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36 votes
Accepted

Cardioid-looking curve, does it have a name?

The name of the curve is cochleoid (= shell-shaped rather than cardioid = heart-shaped). I compare the two below (gold = cochleoid, blue = cardioid). The distinction shell/heart refers to the ...
Carlo Beenakker's user avatar
33 votes

Cardioid-looking curve, does it have a name?

Not an answer - just want to note that the curve has more hidden branches. They can be seen looking at the parametric equation $$ x=\frac{\sin(\varphi)}\varphi,\quad y=\frac{1-\cos(\varphi)}\varphi $$ ...
19 votes

A necessary and sufficient condition for a space curve to lie on a ellipsoid

There is a straightforward way to deduce necessary conditions for a space curve to lie on an ellipsoid, and it's really a matter of calculation to make these conditions explicit in terms of the ...
Robert Bryant's user avatar
14 votes
Accepted

A regular, geometrically reduced but non-smooth curve

I believe a classic example is the curve define in $\mathbb P^2_{\mathbb F_p(t)}$, with coordinates $(x:y:z)$, by the equation $$ t x^p + z^{p-1} y + y^p=0$$ for $p>2$. Differentiating with respect ...
Will Sawin's user avatar
  • 138k
12 votes
Accepted

There exists differentiable curves arbitrarily close to the continuous ones

It turns out that something much more general is true and can be found in the literature. Theorem [Thm 3.3, Hirsch, Differential Topology] Let $M$ and $N$ be $C^s$-manifolds (with boundary), $1\le s\...
Z. M's user avatar
  • 2,053
11 votes
Accepted

Connecting a compact subset by a simple curve

Not always. Let $K$ be a subset of an ambient space $V$ ($V=\mathbf{R}^2$ is fine, but doesn't matter) that is the closure of a discrete subset $D$, such that $K-D$ is homeomorphic to a segment. This ...
YCor's user avatar
  • 60.7k
11 votes
Accepted

Higher order inflection points

(The statements you quote are only true if you are working over a field of characteristic zero or $p > d$. I will continue to make that assumption) The formula is not $I(3)=3d(d-2)$ but rather $\...
Felipe Voloch's user avatar
10 votes
Accepted

A problem of four conics

This is a consequence of the Double Contact Theorem. If $S_1$, $S_2$, and $S_3$ are three conics having the property that there is a point $X$, not on any of the conics, lying on a common chord of ...
brainjam's user avatar
  • 660
9 votes
Accepted

Are any embeddings $[0,1]\to\mathbb{R}^3$ topologically equivalent?

As igorf pointed out in the comment, the answer to the first question is 'no'. A quick counterexample is by looking into the complement of Fox-Artin arc, which is not simply connected. See figure ...
Shijie Gu's user avatar
  • 2,056
9 votes

Difference in length of two dimensional concentric closed paths

if you mean that the outer bicyclist's position $f(t)$ is related to the inner bicyclist's position $\gamma(t)$ (where $\gamma$ is a natural parametrization) by $f(t)-\gamma(t)=Dn(t)$, where $n$ is an ...
Fedor Petrov's user avatar
8 votes
Accepted

Jordan curves admitting only acyclic inscriptions of squares

The recent work by Jason Cantarella, Elizabeth Denne and John McCleary implies (but does not prove) that the answer to all questions is negative, as I conjectured. Specifically, under certain general ...
David G. Stork's user avatar
8 votes

A necessary and sufficient condition for a space curve to lie on a ellipsoid

Robert describes the differential equations which one can write in terms of $\tau$ and $\kappa$, and the inherent limitations in this local approach. But maybe one can find more reasonable or useful ...
Mohammad Ghomi's user avatar
8 votes
Accepted

Can we foliate the space $\mathbb{R}^3$ with Frenet curves whose tangent and normal vectores span a given $2$ dimensional distribution?

The answer is 'no' in general for an arbitrary Riemannian metric $g$ on a $3$-manifold $M$ and $2$-plane field $D\subset TM$. I'll give the argument for the flat metric on $\mathbb{R}^3$ and leave ...
Robert Bryant's user avatar
8 votes
Accepted

Is this curve well known?

