# Questions tagged [curves]

For questions about or involving curves.

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