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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When is $t \mapsto \gamma (t) = \big( (1 + a \cos t) \cos t, (1 + a \cos t) \sin t \big)$ simple and closed?

Show that $\gamma (t) = \big( (1 + a \cos t) \cos t, (1 + a \cos t) \sin t \big), t \in [0, 2\pi]$, where $a$ is a constant, is a simple closed curve if $|a| < 1$ , but that if $|a| > 1$ its ...
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1answer
32 views

Image of a circle under conformal map $1/z$

The image of a circle under conformal map $1/z$ should be a circle, but how to prove it (or how to find the relationship between the two circles)? $z = x + iy = d + a\exp(i\theta)$, where $a$ is the ...
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1answer
78 views

Finding a parametric form for the locus of points for a vanishing determinant

I need to find the locus of points in the real $(x, y)$ plane, in parametric form, satisfied by the equation \begin{equation}\det\begin{pmatrix} 0 & 1 & 1 & 1 & 1\\ 1 & 0 & ...
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45 views

Relationship between discriminants and smoothness of curves

My understanding of the use of the discriminant in elliptic curve theory is to test whether an elliptic curve in Weierstrass normal form over a field not of characteristic either 2 or 3, $y^{2} = x^{3}...
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1answer
22 views

Basic plane question, finding a plane traveling through the heads of 3 given vectors.

Find the equation for the plane passing through the heads of the three given vectors (2, 2, 0) (−1, 2, 1) (1, 1, 4) If I was given 3 points, I know how to do this. Simply find AB x AC and plug one ...
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2answers
35 views

Quick question regarding wording of a homework question

Find the equation for the plane passing through the heads of the three given vectors (2, 2, 0) (−1, 2, 1) (1, 1, 4) Is this just another way of asking what is the plane passing through these ...
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47 views

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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2answers
24 views

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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2answers
2k views

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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37 views

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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1answer
163 views

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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1answer
23 views

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. $\frac{\partial}{dx}=4x$ $\Rightarrow$ $f_x(1)=4$ $\frac{\...
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1answer
34 views

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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1answer
68 views

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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161 views

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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135 views

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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3answers
26 views

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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1answer
17 views

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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1answer
42 views

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 isomorphic?...
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1answer
31 views

The meaning of $\frac{dy}{dx}$ for an implicit curve $F(x, y) = c$ at a point which is not at the curve

What is the meaning of $\frac{dy}{dx}$ for $$xy^2 + x^2 - \frac{y}{x} = 2$$ at the point $(1,1)$ which is not at the curve? I know if the point is on the curve the derivative is the slope of the ...
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1answer
128 views

Envelope of family of curves $x(u,v)=\cos^2(u)\cos(v)+\cos(u)\sin(u)\sin(v)$, $y(u,v)=\cos^2(u)\sin(v)-\cos (u)\sin(u)\cos(v)$

How to generally find singular solution or envelope of a two parameter family of curves $ x(u,v),y(u,v) $ in the plane? The parametric equations $$x(u,v) = \cos^2 (u) \cos (v) + \cos (u) \sin (u) \...
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1answer
50 views

How can I plot curves relative to other curves?

Suppose I have a curve – say a sine curve y=sin(x) Now I want to draw a second curve, y = ½sin(x), but relative to the first – so it has the first one as a “baseline”. I don’t mean just adding the ...
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1answer
70 views

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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1answer
43 views

Are there computational short cuts to calculating the distance from a large number of points to 3 different planes?

I have three planes, and i want to calculate distance of my point to each of them. However, there are 68000 points in the space, so it does not make sense( computationally) to calculate the distance ...
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79 views

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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40 views

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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22 views

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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37 views

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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39 views

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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26 views

advice for curve fitting

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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1answer
48 views

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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1answer
65 views

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 $$γ(t)--->...
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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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12 views

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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1answer
37 views

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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1answer
86 views

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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123 views

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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49 views

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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1answer
18 views

For a regular parametrised plane curve $\alpha$, show that $\langle \alpha''(t),n(t)\rangle =- \langle \alpha'(t),n'(t)\rangle$

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 \alpha''(t),n(t)\...
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175 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 ...
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1answer
78 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 ...
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3answers
69 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 ...
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1answer
27 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 ...
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2answers
143 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 ...
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1answer
40 views

Fitting an ellipse to a point with the first and second derivatives specified

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, $(x,y)$, ...
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16 views

move curve normal to itself

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 ...
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90 views

Existence of a Rectifiable Piecewise Smooth Path

Suppose you have $\gamma(t):[0,1]\rightarrow \mathbb{C}$ simple piecewise smooth, $\gamma(0) = 0$ and $\gamma(1)=1$. Does there exist $\eta:[0,1]\rightarrow \mathbb{C}$, another simple piecewise ...
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85 views

Find the maximum volume of the pyramid bounded by the plane and the coordinate planes?

Surface $\sqrt{c}=\sqrt{x}+\sqrt{y}+\sqrt{z}$ , $(c>0)$ I found that at $(x_{0},y_{0},z_{0})$ a tangent plane to the surface is : $\frac{x-x_{0}}{2\sqrt{x_{0}}}+\frac{y-y_{0}}{2\sqrt{y_{0}}}+\...
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0answers
78 views

Intersecting convex figures

Take in the real plane a finitely long horizontal line segment and connect the two endpoints by a convex path, above the segment, with the property that the only extreme points of the convex hull of ...
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1answer
82 views

Not all tangents to a plane curve are bitangents

I'm struggling on a question from a previous qualifying exam, and I don't see a clean way to do it. Let $X\subset \mathbb{R}^2$ be a connected, 1-dimensional, real analytic submanifold, not contained ...