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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Intersection Multiplicity of Rational Plane Curves

Suppose I have two rational curves in the complex projective plane. I know their parametrizations, $<x_1(t),y_1(t)>$ and $<x_2(t),y_2(t)>$ I know I can use Grobner bases to find an ...
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63 views

How do you find the angle of intersection between two given polar curves?

How does one find the angle of intersection between two given polar curves? For example, between $a^2=r^2\sin(2\theta)$ & $b^2=r^2\cos(2\theta)$
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Intersection point of the normal lines at α(t), α(t+h) converges as h→∞ given α is parametrized by arc lenght and it´s curvature is non zero

Let α(t):I→R2 be a curve parametrized by arc lenght and k(t) (curvaure) be non zero. Need to show the intersection point of the normal lines at α(t), α(t+h) converges to a point in the trace of the ...
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Does every graph have an algebraic form?

Let's say I take a pencil and start drawing a curve on $xy$ plane. The curve is continuous and for each value of $x$ there is only one corresponding value of $y$. So question that interests me is - ...
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Why Are Fresnel Functions Used For Splines?

Why are Fresnel functions still used in the research and implementation of clothoid splines? They cannot represent curves of constant curvature, which has led to a lot of research/implementation ...
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190 views

Measure of curve smoothness

Could someone please give me the intuition behind using integral of squared second derivative as a measure of curve smoothness? I was thinking that since curvature measures how fast a curve changes, ...
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2answers
39 views

Vector analysis: understanding formulas for normal and tangent

I'm studying a chapter on vector analysis and I don't really understand why the following equations are switched for the two situations. (My course is not in English, so I apologize if I translated ...
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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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Why is this the equation of the tangent plane?

I want to find the equation of the tangent plane of the surface patch $\sigma (r, \theta)=(r\cosh \theta , r\sinh \theta , r^2)$ at the point $(1,0,1)$. I have done the following: The point $(1,0,...
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Problem with $arg(\gamma (t))$

I am see my notes about curves on complex spaces and I do not understand why it is so... I need help. I do not understand what way take to do it, I need someone explain slowly please, thanks
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1answer
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Given two tangents $\varepsilon_{1},\varepsilon_{2}$ of the curve $c_{1}$, on two specific points $x_1, x_2$, find the tangents

Let $\varepsilon_1,\varepsilon_2$ be the equations of the tangents to the curve $c: y = 1+x^2$, on the points $x_1 = 2$ and $x_2 = -2$, respectively. Prove that the equations for the two tangents are $...
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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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Genus-Degree formula gives the wrong answer: ordinary points?

I'm trying to compute the genus of the normalization of the curve: $y^5=x(x-1)(x-2)$ Now I calculate the ramification points of the projection x: they are $(0,0),(1,0),(2,0)$ and they are of ...
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Circle of radius of Intersection of Plane and Sphere

The plane $x+2y-z=4$ cuts the sphere $x^2+y^2+z^2-x+z-2=0$ in a circle of radius? I tried putting value of y from plane in sphere but then I get a $zx$ term. How to proceed?
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Is this the correct way to compute the blow up of a curve

I'm trying to calculate the blowup of the curve $y^5=z^2-3z^3+2z^4$ at $(0,0)$ We have the relation $Ay=Bz$, now I split it into two charts: The first chart$(y,a=A/B)$: $y^5=a^2y^2-3a^3y^3+2a^4y^2-y^...
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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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107 views

Circle Rolling on Ellipse

I've gotten interested in describing a circle rolling on an ellipse; specifically, the curve traced out by a point on the circumference of the circle. I want a symbolic solution to the general case, ...
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55 views

What's the area of the shape defined by all points whose distances from two focal points multiply to give the same product?

This shape, which I call the multiplicoid, is the equivalent of, and very similar to, an ellipse. However, instead of the distance between each point and the two focal points summing to a constant, ...
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37 views

Angle between tangent vector and point of a cardiod.

Consider the cardioid $\rho=2a\left( 1 - \cos \phi \right) $. Show that the angle between the tangent vector and an arbitrary point (different from the origin) of the curve is half the polar angle. ...
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Characterizations of cycloid

There are several motions that create a cycloid. I have some examples here. Are there any others? Trace of a fixed point on a rolling circle Evolute of another cycloid (the locus of all its centers ...
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76 views

Focus of a rolling parabola traces a catenary - geometric explanation

It is known that the focus of a rolling parabola along the x-axis traces a catenary. I'm interested in a geometric explanation. But I don't get why $\cos \angle PFK = \frac{dx}{ds}$. Can someone ...
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145 views

Roll one ellipse on another: Locus of center ever a circle?

Let $E_1$ be an ellipse fixed in the plane. Let $E_2$ be a second, possibly different ellipse, which rolls around without slippage outside $E_1$, touching perimeter-to-perimeter. Let $c_2(t)$ be the ...
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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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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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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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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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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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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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158 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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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. $\frac{\partial}{dx}=4x$ $\Rightarrow$ $f_x(1)=4$ $\frac{\...
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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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67 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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132 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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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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arithmetic and geometric genus for a reducible plane curve

If $C$ is an irreducible plane curve we have the well known formula relating the airthmetic (obtained via the degree-genus formula) and the geometric genus $$\frac{(d-1)(d-2)}{2} - \sum \frac{r(r-1)}{...
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65 views

A point of infinite curvature on a curve

Let $\gamma(t)$ be a $C^r$ smooth curve in the plane. Suppose $r\ge 2$, so that one can define the curvature $\kappa(t)$ at $\gamma(t)$. For example, $\kappa(t)=0$ means that the curve is kind of flat ...
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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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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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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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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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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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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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2answers
130 views

Curvature of plane parametric curves

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