1, -1], [-1, 0, -1]),
    ([0, -1, 1], [-1, -1, 0], [1, -1, 0]),
    ([0, -1, -1], [1, -1, 0], [-1, -1, 0]),
    ([-1, 1, 0], [-1, 0, -1], [-1, 0, 1]),
    ([-1, -1, 0], [-1, 0, 1], [-1, 0, -1]),
    ([1, 1, 0], [1, 0, 1], [1, 0, -1]),
    ([1, -1, 0], [1, 0, -1], [1, 0, 1]),
    ([0, 1, 1], [-1, 1, 0], [-1, 0, 1]),
    ([0, 1, 1], [1, 0, 1], [1, 1, 0]),
    ([0, 1, -1], [-1, 0, -1], [-1, 1, 0]),
    ([0, 1, -1], [1, 1, 0], [1, 0, -1]),
    ([0, -1, -1], [-1, -1, 0], [-1, 0, -1]),
    ([0, -1, -1], [1, 0, -1], [1, -1, 0]),
    ([0, -1, 1], [-1, 0, 1], [-1, -1, 0]),
    ([0, -1, 1], [1, -1, 0], [1, 0, 1]),
]

cuboc = magpy.Collection(style_label="Cuboctahedron")
for ind, vert in enumerate(vertices):
    cuboc.add(
        magpy.misc.Triangle(
            magnetization=(100, 200, 300),
            vertices=vert,
            style_label=f"Triangle_{ind+1:02d}",
        )
    )

magpy.show(
    *cuboc,
    backend="pyvista",
    style_orientation_size=2,
    style_orientation_color='yellow',
    style_orientation_symbol='cone',
    style_magnetization_mode="arrow",
    jupyter_backend="panel", # better pyvista rendering in a jupyter notebook
)