# 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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### Techniques for finding functions with known values and derivatives at two points

I want to find a function $\gamma(t) = (x(t),y(t))$ such that for two values of $t$, we have $\gamma'(t)$ and $\gamma(t)$ have some value, and at no point does the curvature ever exceed $r$. What ...
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### Algebraic Curve

I know that a curve of the type $$\vec{\sigma}(t)=\cos(mt)\hat e_1+\cos(nt)\hat e_2$$ with $m,n\in\mathbb{Z}$ is algebraic. My question is: what is the polynomial that define this curve?
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### Equation for sickle shaped plane curve

Is there a parametric equation for a plane curve with the shape of a sickle cell, e.g. half nephroid and half circle? I couldn't find one so far. Thanks! I'm looking for an equation consisting of ...
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### Can a curve be rotated and translated to 'fit inside itself'?

Working in $R^2$, consider any continuous curve $C$ with endpoints $a$ and $b$, such that the curve does not intersect the line formed by $a$ and $b$. Then the line $ab$ and $C$ form an enclosure, $E$....
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### 3-D Geometry Problem. Find a curve which touches the straight line.

If two perpendicular tangent planes to paraboloid $x^{2}+y^{2}=2z$ internsects in a straight line in the plane $x=0$, obtain the curve to which the straight line touches. I don't know how to ...
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### Is the plane curve $y^3=x^4+x^3$ an irreducible algebraic affine set?

I'm dealing with the plane curve $C=\{(x,y)\in k^2:y^3=x^4+x^3\}$. I want to know if this curve is irreducible, where $k$ is a commutative field. I know this is equivalent to the ideal $\sqrt{I}$ ...
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### Direction of a curve given by ODE

Let $a,b \in C(\mathbb{R^2})$ be bounded and $(x_0,y_0) \in \partial B_1(0)$. Consider the ODE system $$\begin{cases} x'(t)=a(x,y) \\ y'(t)=b(x,y) \\ x(0)=x_0 \quad y(0)=y_0 \end{cases}$$ We know ...
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### Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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### The plane curve of maximum average speed under constant gravitational force

A line gives us the minimum distance from $A$ to $B$. A cycloid gives us the minimum traveling time of a point mass from $A$ to $B$ (under constant gravitational acceleration $g$). What about the ...
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### Convex, closed plane curve is a Jordan curve

The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it'...
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### What is the definition of a path along a multivariable function?

I'm taking a class equivalent of Calculus III, and we saw how to prove continuity of a multivariable function. Recently we looked at the following example: \begin{align} f(x,y) = \begin{cases} \frac{...
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### S-shaped Reverse Logistic Curve

Is there any curve that grows very slow at the beginning then growth picks up exponentially before hitting the wall. I need sort of reverse behavior of the logistic curve. Here's raw idea how it ...
I have two lines with the following equation $$D1 : (x, y, z) = (2,0,0) + k(0,3,0)$$ $$D2 : (x, y, z) = (2,0,2) + k(0,0,1)$$ and I must find out the equation of the plane that they make. I ...