# Tagged Questions

Plane curves are continuous (or smooth) functions $\gamma\colon I\to\mathbb R^2$ from a real interval to the plane. Sometimes also the image $\gamma(I)$ is called curve.

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### Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
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### Curve fitting: How to identify the appropriate function for a beat-like phenomena?

I have a time series data which shows some beat like behaviour. The envelope does not look exponentially decreasing, as it is impossible from a physics point of view. The envelope is likely to be a ...
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### Proof of the formula that computes the genus of smooth projective plane curve

I was searching for a proof of the formula that computes the genus of a smooth projective plane curve of degree $d$: $$g = \frac{(d-1)(d-2)}{2}$$ which do not make use neither of triangulation or ...
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### Dichotomy in the number of regions on a plane formed by an infinite number of lines

I'm reading Knuth's Concrete Mathematics and we are dealing with recurrence relations. He proves that the number of regions $L_n$ formed by $n$ lines on a plane is $L_n=\frac{n(n+1)}{2}$. I don't ...
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### What is the rotation index of a figure 8?

Is it 0 since the total turning angle covers one clockwise circle and one counterclockwise circle thus making the total 0 and the rotation index 0?
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### Existence and uniqueness of a point with horizontal tangent in a convex curve

I had a look on the proof by E. Schmidt of the Schur's Theorem about arcs of convex curves. It states the following: Let $C$ and $C'$ be two arcs of the same length with the endpoints $a,b,a',b'$ ...
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### Understand if a curve is parametrized by arc length or not

Show that the curve $$\alpha(t)=(t,1+\frac{1}{t},\frac{1}{t}-t), \quad t\in(0,\infty)$$ is a plane curve. I know $\tau$ must be zero for curve being plane. However, I want to determine the ...
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### Maximum of the length of a curve and its curvature

I'm trying to solve this problem but I can't find any way to do it. Some hints or helps will be very useful and I'll be very thankful. The problem says: Let $\alpha : (a,b) \rightarrow \mathbb{R}^2$ ...
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### Are involutes always regular?

Suppose the curve $\alpha:I \rightarrow \mathbb{R}^2$ is an involute of a regular curve. Does $\alpha'(t)\neq 0$ hold for all $t\in I$?
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### Is the evolute always a regular curve? [closed]

Is the evolute always a regular curve? (that is, the tangent vector of this curve is nonzero at any time)
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### Differentiable, Parametrized Curve for Trace $y = |x|$

my professor gave practice problems for an upcoming midterm, and one is to find a differentiable parametrized curve with trace $y = |x|$ with $-1 \leq x \leq 1$. My thinking is that I should find a ...
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### Are all curves with equation of the form $(\xi x +n) \cdot x = \text{const}$ circles?

Let $x(t)=(x_1(t),x_2(t))$ with $t\in [a,b]$ be a smooth curve in $\mathbb{R}^2$ and $\xi \in \mathbb{R}$ such that $$(\xi x +n) \cdot x = \text{const}$$ Here $n$ is the unit normal to the curve. Is ...
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### Examples of functions whose arc-length from the origin is given by their derivative

I'm looking for functions $y:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\int_{0}^{a} \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} dx = \frac{dy}{dx}\Bigg|_{a}$$ (this kind of feels like a calculus-of-...
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### Are all rationally parametrized plane curves algebraic? How does one find their degree?

Suppose a plane curve is given parametrically by $x=p(t),y=q(t)$, where $p,q$ are rational functions. I originally assumed that this means that the parametrized curve is algebraic, i.e. that it is the ...
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### Prove that for any piecewise smooth curve it is possible to find the parametrisation

Prove that for any piecewise smooth curve it is possible to find the parametrisation $\phi$ that is consistent with its length, ie. length of a curve segment between $\phi(a)$ and $\phi(b)$ is equal ...
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### Equation of the locus of centre of the ellipse?

An ellipse slides between two perpendicular lines. To which family does the locus of the centre of the ellipse belong to?
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### Geodesics on surface of revolution of regular curve

I was recently presented with this in differential geometry stating the following: Let us define the regular curve on the XZ plane as: $\gamma (t) = (sin(t)+2,0,t)$ on XZ plane for $t \in R$, ...
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### Can a curve cross its asymptote infinitely many times?

Can a curve cross infinitely many times its asymptote? If so, is there a special name for this behaviour?
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### Can a surface of revolution be built from a self-intersected curve?

I'm reading "Differential Geometry of Curves And Surfaces" of Manfredo Do Carmo. There's a point in his book about Surfaces of Revolution which confuses me a lot. Here is the part: The part ...
### Problem about curves. A particle is running along circumference $x^2+y^2=25$
I'm considering a problem about curves. A particle is running along circumference $$x^2+y^2=25$$ with a costant modulus speed compliting a turn in 2 second. I need to determinate the acceleration in ...