# Questions tagged [curves]

For questions about or involving curves.

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### What shape is the waterfall start line on a track?

The most common shape of a 400m outdoor running track is called a stadium in mathematics and consists of a rectangle capped off by semicircular ends. The length of the straight and the radius of the ...
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### Elegant way to see that $(A\cos(t+\alpha), B\sin(t+\beta))$ always defines an ellipse?

Let $A,B>0$ and $\alpha,\beta\in\mathbb{R}$ and consider the curve $$\gamma(t)=(A\cos(t+\alpha), B\sin(t+\beta)).$$ Is there a nice way to see that it is always an ellipse centered at the origin? ...
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### Can 2 curves be orthogonal at one point but not at another [closed]

My teacher gave us a note saying : if 2 curves intersect orthogonally at one point then they are orthogonal at all points of intersection I do not think that is true. Can someone explain how it is ...
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### Is every geodesic constant-speed?

I am learning differential geometry with "Pressley, Elementary Differential Geometry". He establishes that (1) geodesics are curves of constant speed, and that (2) every meridian on a ...
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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 ...
1 vote
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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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1 vote
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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. ...
1 vote
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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 ...
1 vote
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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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### Finiteness of the intersection number [duplicate]

I am taking a course on Algebraic Curves following Gathmann and I am trying to solve exercise 2.7(b) which reads as follows: $F,G$ two curves with no common components through the origin, then every ...
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### Why is a crossing of 2 arcs defined like this here?

The below text defines a crossing for 2 arcs, where $a_ib_j$ is a Jordan arc of finite length terminating at 2 distinct points $a_i$ and $b_j$: Two arcs $a_i b_j, a_k b_l(i \neq k, j \neq l)$ are ...
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### Tangent developable is locally isometric to an open set in ℝ²

I'm trying to solve this problem about tangent developable from a differential geometry exam $\sf 1996\ Q3$: My working: First part $\ldots\ldots$deduce that $Σ$ is ruled (that is, each point of $Σ$ ...
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### Let C be the path in $\Bbb R^2$ from from $(0,0)$ to $(2,1)$ defined by the equation $x^4-6xy^3-4y^2=0$. Find $\int_C (10x^4-2xy^3)dx -3x^2y^2dy.$ [duplicate]

Let C be the path in $\Bbb R^2$ from from $(0,0)$ to $(2,1)$ defined by the equation $x^4-6xy^3-4y^2=0$. Find $\int_C (10x^4-2xy^3 ) dx -3x^2y^2dy$. I have no idea how to do this line integral. In our ...
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### Exercise about contact between a curve and a surface in $\mathbb{R}^3$

This excercise is taken from do Carmo, Differential geometry of curves and surfaces, section 3.3. A Curve $C$ and a Surface $S$ have contact of order $\ge n$ in a common point $p$ if there exists a ...
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### Arc length of a polar curve $r = -8\cos(t)$

If I am being asked to find the arc length of the polar curve $r = -8\cos(t)$ when I use the integral formula it gives me $16 \pi$. But since this polar curve represents a circle with radius 4, should ...
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### Differential Geometry of Curves and Surfaces from Riemannian Geometry

I'm a relativist. Hence, I have a working knowledge of Riemannian geometry, but I never really studied differential geometry of curves and surfaces. I know the traditional path is to start with curves ...
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### Confused on the result of Sequence of Geometric Transformations

Here is the question. Consider the parent function $f(x)=\frac{1}{x}$. Now do the following sequence of transformations. $1.$ Shift up by $4$ units. $2.$ Shift left by $2$ units. $3.$ Vertically ...
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### Lemniscate of Bernoulli using Watt's linkage

The lemniscate of Bernoulli is a curve which can be defined as all points $P$ with $\overline{PA} \cdot \overline{PB}=2c$ with two given points $A$ and $B$ at distance 2c (see wikipedia). One way to ...
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### Two injective paths $\gamma_1$ and $\gamma_2$ with the same curve can be expressed as $\gamma_1 = \gamma_2 \circ \alpha$ for some $\alpha$

My professor provided a proof to the following theorem that I am not quite able to understand. Any help would be greatly appreciated since I already spent a lot of time deciphering this (not very long)...
1 vote
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### Can you identify this curve generated by a rotating Reuleaux triangle?

The image below shows the locus of the centroid of a Reuleaux triangle as it rotates and revolves about a stationary one. The curve is shown as blue dots. I'm trying to determine if the curve has been ...
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### Pseudoeffectivness (or nefness) on a curve.

In a proof about varieties of higher dimension we do an induction, so we consider the case where the dimension of a variety is $n=1$. The thing is that we are supposing that some sivisor $D$ on our ...
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### Area of Cassini Oval

I am trying to find the area of the cassini oval, whose parametric and polar equations can be found here. To verify the result of area of Cassini Oval in the case $b>a$ written in Wolfram, I ...
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### Parameters behind non-symmetric Lissajous loop?

I'm trying to guess what kind of two signals can create this kind of Lissajous curve: However I can't figure out what are the parameters that break the symmetry of the curve. The relative phase ...
1 vote
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### Can we set the components of a parametrized equation to be vectors?

Assume I have a polynomial function $y=f(x)=ax^3+bx^2+cx+d$ for $a,b,c,d \in \mathbb{R}$. In order to investigate the curve, I parametrize the equation as such $(t,x(t),y(t))$. Now, If I were to ...
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### Folium of Descartes - what is this point P?

I came across this example in an old book. I have this question here. What is this point P, how is it defined? I did some calculations (implicit differentiation) and it seems to me it's the point ...
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I am preparing to teach 'cross-section of solid structures'. Currently I am studying the cross section of a cylinder. I take one dot $A$ on the top circle, and dot $B$ on the bottom circle. I think ...