It is not clear what you mean by "known" but this system can be solved explicitly, in quadratures of elementary functions. Set $x'=u,\; y'=v,\; g(t)=a\sin t+b$. Then your system becomes $$u'=...
Alexandre Eremenko's user avatar
8 votes

Points on curves of genus 3

No. Note that $P\neq Q$ since $i$ is fixed-point free. Since $i^*K=K$, one would have $5P+3Q\equiv 3P+5Q$ (where $\equiv$ means linear equivalence), hence $2P\equiv 2Q$, so $P$ and $Q$ are Weierstrass ...
abx's user avatar
  • 37.4k
7 votes

A variation on four-vertex theorem

Yes, this is true. More generally, if a smooth closed strictly convex curve intersects some circle in $2n$ points, then it has at least $2n$ vertices. This is stated in Blaschke's book "Kreis und ...
Ivan Izmestiev's user avatar
7 votes

Generating Random Curves with Fixed Length and Endpoint Distance

Here are some random "scribbles," based on Bjørn Kjos-Hanssen's idea (but not following his specifications exactly), mixing with Izaak Meckler's comment: What you see is points of a random walk fit ...
Joseph O'Rourke's user avatar
7 votes

Surprising properties of closed planar curves

The Four Vertex Theorem, due to Mukhopadhyaya in 1909, states that a plane closed simple smooth curve with positive curvature has at least four vertices, where a vertex is a local maximum or minimum ...
Gerry Myerson's user avatar
7 votes

Does this property characterize straight lines in the plane?

This is not a complete answer, but just some extended comments: 1. Existence of an inflection point inside every circle Suppose that $\gamma$ is $C^2$ so that its curvature is everywhere well-defined....
Mohammad Ghomi's user avatar
7 votes

Does this property characterize straight lines in the plane?

Edit: the following argument is incomplete insufficient, since it doesn't prove that the direction of $S_x$ remains the same, or at least has vanishingly small perturbations, when $x$ goes to $\infty$....
Ilkka Törmä's user avatar
7 votes

Snake algorithm that minimizes straight lines

This problem is a special case of the quadratic traveling salesman problem in which the cost for traversing three consecutive nodes that have no turn is $1$ and the cost is $0$ otherwise. Because ...
RobPratt's user avatar
  • 5,179
6 votes

Axioms of length

I would suggest the following axioms. The length in a metric space $X$ is a function $\ell:c(X)\to[0,+\infty]$ defined on the family $c(X)$ of all connected compact subsets of $X$ that satisfies the ...
Taras Banakh's user avatar
6 votes

Jordan curves admitting only acyclic inscriptions of squares

I am not a member here and so could not provide a comment. The claim that it has recently been solved is inaccurate. Green and Lobb solve the $\textbf{smooth}$ version of the Rectangle Peg Problem. ...
Jack L.'s user avatar
  • 1,433
6 votes

Surprising properties of closed planar curves

Chakerian's Theorem (proved in this paper) that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L. (In this later paper Chakerian gave a simpler proof and ...
Ivan Meir's user avatar
  • 4,802
6 votes

Surprising properties of closed planar curves

One result that I initially found a bit surprising is Grayson's theorem. It's a little bit of a different flavor than the other examples but I think it's interesting and worth a mention. Given a ...
Gabe K's user avatar
  • 5,394
6 votes
Accepted

Grand tour of the special orthogonal group

[EDIT: Dan Asimov notified me that this construction is similar to a construction in his 1985 paper entitled "The Grand Tour: a Tool for Viewing Multidimensional Data". The construction in ...
Adam P. Goucher's user avatar
6 votes
Accepted

Is a simple closed curve always a free boundary arc?

The answer is positive. For any Jordan curve, there is a homeomorphism of the Riemann sphere, which sends this curve to a circle. Your statement immediately follows. One proof of this statement ...
Alexandre Eremenko's user avatar

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