Questions tagged [curves]

For questions about or involving curves.

649 questions
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Reparameterisation of Curve as a Regular Curve (Topology)

There is a result that a curve or topological path can be reparameterized as a regular curve contained in the paper "Reparametrizations of continuous paths - Ulrich Fahrenberg and Martin Raussen" ...
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Orthogonal monomials on a curve

Let $\Gamma \subset \mathbb{C}$ be a smooth curve such that monomials are orthogonal on it, i.e. with $n,m \in \mathbb{N} \cup\{0\}$ \int_{\Gamma} z^n \overline{z^m} |dz| = 0, \qquad \qquad \forall ...
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What is the definition of ''geometrically irreducible closed curve''?

In the algebraice geometry, one says about "geometrically irreducible closed curve" over field $k$. For example, the theorem 5.4.5 (pp. 147) of ''Heights in Diophantine Geometry'' of E. Bombieri wrote ...
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How to prove that Frenet frame is independent of the choices of parameters?

When I am reading ''A course in differential geometry'' of Klingenberg, I cannot be sure the Frenet frame defined in this book is independent of the choice of parameter of a curve. As a result, the ...
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Inscribed Rectangle Proof (Basic Question)

Here's a video on the problem that I am referencing (it's well worth the watch either way) https://youtu.be/AmgkSdhK4K8 The problem asks whether you can find an inscribed rectangle on any Jordan ...
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Osculating circle at curvature minimum point of simple closed curve encloses the curve

The story begins with seemingly unrelating situation. I was trying to find out an elementary solution of the following problem. A circle is called a separator for a set of five points in a plane if ...
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Closed curve $\gamma$ such that least area of a surface with boundary $2\gamma$
What does the following statement mean? Any geometric intuition would be very helpful. L.C. Young constructed a closed curve $\gamma$ in the Euclidean space $\mathbb{R}^4$ such that the least area ...