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While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip using implicit functions?

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  • $\begingroup$ One obvious problem is how to incorporate the edge: that is, how to stop your surface going off to infinity, rather than covering only a finite region. $\endgroup$ – Chappers Apr 27 '15 at 15:44
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Chappers pointed out a problem with edges; those could be handled by imposing an inequality constraint. That is, it is conceivable to have an implicit equation for the Möbius strip of the form $$M=\{(x,y,z) : F(x,y,z) = 0, \ \text{some inequality with x,y,z}\}$$ Another issue is the non-orientability. If the gradient of $F$ did not vanish anywhere on $M$, it would define a nonvanishing normal field, contradicting the non-orientability. Even allowing $\nabla F$ to vanish in some places leaves a sticky issue: if we go off the surface by a small amount, $F$ will be either positive or negative... how to deal with transition from positive to negative? There are two ways:

  1. No transition: use $F$ such that $F\ge 0$ everywhere. For example, $F(x,y,z)$ could be the squared distance from $(x,y,z)$ to the set $M$. Then, by construction, the equation $F=0$ describes $M$. Drawback: there is no hope of getting an explicit form of $M$. This cheap, trivial, definition of $F$ does not yield any insights about $M$.

  2. Transition via self-intersection: the surface $\{F=0\}$ crosses itself, as on the picture below. Drawback: only a part of $\{F=0\}$ is a Möbius strip (the part highlighted in red); the intersecting piece is extraneous.

mobius

The blue surface above has a pretty simple implicit equation: $$y(x^2+y^2+z^2-1)-2z(x^2+y^2+x) =0 \tag{1}$$ One can check that the standard parametrized Möbius strip $$ \begin{split}x&=(1+s\cos(t/2))\cos t \\ y&=(1+s\cos(t/2))\sin t \\ z& = s\sin(t/2) \end{split} \tag{2}$$ is contained in (1); that is, plugging (2) into (1) yields $0=0$. But (1) does not exhaust all solutions of (2).

References:

  1. Wolfram Mathworld
  2. Encyclopedia of Analytical Surfaces by S. Krivoshapko and V.N. Ivanov; page 419.

The latter refers, concerning equation (2), to Willy Dedonder, "La surface de Moebius...", Industries et sciences. 1987; 63, no.2, p.2-8.

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