# Questions tagged [curves]

For questions about or involving curves.

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### Are the tangent lines to a helix disjoint?

Let $H$ be the helix parametrized by $t \mapsto (\cos(t), \sin(t), t) : \mathbb{R} \to \mathbb{R}^3$. It seems like any two distinct tangent lines to $H$ will be disjoint – in fact I thought this was ...
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### How do I find the intersection of an involute gear's involute face curves and trochoid root curves?

This question is very similar to another question on this site, but I can't figure out how to use the answer to that question to solve my problem. I am working on parametrically generating involute ...
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### Matrix Representation for Clamped b-spline

I wanted to know how can I write a matrix representation of a clamped b-spline. Following this document, https://ieeexplore.ieee.org/document/731996, I get matrix representation to write b-spline ...
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### Determining how well a curve approximates another curve

Let's say that I have a function $y = f_1(x)$. I have two other functions, $y = f_2(x)$ and $y = f_3(x)$, that I want to compare with the first function. Specifically, I want to see which one of $f_2$ ...
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### Cauchy integration theorem proof

My textbook reports the following result: Given $f\colon A \subseteq \mathbb{C} \to \mathbb{C}$ holomorphic in the open simply connected set $A$, and given $\gamma\colon [a,b] \to A$ a simple closed ...
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### Find a vector function that represents the curve of intersection of two surfaces and the tangent vector [duplicate]

Consider the curve $C$ obtained by intersecting the surfaces defined by $$x^2+y^2 + z^2=3$$ and $$x^2-y^2+z^2 =1$$ At the point $(1,1,1)$, which of the following is a tangent vector to the curve? The ...
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### Find $a,b$ for which the given curve lies on a plane

Given the curve defined as: $$\mathbf{r}(t)\,=\,{\begin{pmatrix}2+\sqrt{2}\cos\left(t\right)\\1-\sin\left(t\right)\\3+\sin\left(t\right)\end{pmatrix}}$$ Does there exist real $a,b$ ...
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### Find the length of the given curve

Given a curve defined as: $$\mathbf{r}(t)\,=\,{\begin{pmatrix}\left(1-\cos\left(t\right)\right)\cos\left(t\right)\\\left(1-\cos\left(t\right)\right)\sin\left(t\right)\end{pmatrix}}$$ ...
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### Find area of this bounded curve. I don't know how to start? This question was asked in Mathematical belarusian olympiad. [closed]

Find the area bounded by the curve: $$\left(\frac x2 + \frac y3\right)^4= 4xy$$ Tried using subtitutions but couldn't figure it out.
### Finding curvature for $y = \sin ( -2x )$ at $x =\pi/4$
The answer is 4, but I got 1. I said $r(t)=<t,\sin(-2t)>$ and the $|r^\prime(\frac{\pi}{4})|$ is equal to 1. I also got 1 for $T^\prime(\frac{\pi}{4})$ but none of this information matters ...