Questions tagged [curves]

For questions about or involving curves.

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Is there a function that is the envelope of the sum of ceilings of reciprocal functions

TL;DR: Given a sum of ceilings of reciprocal functions $$y_1 = T = \sum^{n-1}_i \Big\lceil \frac{p_i}{k} \Big\rceil$$ is there a corresponding form for a function that envelopes the $T$ on the left? ...
26 views

Characterization of arclength as unique function on continuous curves that satisfy certain conditions (resolution of "$\pi=4$ paradox")

I was again thinking about the famous $\pi=4$ paradox, and this question in particular: How to convince a layperson that the $\pi = 4$ proof is wrong?, about why the standard sup over polygonal ...
6 views

Singularity of the curves in affine form vs in projective form

I have just started learning about elliptic curves and I have this thought about curves in affine form and projective form. Apologies in advance if the question sounds silly, admittedly I am not ...
26 views

Does a linear homotopy between two simple closed curves cover the area between them?

Consider two closed simple curves parametrised by $\gamma_1,\gamma_2: [0,1]\rightarrow \mathbb{R}^2$ that are smooth functions. Both curves $\gamma_i$ encircle a bounded subset $A_i$ of $\mathbb{R}^2$,...
31 views

Smoothness of reparameterization

Suppose we have a parametric curve in $\mathbb{R^2}$, i.e. $\phi: [a, b] \subset \mathbb{R} \to \mathbb{R}^2$, $\phi(s) = (\phi_1(s), \phi_2(s))$ for $s \in [a, b]$. Suppose there is a different ...
40 views

Finding asymptotes for the given curve of two variables

I am looking for the asymptotes for the curve $x_1=\frac{C}{x_{1}^{2}-3x_1x_2+3x_{2}^{2}}$ where $C\in \mathbb{R}$ and $x_1$ and $x_2$ are the set variables. Oftentimes the asymptotes are found by ...
39 views

Intersection of a sphere and a cylinder

Take the Viviani curve intersection of a sphere and a cylinder. Analytically explain what happens to the intersection curve if you keep the radius of the sphere constant, fix one side of the cylinder ...
84 views

Why is it called an 'integral' curve?

The concept of a integral curve is relatively easy to understand as path through a vector field which is tangent to the field at each point. But why is it called an "integral" curve? It ...
56 views

in the sphere or disc,there is no essential simple closed curves?

How we can show in the sphere or disc,there is no essential simple closed curves ? In the mapping class groups By Benson farb , definition of essential closed curve is : a closed curve is called ...
57 views

Find the area inside the curve $r^2=2\cos(5\theta)$ and outside the unit circle.

I found the area of one full rose-petal($A_1$) and the area enclosed by the petal and the unit circle($A_2$), subtracted these from one another to get the area enclosed by the curve outside of the ...
61 views

How can I generate this wave with a formula? [closed]

I am not a mathematician, so please forgive my lack of appropriate mathematical terms! I'd like to generate the below curve procedurally. I had a go, but I couldn't quite get there and I feel like I'...
32 views

Curve Discussion with $f(x) = \frac{1}{x^2+r}$ and $r > 0$ with r as a constant. Need guidance

As stated, i have $$f(x) = \frac{1}{x^2+r}$$ with r being a constant and $r > 0$ I am familiar with curve discussion normally, but confused by the constant r. How do i properly calculate this, ...
25 views

Arc length parameter relation with fourth derivative

Let non-planar curve $\gamma:I \rightarrow \mathbb{R^{3}}$ with arc length parameter $s$. Find $a,\,b,\,c$ such that $\gamma^{(4)}(s)=a\gamma^\prime(s)+b\gamma^{\prime\prime}(s)+c\gamma^{(3)}(s)$. I ...
19 views

Find a tangent line of a given parabola perpendicular to another given line

The question from my textbook: Find the equation of the tangent line to the curve $y = x²-2x$ that is perpendicular to the line $x-2y = 1$ The derivative of the parabola: $2x-2$ It's normal slope of ...
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complex integral over a spiral

Good morning everyone, I am not sure how to solve the following integral, can anybody help me? $$\int_\gamma \frac{1}{z}dz$$ with $\gamma (t) = (t+1)e^{it}$ and $t\in [0, 2\pi]$ I split the curve in ...
41 views

curve between 0:0 and 1:1 [duplicate]

I want to describe a single smooth curve between 0:0 and 1:1, where a curvature c describes the curve. When c=1/2, the 'curve' is just a straight line (f(x)=x) ...
29 views

Computing the steepness of a curve on a surface

Given a curve $x(s),y(s)$ where $s$ is also the arc length. Let $u(x,y)$ be a function such that $u(x(s),y(s))$ forms a surface $D$. Now I want to compute the steepness $\theta$ of that curve on $D$ ...