Questions tagged [curves]

For questions about or involving curves.

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2
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1answer
34 views

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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0answers
29 views

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 ...
0
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0answers
22 views

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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1answer
34 views

Proving that a given curve is simple ( injective )

I have been reading Vector calculus by Peter Baxandall. It defines : A curve $C$ in $\Bbb R^n$ is a simple arc if $C$ has a $1-1~~ C^1$ parametrization of the form $f:[a,b] \subseteq \Bbb R \...
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0answers
17 views

Using JS (or PHP) draw curve based on $f(x,y) = 0$

If I have $y = f(x)$, I know how to do it. Basically, just let $x$ in a range, and calculate $y$ basing on $x$. After having $(x,y)$, draw the dot. BUT I only have $f(x, y) = 0$, and it's NOT easy to ...
1
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1answer
30 views

Prove: For a smooth curve $C$ parameterized by $r(s)$ where $s$ is arc length,$r′(s)$ satisfies $|r′(s)| = 1$.

Prove: For a smooth curve $C$ parameterized by $r(s)$ where $s$ is arc length,$r′(s)$ satisfies $|r′(s)| = 1$. I understand that $|r'(s)|$ should equals to one because it's the magnitude of change in ...
0
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1answer
23 views

How can I find the formula for a cycloid with a given speed?

I know that for a cycloid with radius $R$ and time $t$, it can be defined as $x = R(t - \sin t)$ and $y = R(1 - \cos t)$. However what if it's not at unit speed and we have a speed $z$/sec such that ...
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0answers
20 views

Sketch the vector field on R2: X(p) = (p, X(p)) where X(p) = -p [closed]

Sketch the vector field on R2: X(p) = (p, X(p)) where X(p) = -p
0
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1answer
38 views

Why do circles (of the same sized circumference) not tesselate?

If our test for whether a regular polygon can tesselate with itself is whether the degrees of an individual interior angle can divide 360 to yield an integer (some examples of these integers are 6 [...
8
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2answers
123 views
+50

Epitrochoids and adjacent loop touching

Consider the pair of parametric equations which describe a (simplified) epitrochoid: \begin{align} x(t) &= \cos (t) - a \cos (\alpha t)\\ y(t) &= \sin (t) - a \sin (\alpha t). \end{align} Here ...
2
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2answers
85 views

Finding the asymptotes of the polar curve $ r=\frac{ 5\cos^2 \theta+3 }{ 5\cos^2 \theta-1 }$

Find all asymptotes ( their equations in polar or cartesian form) for the polar curve $$ r(\theta)=\dfrac{ 5\cos^2 \theta+3 }{ 5\cos^2 \theta-1 }$$ When denominator goes to zero we have one set, but ...
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0answers
18 views

tame ramification and separability of the reduction

Let $K$ a local field, with ring of integer $\mathcal{O}_K$ or $\mathcal{O}$ ,of uniformizer $\pi$, residue field $k$ of caracteristic $p$. Let $\varphi:\mathbb{P}^1_K\to\mathbb{P}^1_K$ finite ...
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2answers
33 views

Area Under a Curve - Right points

I'm really conflicted about whether I did this correctly. Feedback would be appreciated! 2nd Attempt: I'm still really confused because I've tried so many variations of this problem and still haven't ...
2
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4answers
70 views

Proving the existence of minimum distance between two curves

Consider the following subsets of the plane:$$C_1=\Big\{(x,y)~:~x>0~,~y=\frac1x\Big\} $$and$$C_2=\Big\{(x,y)~:~x<0~,~y=-1+\frac1x\Big\}$$Given any two points $P=(x,y)$ and $Q=(u,v)$ of the plane,...
1
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1answer
18 views

Textbooks with non-trivial $2D$ Cartesian curve sketching exercises

I'm looking for textbooks that have a decent amount of curve sketching exercises. I have been disappointed with how much this topic has been skipped over lightly by most textbooks, often summarised in ...
3
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1answer
60 views

Reparametrization theorem

The reparametrization theorem says the following: If $α:I\to\mathbb{R}^n$ is a regular curve in $\mathbb{R}^n$, then there exists a reparametrization $\beta$ of $\alpha$ such that $β$ has unit speed....
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3answers
65 views

