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Questions tagged [curves]

For questions about or involving curves.

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17 views

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 ...
0 votes
1 answer
39 views

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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0 answers
26 views

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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0 answers
57 views

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 ...
-1 votes
1 answer
21 views

What does it mean for a curve to be another parametrization of a path? [closed]

I think I am somewhat struggling understanding paths. I have a curve $\phi$ defined as closed and simple which gives the path $C_\phi$. There is another curve $\psi$ that is defined as another ...
-1 votes
1 answer
33 views

Arc length of a polar curve r = -8cos(t)

If I am being asked to find the arc length of the polar curve $r = -8cos(t)$ when I use the integral formula it gives me 16 $\pi$. But since this polar curve represents a circle with radius 4, should ...
-1 votes
0 answers
29 views

Fundamental theorem of space curve_Existence Theorem [closed]

I was trying to learn the proof of the Existence Theorem for space curves. Can someone give me a specific example of two different curves and tell me how to prove the existence theorem for them curves....
1 vote
1 answer
1k views

Prove that the extremity of the polar subtangent of the curve $u+f(\theta)=0$ is $u=f'(\theta+\frac{\pi}{2})$ where $u=\frac{1}{r}$

Prove that the extremity of the polar subtangent of the curve $$u+f(\theta)=0$$ is $$u=f^{\prime} \left(\theta+\frac{\pi}{2} \right)$$ where $u=\frac{1}{r}$ I am confused and I think that it should ...
2 votes
3 answers
201 views

Criticizing 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 ...
3 votes
1 answer
3k views

Proof of geodesic has constant speed

So I'm working with differential geometry. So my book claim that "any geodesic has constant speed". And the proof is left as an exercise and I found the exercise in the book. Exercise: "Prove that ...
4 votes
1 answer
132 views

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 ...
2 votes
1 answer
19 views

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 ...
0 votes
1 answer
29 views

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 ...
2 votes
1 answer
74 views

Curvature of an Embedded Curve

If one has a a compact Riemannian surface $\Sigma$ (ie. a Riemannian manifold of dimension $2$), the Ricci curvature can be written as $R_{ij} = \frac{1}{2} R g_{ij}$. In other words, the curvature ...
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0 answers
19 views

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)...
2 votes
1 answer
72 views

Maxima and Minima of a Funny Looking Graph

I was messing around with the function $$f_k(x)=\frac{1}{k+1}\prod_{n=1}^{k} (x-n)^{\frac{1}{2n-1}}$$ because it has a funny looking graph. It was easy to determine that the zeroes are at $$x=1, 2, 3, ...
1 vote
1 answer
80 views

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 ...
1 vote
0 answers
1k views

Frenet-Serret Formulas for Arc Length and Regular Parametrization

Frenet-Serret formulas for arc length parametrization are in matrix form where $\mathbf{ \widetilde T}(s)$, $\mathbf{ \widetilde N}(s)$, $\mathbf{ \widetilde B}(s)$ are unit tangent, normal, binormal ...
1 vote
1 answer
58 views

Minimum squared curvature shape

[Related but different: https://math.stackexchange.com/questions/2989908/which-shape-does-an-elastic-rod-take-as-its-ends-are-getting-closer - different functional] I'm looking for a shape that ...
0 votes
2 answers
976 views

Do all the curves become straight line in limit [closed]

Do all the curves become straight line in limit of their length to zero. I know $$ \triangle L=\sqrt{(\triangle x)^2 + (\triangle y)^2}$$ And when $ \triangle L ,\triangle x , \triangle y$ $\...
2 votes
2 answers
467 views

Trace of the cissoid of Diocles

I want to show that you can parameterized the cissoid of Diocles using $$ \alpha(t) = \left(\frac{2at^2}{1+t^2}, \frac{2at^3}{1+t^2}\right),\quad\text{where }t=\tan\theta. $$ Here is an image of the ...
3 votes
1 answer
379 views

Show that the genus of the projective curve $x^n_1 = x_2x^{n-1}_0 - x^n_2$ is $\frac{(n-1)(n-2)}{2}$.

