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 ...

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
$$
x=\frac{\sin(\varphi)}\varphi,\quad y=\frac{1-\cos(\varphi)}\varphi
$$
...

Community wiki

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 ...

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 ...

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\...

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 ...

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 $\...

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
...

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 ...

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 ...

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 ...

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 ...

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 ...

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'=...

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 ...

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 ...

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 ...

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 ...

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....

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 $\infty$....

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 ...

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 ...

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. ...

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 ...

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 ...

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 ...

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 ...

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