Find $a,b,c$ and intersection of $y = ax^2 + bx + c$ with $x$-axis

Known: max at $x = 1, y = 16$, intersect $x$-axis with the length of the section connecting the two points is $8$. What I tried is this: $f'(x) = 2ax + b$, max at $x = 1, so -b = 2a, y = 16 \...
0
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1answer
24 views

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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0answers
49 views

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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2answers
61 views

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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2answers
44 views

Find $a,b$ for which the given curve lies on a plane

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

Find the length of the given curve

Given a curve defined as: $${\displaystyle \mathbf{r}(t)\,=\,{\begin{pmatrix}\left(1-\cos\left(t\right)\right)\cos\left(t\right)\\\left(1-\cos\left(t\right)\right)\sin\left(t\right)\end{pmatrix}}}$$ ...
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2answers
100 views

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.
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3answers
54 views

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 ...
0
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1answer
34 views

Finding different phase shifts

Find the two different phase shifts that translate the cosine curve onto the sine curve. Write an equation for both new cosine curves and each phase shift. I am not sure about the answer. Thank you! $...
1
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2answers
18 views

Show $C(t)=(2\cos^3(t),\sin(2t),2\sin(t))$ is on the intersection of $x^2+y^2+z^2=4$ and $x^2+y^2=2x$

Given a curve defined as: $$C(t)=(2\cos^2(t),\sin(2t),2\sin(t))$$ for $0\le t\le2\pi$ Show that the curve is on the intersection of the sphere $x^2+y^2+z^2=4$ and $x^2+y^2=2x$. The intersection of ...
0
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1answer
43 views

Convert cubic spline to Bézier curve and get control points

There is a cubic spline represented by the standard equation: $$ f(x) = a + b (x - x_0) + c (x - x_0)^2 + d (x - x_0)^3 $$ and 2 endpoints: $P_0~ [x, y]$ - starting point $P_1~ [x, y]$ - end ...
1
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8answers
65 views

Show that $r(t)=(3t^2-9t,3t-3t^2,2t^2-3t+5)$ is on a plane

Given a curve defined by :$$r(t)=(3t^2-9t,3t-3t^2,2t^2-3t+5)$$ Show that the curve is flat (it lies on a plane) and then find the equation of th eplane. If the curve lies in a plane,then : $$A(3t^2-...
1
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2answers
57 views

What is the Lebesgue measure of the Koch and the Minkowski curves?

Are the Koch curve and the Minkowski curve Lebesgue measurable? (I believe they are.) If so, what are their measures? (Intuitively, it would seem to be zero.) I unfortunately can't seem to find much ...
-3
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0answers
29 views

Finding the radius of curvature of the trajectory of a projectile.

The parabolic trajectory of a projectile has different radius of curvature at different points of time. Is there a way to find R of C for a simple projectile, thrown at an angle θ and initial ...
0
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0answers
19 views

Fitting a curve using gsl interpolation with smooth first derivative

I want to compute an analytic representation of a graph passing through a given dataset. See the image below for an example: I used GSL cubic spline interpolation (https://www.gnu.org/software/gsl/...
0
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3answers
67 views

What is the surface $25x^2 + 75y^2 + z^2 = 0$?

Find the intersection curves of the surfaces $$25x^2 + 25y^2 - z^2 = 25$$ and $$25x^2 + 75y^2 +z^2 = 0\,.$$ I am fairly certain the first surface is a hyperboloid of one sheet, but I cannot figure ...
2
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0answers
38 views

Torsion As The Rate Of Change Of An Angle

Just as we define the signed curvature of a plane curve as the rate of change of the angle through which a constant vector must be rotated to bring it into coincidence with the tangent vector, is it ...
1
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0answers
42 views

Find the equation of the curve that passes thought points

I need to find the equations of the curves that passes thought (x,y) and (x,z), where x,y and z are: ...
0
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1answer
59 views

Blue tooth's curve and integral inequality

Well playing with geogebra I have discovered (maybe not) the blue tooth's curve see the picture below : It's well define by the function : $$f(x)=(π x)^{2 (1-π x)}+(π (1-x))^{2 (1-π (1-x))}$$ I want ...
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0answers
43 views

Relationship between algebraic multiplicity and pertubative interpretation

I'm reading Fulton's book on Algebraic curves and he gives an algebraic definition of the multiplicity of an intersection of curves using local rings (see http://www.math.lsa.umich.edu/~wfulton/...
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0answers
55 views

How to parametrize the intersection of an ellipsoidal surface and a sphere?