Suppose that we have a curve $X \subset \mathbb{P}^2$ given by $x^n_1 = x_2x^{n-1}_0 - x^n_2$ . How do we show that the curve has genus $\frac{(n-1)(n-2)}{2}$ (whenever the curve is smooth). I should ...
2 votes
1 answer
57 views

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 ...
0 votes
0 answers
24 views

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 ...
1 vote
1 answer
48 views

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 ...
1 vote
0 answers
32 views

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 ...
1 vote
1 answer
182 views

Rank of elliptic curve over $\Bbb{Q}(2^{1/3})$

Rank of elliptic curve over quadratic extension $K=\Bbb{Q}(\sqrt{D})$ is calculated by a formula $rank(E/K)=rank(E/\Bbb{Q})+rank(E_D/\Bbb{Q})$(this is easy to prove). Thus rank over quadratic field ...
0 votes
1 answer
1k views

Finding Area under Curve Bounded by both x and y axis.

I am trying to find the area under the curve $f(x)= C - x^4$ bounded by the $x$ and $y$ axes, where $x$ is any real positive number. I know to find the area under the curve I need to evaluate the ...
2 votes
1 answer
4k views

Torsion formula derivation

I am trying to derive the formula for torsion of a curve. In many example proofs I've seen the final step is that $\kappa=\|\dot{\lambda}\times\ddot{\lambda}\|$. However I thought curvature was ...
3 votes
1 answer
197 views

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 ...
0 votes
1 answer
38 views

When shall a curve that connects two points on a surface be a line?

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 ...
0 votes
2 answers
42 views

Curve is traveled clockwise or anti-clockwise

Given the curve $$ \vec{\mathbf{r}}(t) = \bigl(t - \tfrac{1}{3}t^3 \bigr) \, \vec{\mathbf{i}} + (4 - t^2) \, \vec{\mathbf{j}}, $$ how can I tell whether it's traveled clockwise or counterclockwise? ...
1 vote
0 answers
28 views

Why is the order of $dz$ on Riemann sphere at $\infty$ equal to $-2$? [duplicate]

The setting is as follows: We consider $X= \mathbb{C}_\infty$, the Riemann sphere with coordinate $z$ on $\mathbb{C}$. Let $\omega = dz$ (this is a $1$-form). On page 131 of Rick Miranda's book "...
0 votes
0 answers
14 views

How do I identify which vertex to start with when constructing an irregular curve of constant width?

A Curve of Constant Width (CoCW) is a closed, convex curve whose width (measured as the distance between parallel supporting lines) is the same in all directions. A CoCW can be constructed from any ...
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0 answers
19 views

Tangent vectors to regular curve at constant angle to a fixed vector

Problem: Show that the tangent vectors to the regular curve $x(t)=(3t,3t^2,2t^3)$ make a constant angle ($\theta$) with the vector $a:=(1,0,1)$. We have $b:=x'(t)=(3,6t,6t^2)$. I worked out that $b\...
1 vote
2 answers
280 views

Question about if something would be visible from the surface of a sphere. [closed]

So, I am trying to see if something would be visible to someone standing on the surface of a planet or the top of mountain on it. So, imagine a perfect sphere for the planet, then imagine a moon which ...
0 votes
2 answers
74 views

How do I find the corner shape of the bounding box of a smooth curve of constant width?

