Questions tagged [curves]
For questions about or involving curves.
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Optimal Value t for Subdivision of Cubic Bézier Curve and How to Calculate It
In Gabriel Suchowolski’s paper, “Quadratic bezier offsetting with selective subdivision”, he explains how the midpoint—or better said, a parameter $t$ of 0.5—is often not the optimal* point on a ...
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Resolution to the "Ladder Movers' Problem"?
A problem that has been discussed before on this site has recently resurfaced on X. Namely:
Two painters are carrying a 20-foot ladder, one at each end, along a garden path which begins and ends with ...
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How are these electronics PCB design constants calculated or decided on?
I've asked this in the Electronics Stack exchange with no real answers so maybe a mathematician could help! PCB trace width calculators like this one and many others like it always quote at the bottom ...
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Determining the significance of a curve's factors
Given the equation $x^2+x+1$ you could easily determine that $x^2$ will have the greatest overall impact on the curve--then $x$ and finally $1$. And this holds true for any coefficients present as the ...
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Convex combination of equidistant curves
Say we have three curves $\gamma, \delta, \varepsilon : \mathbb R \to \mathbb R^n$ such that the distances $\lVert \gamma(t) - \delta(t) \rVert$ and $\lVert \gamma(t) - \varepsilon(t) \rVert$ are ...
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Curves on a trousers space. [closed]
How does one go about defining curves on a trousers space? I want to define two curves evolving cyclically around a cylinder and then at some time let one of the curves evolve on the other cylinder. ...
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Genus of a smooth curve on blowup of $\mathbb{P}^2$ at some points
Everything here takes place over $\mathbb{C}$.
Let $p_1, \ldots, p_n$ denote distinct points on $\mathbb{P}^2$ and let $\pi: S\to \mathbb{P}^2$ denote the blowup of $\mathbb{P}^2$ at these points. Let ...
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$\langle v'',v\times v'\rangle $ is constant. Can you prove $v$ lies on a plane?
I tried to make the title short but of course there are additional hypothesis.
Let $v:I\to \mathbb{S}^2$ be a regular curve parametrized by arc-length. This is to say that the tangent $v'$ is also a ...
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Smoothing projective nodal curve, is the general fiber smooth?
Proposition 29.9 of Hartshorne's Deformation theory states the following:
A reduced curve Y in $\mathbb{P}^n$ with locally smoothable singularities and $H^1(Y,O_Y(1)) = 0$ is smoothable. In particular,...
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conductors of representations coming from jacobians of curves
Let $C$ be a curve defined over $\mathbb{Q}$, and we denote by $J:=Jac(C)$ its Jacobian.
For a prime $l$, we define by $V_l(J)=T_l(J)\otimes \mathbb{Q}_l$. There is a natural action of the absolute ...
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How to prove an Identity regarding the Norm of the Second Derivative of a Curve
Let $\gamma:A\rightarrow\mathbb{R}^{3}$ be a normal differentiable curve.
Suppose $\gamma(t)=\left(\begin{matrix}x(t)\\y(t)\\z(t)\end{matrix}\right)\in{C}^{1}(A)\;$. Suppose also that $\forall{t}\in{A}...
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What is a curve at $y=\infty$ mean?
On the wikipedia for the trident curve, $xy+ax^3+bx^2+cx=d$, two graphs are shown:
Both are for the case where $a=b=c=d=1$, with the first matching what I find in desmos, but the latter being the '...
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Derivation of torsion formula - do Carmo exercise 1.5.12
The problem: Let $\alpha \colon I \to \mathbb{R}^3$ be a regular parametrized curve
(not necessarily arc length) and let $\beta \colon
J \to \mathbb{R}^3$ be a reparametrization of $\alpha(I)$ by
the ...
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On the Fundamental Theorem of the Local Theory of Curves
The Fundamental Theorem of the Local Theory of Curves is often states as the following:
The last line of the theorem states "with positive determinant". What is the significance of this and ...
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Connect two points on an annulus
I want to prove that the annulus given by the set
$$
B(a;R_1,R_2) =\{{z \in \mathbb{C} : R_1 < |z-a| < R_2\}}
$$
is a connected space. As I am working in $\mathbb{C}$ it is enough to prove it to ...
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