Suppose you have an ellipsoid given by the set, $$\left\{ x \in\mathbb{R}^3 \mid x^TQx = 1 \right\}$$ where $Q = \mbox{diag}(a,b,c)$. Is there a way to parametrize the set $$\left\{ x \in \mathbb{R}^3 ...
2
votes
1answer
64 views

Cat's curve and some properties

Well, it's my cat which inspired me today . The goal was :find a curve using elementary function which looks like my cat and I have found this : let $0<x<1$ the cat's curve is defined by the ...
0
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0answers
38 views

What is the maximum number of intersections of two sigmoid curves?

I have two specific sigmoid functions with domain in $[0,\infty)$ that depend on a common set of parameters. I was able to show that they have the same limit at infinity and that the dashed curve is ...
2
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1answer
80 views

Critiquing the notion of a tangent as a line that “just touches” a curve.

The most common answer to this question "What is tangent to any curve?" is as follows: "Tangent to a plane curve at a given point is the straight line that "just touches" the ...
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0answers
26 views

Help with understanding the following iterations of a Peano Curve

I have recently read about the Peano Curve and I have found and understood the iteration using a square and the subdivisions of it into 9 other squares. So the process I understand is in the figure ...
0
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1answer
38 views

Does rectifiable curve in $\mathbb{R^n}$ must be piecewise smooth?

For a curve in $n$ dimension Euclid Space $\mathbb{R}^n$, if it is rectifiable, then does it have to be a smooth curve, or piecewise smooth. And of course, how to prove it. To define smooth, there ...
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2answers
17 views

(c) Find the area contained between the curve, the y-axis, the line t = 1 and the asymptote to the curve which is parallel to the t-axis. [closed]

Part (a) and (b) are fine, and I believe (c) is an integral, but i'm not quite sure how to go about solving said integral with the given parameters. Mainly the vertical limits, as i'm sure the ...
0
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2answers
27 views

The equation of a curve which is almost similar to Gaussian curve (normal distribution curve) with $2$ asymptotes

We know that the equation of the following curve could be of the form $y=A+Be^{ax^2+bx+c}$ where $A,B,a,b,c$ are constants. Which has one horizontal asymptote. I have (almost) same kind of curves (...
1
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1answer
44 views

Is it true that every curve defined by a graph of polynomial is regular?

A Regular point on a curve defined as the following Def.1 (Singular and Regular points of planar curves) Suppose that $S$ is a curve in $\mathbb{R}^2$ and $a\in S$. If, for every $r>0,S\cap B(a;r)$...
2
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2answers
113 views

What is the locus of points equidistant from two circles?

What is the locus of points equidistant from two circles? $$ x^2+y^2+ 2 h x + 2 g y + c =0 ;\; C =0 ;$$ Construction of circles with $(a,b,2h)= (3,2,3.6)$ Please help finding equation of the locus ...
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0answers
20 views

Find the equation of the osculating circle of Cycloid

Find the equation of the osculating circle of the curve defined parametrically by : $${\displaystyle \mathbf{r}(t)\,=\,{\begin{pmatrix}r\left(t-\sin\left(t\right)\right)\\r\left(1-\cos\left(t\right)\...
0
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1answer
28 views

To find a smooth planar curve starting at $\vec{r_0}$ stopping at $\vec{r_1}$ with some additional constraints.

To find a smooth planar curve starting at $\vec{r_0}$ stopping at $\vec{r_1}$ whose unit tangents at start and stop are $\hat{v_0}$ and $\hat{v_1}$ and has the minimum length. Let us assume that the ...
0
votes
1answer
58 views

How do I find the intersection of two opposing involutes?

I have two involutes of a circle. They are drawn from the same base circle, but one is offset from the other and faces the opposite direction (see plot below). The parametric equations for the first ...
0
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0answers
38 views

Find the equation of the osculating circle of one of the special cases of Lissajous curve

Find the equation of the osculating circle of the curve defined parametrically by : $${\displaystyle \mathbf{r}(t)\,=\,{\begin{pmatrix}\cos(3t)\\\sin(2t)\end{pmatrix}}}$$ Which is indeed a special ...

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