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FALSE: the patches of a finite presentation can always be reoriented to make their normals agree on overlaps
Statement
False claim: given any finite patch presentation of a surface, one can reorient the patches so that their induced normals agree on every overlap.
Facts & Assumptions
Given: The two-patch Möbius-band presentation of The Mobius band presented by two regular patches, with normal comparison on the interiors of the overlap components.
On that presentation, the induced normals agree on one overlap component and are opposite on the other (The Mobius band presented by two regular patches, with normal comparison on the interiors of the overlap components).
In a compatible finite patch presentation, induced normals must agree at every overlap point coming from interior parameter points of both patches (Finitely patched regular surfaces, their area, scalar integrals, and flux).
Choosing rather than is an orientation (Unit normal fields, orientations, and flux through a regular surface patch).
A regular patch has no interior parameter point sharing its image with a distinct point of the parameter region (Regular parametrized surface patches on compact Jordan parameter regions).
A regular surface reparametrization is orientation-preserving when its parameter Jacobian determinant is positive and orientation-reversing when it is negative (Surface reparametrizations and their orientation sign).
The cross product is that of The cross product in .
Refutation
Reorienting a patch means replacing its induced normal by the opposite one. In coordinates, swapping the two parameters reverses the sign of the oriented area vector, so by [F2], [F4], and [F5] a reorientation changes nothing but the sign of the normal on that patch.
For two patches there are exactly four orientation choices, and whether the two normals agree at an overlap point depends only on the product of the two chosen signs.
By [L1] and [F1], the first overlap component of the Möbius presentation demands a positive sign product while the second demands a negative sign product. No one sign product can satisfy both.
Enumerating the four choices confirms it: the two like-sign choices preserve agreement on the first overlap and fail on the second, while the two mixed-sign choices do the opposite.
Therefore no reorientation of this finite patch presentation makes the normals agree on every overlap, so the claim is false. The compatibility clause is a genuine restriction and not a normalization.
Remarks
- The refutation is presentation-level, exactly as intended on this page. It does not claim that every presentation of every nonorientable surface fails in the same two-component way.
Depends on
- The Mobius band presented by two regular patches, with normal comparison on the interiors of the overlap components
- Finitely patched regular surfaces, their area, scalar integrals, and flux
- Unit normal fields, orientations, and flux through a regular surface patch
- Regular parametrized surface patches on compact Jordan parameter regions
- Surface reparametrizations and their orientation sign
- The cross product in $\mathbb R^3$
- The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- University of Toronto MAT237 notes, Section 5.3 (standard reference, not scraped)
- M. E. Taylor, Introduction to Analysis in Several Variables, Section 3.2 (standard reference, not scraped)