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A defect localized at a boundary of an ordered medium. A spherical inclusion with degenerate planar anchoring in a nematic liquid crystal can have a bipolar texture with two surface boojums, unlike a bulk point-defect pair.

planar anchoring favors the nematic director tangent to a surface. A flat wall at $z=0$ permits uniform $\mathbf{n}=\hat{\mathbf{x}}$, as in the sketch. A rubbed surface can select one in-plane axis; degenerate planar anchoring selects only the tangent plane.

For the degenerate case, $f_s=(W/2)(\mathbf{n}\cdot \mathbf{\nu }{)}^2$ has minima at $\mathbf{n}\cdot \mathbf{\nu }=0$. A remote incompatible boundary produces elastic distortion. Tangency alone does not specify an azimuthal direction within the surface.

A sphere with degenerate planar anchoring can have a bipolar texture with two surface boojums at opposite poles along the far-field axis. Surface director lines remain tangent, curve between the polar cores and relax toward uniform alignment far away. The original cross-section shows both cores.

A globally regular tangent orientation field cannot cover a sphere; the associated surface winding indices sum to its Euler characteristic, two. These are surface defects, not two arbitrarily placed bulk hedgehogs. Finite anchoring allows deviations near the cores, while an azimuth-selective coating could favor a different texture.

With planar anchoring on the cylindrical side wall, uniform axial alignment $\mathbf{n}=\hat{\mathbf{z}}$ is a simple defect-free profile. The axial direction is tangent at every wall point, so it satisfies the lateral boundary everywhere without elastic distortion.

This assumes degenerate planar anchoring, or an azimuthal preference compatible with axial alignment. End constraints, selected circumferential anchoring or imposed twist can favor another state. planar anchoring alone does not require a circumferential vortex. The original axial-section sketch shows the uniform axial option.