Given the functions $$ p(θ) = \frac{S}{2} × \frac{\cos\bigl(n × (θ - α)\bigr)}{n^2 - 1} + \frac{W}{2}\\ \begin{align} X(θ) = \cos(θ) × &p(θ) - \sin(θ) × p'(θ) - \left(p\left(\frac{0}{2}\right) - \...
3 votes
0 answers
35 views

Why does the path traced by a point reflect on a tangent of a circle create a cardioid

I was messing around in GeoGebra and noticed that this construction creates a shape that, at least, looks like a cardioid. Since both $B$ and $B'$ are equidistant from point $C$, they must be on the ...
0 votes
0 answers
77 views

Can we find a "s", "t" parametrization of any planar simple curve?

Let $$r: [0,1] \to \mathbb{R}^{2}, $$ be a simple curve. Denote by $\Omega$ the interior of the curve defined by $r(t)$. Without loss of generality, let us assume that $(0,0) \in \Omega$. Does there ...
1 vote
1 answer
2k views

Average Value of a Line Integral

I'm having quite a hard time calculating the average value of a line integral. Given the surface $f(x,y) = \sqrt{16 + 36y^{2/3}}$ and the curve $y = x^{3/2}$, I need to calculate the average value of ...
2 votes
1 answer
71 views

Complex fundamental theorem of calculus

The fundamental theorem of calculus for complex functions (as well) states that $\int_{\gamma} f(z)dz := \int\limits_{a}^{b} f(\gamma(t)) \gamma'(t) dt = F(\gamma(b)) - F(\gamma (a)) = \int\limits_{\...
0 votes
2 answers
54 views

Find loop passing through two points with length $L\pi$

Problem: Find a nice simple closed curve other than circle which passes through the points $(0,0)$ and $(1,0)$ on the Cartesian plane and whose length is $L\pi$. If the given condition is not the loop ...
2 votes
0 answers
51 views

Projective curve covered by two affine pieces

A non-singular projective curve $X$ is covered by two affine pieces (with respect to different embeddings) which are affine plane curves with equations $y^2 = f (x)$ and $v^2 = g(u)$ respectively, ...
0 votes
1 answer
42 views

Find the slope at a given time value of this equation of a heart [closed]

The following equation draws the curve of a heart: $x(t)=12sin(t)-4sin(3t)$ $y(t)=13cos(t)-5cos(2t)-2cos(3t)-cos(4t)$ Image https://de.wikipedia.org/wiki/Benutzer:Georg-Johann I would like to find the ...
16 votes
1 answer
933 views

Exercise 1.24 from Joe Harris' Algebraic Geometry: First Course

I have a question about solvability of Exercise 1.24 (p. 14) from Joe Harris' Algebraic Geometry: A First Course and correctness of following 'synthetic' construction which according to the book (or ...
2 votes
0 answers
45 views

How to generalize curvature to n dimensions parameterized by time instead of arc length?

I am a novice in mathematics in general and even more so in differential geometry. Currently, I am looking to generalize the Frenet-Serret formulas to $n$ dimensions. At the moment, I am interested in ...
0 votes
0 answers
41 views

The question of a curve is uniquely determined by an number of points

It is known that a line is uniquely determined by two of its points The circle is uniquely determined by three points, while the conical cut is determined by five points, and the cubic curve is ...
1 vote
1 answer
54 views

Calculating the area enclosed by three graphs, but two out of them already enclose a different area.

I have three functions : $$y = 2x^2 + 2x \tag{1}$$ $$y = -x - 1 \tag{2}$$ $$x = 0 \tag{3}$$ Exercise asks for the area enclosed by three of those functions. It is clear that the area from $x = \frac{-...
0 votes
0 answers
48 views

Curvature of a product of two functions

Can we estimate how the curvature of a function changes if "bent" over another function? Example: $y_1(s)$ - is sinusoidal function, e.g., $a\cdot \sin(b\cdot s), \; s \in [0, L]$ $y_2(t)$ -...
4 votes
1 answer
632 views

A convex closed plane curve never intersects itself?

Intuitively, I think that a convex closed curve has to be simple (i.e. cannot intersect itself except at the starting and the ending points). How one can prove it rigorously? My attempt: Suppose it ...

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