# 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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### Rational maps between affine varieties

If I want to check that the map $$\phi:C_1\rightarrow C_1,\hspace{0.5cm}\phi(x,y)=(\phi_1(x,y),\phi_2(x,y))$$ between two affine plane curves is rational I just should check that $\phi_1$ and ...
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### Find real constants $c$ and $k$ such that $y=cx^k$ passes through point $(a, b)$ with slope $m$

In the Cartesian plane, can a power function of the form $y=cx^k$ (where $c>0$ and $k>1$, not necessarily an integer) be found such that its graph passes through any arbitrary point $(a, b)$ ...
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### Normal to a parametric curve: $x=2t+3$, $y=2/t$ [closed]

A curve is given by the parametric equations $x=2t+3$, $y=2/t$. Find the equation of the normal at the point on the curve where $t=2$. I honestly do not understand how to do this question.
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### Find the Equation of the Envelope of a Family of Line (Plane) Segments

Consider the first quadrant in the $OXY$ plane in $\mathbb{R}^2$. Point $O$ is the origin and the points $P$ and $Q$ are chosen on the $y$-axis and the $x$-axis, respectively as it is showed in the ...
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### Find the angle between two planes using their normal vectors

The angle between two intersecting planes is defined to be the angle between their normal vectors. Find the angle between the planes $x – 2y + z = 0$ and $2x + 3y – 2z = 0$. Find the parametric ...
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### Does a plane curve with polar equation $r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$ have a name?

Does a plane curve with polar equation $$r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$$ where both $\lambda_i>0$ have a name? It's very similar to hippopede, also known as lemniscate of Booth, ...
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### Find an equation for the plane tangent to $2x^2+3y^2−z^2=4$ at $(1,1,−1)$

Find an equation for the plane tangent to $2x^2+3y^2−z^2=4$ at $(1,1,−1)$ So I started by taking the partial derivative for each term. $\frac{\partial}{dx}=4x$ $\Rightarrow$ $f_x(1)=4$ ...
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### For all unit vectors $\mathbf u$ and a positive definite $\mathbf C$, what surface do vectors $\mathbf u \mathbf u^\top \mathbf C \mathbf u$ form?

Let $\mathbf C$ be a positive-definite $k\times k$ matrix. For all vectors $\mathbf u\in \mathbb R^k$ of length $\|\mathbf u\|=1$, consider vectors $\mathbf {uu}^\top\mathbf{Cu}$; they form a surface ...
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### Convexity criterion for piecewise regular planar curves

A theorem of classical differential geometry states, that a simple, closed and regular $\mathcal{C}^2$-curve $\gamma:[0;1]\to\mathbb{R}^2$ is convex, iff its signed curvature doesn't change sign. ...
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### What is the circumference (arc length) of $x^4 + x^2 + y^4 + y^2 = 2$?

Consider $$x^4 + x^2 + y^4 + y^2 = 2$$ It is a smooth non-intersecting circle like curve in the plane. A bit like a Hyperellipse. See https://en.m.wikipedia.org/wiki/Superellipse What is the ...
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### Can an elliptic curve be singular?

Is an elliptic curve strictly a non-singular cubic curve or can it have singular points? According to Wolfram Alpha, "an elliptic curve over a field K is a nonsingular cubic curve in two variables" ...
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### Divergence in Definition of Laplace-Beltrami Operator

I am trying to derive an explicit formula for Laplace-Beltrami operator in global Cartesian coordinates for a special case of plane curve. I have found this article, and I would like to match their ...
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### Finite open covers of a complex $C^{(1)}$ curve.

Consider a complex curve $\gamma \subset \mathbb{C}$, parametrized by $\alpha: [a,b]\to \mathbb{C}$, with $\alpha \in C^{(1)}$. Further, consider an finite open cover $\Phi$ of $\gamma=\alpha([a,b])$. ...
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### Rotation of a Line Intersecting the Curve $y = x – \log(x)$ as $x \rightarrow \infty$.

Let a straight line ("line 1") in the $xy$-plane have one end fixed at the origin $(0,0)$, and the other at a variable point $(x, x – \log(x))$ on the curve $y = x – \log(x)$. The domain of $x$ is the ...
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### Set-theoretic equality in double dual graph

Someone in the math fandom on tumblr was explaining the concept of a dual graph, and he was asked a tough question: Is the dual of the dual [equal to] the original graph always, or just ...
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### If three points on a quadric surface then the line going through them is contained in the quadric

I am having trouble understanding a step in my Professor's Lecture notes She shows that Lemma 2.2.4 Let $P_1,\ldots,P_5$ be distinct points in $\mathbb{P}_k^2$. There exists a conic in ...
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### Logarithmic spiral characterized by signed curvature and arc length parameter.

