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?

  • $\begingroup$ 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

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.


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).


  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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.