This is a homework problem I am having trouble with: Show that if a planar unit speed curve $q(s)$ satisfies $$\kappa_s = \frac{1}{es+f}$$ for constants $e, f >0$, then the curve is a logarithmic ...
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### Looking for rounded corner plane curve with certain properties (SIDESTEPPED)

For a project involving simulating traffic lights, I am currently looking for a formula to get a rounded 90-degree corner (to describe the path of a turning car) with certain properties: Defined in ...
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### Involute of a Curve

A string of length ℓ is attached to the point γ(0) of a unit-speed plane curve γ(s). Show that when the string is wound onto the curve while being kept taught, its endpoint traces out the curve ι(s) = ...
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### Describe a curve by other than a fomula, fitting or interpolation

I have a curve defined by a set of $(x,y)$ given points. After ploting these points I get : My aim is to study the sensitivity of the operation that results these points. In order to do that, the ...
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### Finding curves whose tangents intersect with the x-axis at $(\frac{x}{2},0)$

I have to find the family of curves in $\mathbb{R}^2$ with this property: The tangent in an arbitrary point on the curve does intersect with the x-axis in $(\frac{x}{2}, 0)$. I think I have to make ...
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### Curvature of plane parametric curves

What is the neatest way to derive the following formula for the curvature of a parametric curve? $$\kappa =\frac{\|y'x''-y''x'\|}{(x'^2+y'^2)^{\frac{3}{2}}}$$
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### How to find tangent line given several variables

I have a question that I'm having difficulty on. I can solve these normally, but I'm having a bit of a challenge dealing with these extra terms: "Find the equation of the tangent line to the ellipse ...
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I have numerically obtained some curves, corresponding with it I have also obtained some roots. I strongly believed these curves can be fitted with some (elliptic) functions taken the roots as ...
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### What is my line integral answer incorrect?

EDIT: Is my computation not correct, possibly because the parametrization that I used requires x,y to be on the xy-plane? If so, can I adjust from here, and not start over? I.e., is there some ...
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### Prove it is not a closed Curve

I wanna prove that $(cos(t^3+t),sin(t^3+t))=γ(t)$ which is a reparametrization of a circle .Is not a closed curve like the circle. WHat i did is take this problem to the Complex plane so ...
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### How to relate between tangents of two parallel curves?

I am solving a problem about the relationship between the curvatures of two parallel curves. Along the way, I encountered a problem which seems intuitively correct but failed to show it rigorously. ...
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### Proving that an oblique cycloid cannot be tautochrone

Someone asked me if the tautochronicity property of a cycloid would still hold if the cycloid were rotated, so that its lowest point (the equilibrium point) be no more the vertex. If $V$ is the ...
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### Showing that a particular area is small

Suppose that $F(x,y) \in \mathbb{Z}[x,y]$ is an irreducible binary form of degree $d \geq 3$. Let $B$ be a positive parameter, considered to be large (i.e. one can consider it to be sufficiently large ...
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### Parametric equation of line?

I have an assignment I'm doing where I am supposed to determine a parametric equation of a line orthogonal to a segment , let's say OP, and passes through the midpoint of this segment. What I have ...
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### Point normal equation of plane

Im doing an homeassignment in linear algebra! This is the question I'm having problems with! The plane M contains two lines; l1 : (x, y, z) = t(2, −1, 0), t ∈ R, l2 : (x, y, z) = t(0, 1, ...
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### how to prove plane $ax+by+cz = d$ has normal vector $(a,b,c)$

Given a plane function $ax+by+cz=d$, how can one prove that unit normal vector is $$n = \pm \dfrac{a\boldsymbol{i}+b\boldsymbol{j}+c\boldsymbol{k}}{\sqrt{a^2+b^2+c^2}}$$
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### Show that $|\alpha'(t)|^2-1=0$ for any arbitrary parameter $t$

There is this supposed to be not so hard question but concept of re-parametrisation by arc length bothers me a bit. Let $\alpha:I\to R^2$ be a regular paramatrised plane curve (arbitrary parameter) ...
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When I was proving some properties of regular parametrised plane curve $\alpha:I\to R^2$ which has a normal vector $n(t)$, I encountered the need to prove the following: $$\langle ... 2answers 88 views ### Help in showing that an evolute is the envelope of the normals to a curve Let \alpha:I\to R^2 be a regular parametrised plane curve (arbitrary parameter), and define n=n(s) and k=k(t) to be the normal and curvature respectively. Assume k(t)\neq0, t\in I. In this ... 1answer 39 views ### Find a vector function represented by the curve of intersection? I'm struggling with the following problem: Given \, z = \sqrt{x^2 + y^2}\, and \, z = y+1\, find the vector function represented by the curve of intersection of the surfaces using the ... 3answers 58 views ### Parametrization of Hyperbola I "know" that a parametrization of an Hyperbola (x^2-y^2=1) is given by:$$\gamma(t)=(\sec(t),\tan(t)),t\in\mathbb{R}$$I know that x=\sec(t) and y=\tan(t) is a solution of the equation. How ... 1answer 25 views ### When working with trochoids what does θ stand for? These are the formulas with which you can draw trochoids. x = aθ - b sin(θ) y = a - b cos(θ) I'm trying to make trochoids but I got hung up on this symbol θ, what is it and how do I use it, I ... 2answers 66 views ### How to find a plane that is tangent to 3 spheres? So there are spheres with radius of 1 centered at (1,2,0), (4,5,0) and (1,3,2). How can one find a plane that is tangent to all 3 spheres? Visually, it looks like as if the spheres are sitting on a ... 13answers 17k views ### Derivation of the formula for the vertex of a parabola I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form$$y = a x^2 + b x + c$$My teacher gave me the formula:$$x = -\frac{b}{2a} as the ...
I am trying to fit an ellipse to the end of a curve, $y(x)$, such that first and second derivatives of the curve, $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$, are preserved at the contact point, $(x,y)$, ...
I have a plane curve given by $y = f(x)$. At every point on this curve, I construct a normal direction to this curve and move the point a fixed distance $s$ along the normal to a new point. What is ...