Questions tagged [plane-curves]

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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Limit of a Family of Quartics

I've been trying an exercise from a book I'm self-studying, and I can't seem to get the blowup right. I'm not sure where I'm going wrong, so a nudge in the right direction would be helpful. $\textbf{...
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76 views

Introduce a generalization of the Pappus theorem and the Pascal theorem

The theorem proved in A chain of six circles associated with a conic Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a hyperbola Let $1'$ be arbitrary point in the hyperbola. The circle $(121')$ ...
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48 views

Minimal surface having a Jordan curve as boundary

I'm trying to prove that if $\gamma$ is a Jordan curve (contained in the plane $\pi$) and $\Sigma\subset\mathbb{R}^3$ is a minimal surface with $\partial\Sigma=\gamma$, then $\Sigma$ must be the plane ...
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343 views

Dehomogenization and finding multiplicities

I'm trying to find the simple/multiple points of the projective curve with the defining polynomial $F(X,Y,Z)=XZ-Y^2$. As far as I understand, I need to dehomogenize $F$ in each of the three standart ...
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1answer
52 views

Showing that $\alpha$ is a straight line

Let $\alpha : I \to \mathbb{R^2}$ be a regular curve. Suppose that all the tangent lines intersect in a fixed point. Show that $\kappa = 0$. My attempt: Let $q$ be the fixed point. So there exists ...
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51 views

Is this curve known?

A median of a triangle through mid-point of $(-c,0),(c,0) $ is such so that ratio of cosines of angles between sides/median $$ \cos \phi/\cos \psi =e $$ is a constant. Is the curve known? $$ \frac ...
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26 views

What does it mean to say a point is uniquely mapped?

I am looking at space filling curves. Essentially their is a mapping $f: I \to \mathcal{Q}$ where I is an interval in $\mathbb{R}$ such as $[0,1]$ and $\mathcal{Q}$ is a square $[0,1]^2$. For the ...
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157 views

Homotopy of closed curves is also a closed curve?

I'm trying to prove the following statement: Let $\gamma_1$ and $\gamma_2$ be two closed curves from $[a, b]$ to $\mathbb{C}$, and let $h: [a,b]\times[0,1] \to \mathbb{C}$ be a homotopy between ...
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74 views

Deviation of a plane curve from the $x$-axis

I have a smooth plane curve $\gamma$ enclosed in a circle of radius $R$. See Figure 1. I'm interested in measuring the deviation of this curve from the $x$ axis, denoted $\Delta(t)$ in the figure, as ...
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2answers
88 views

Determine equation of tangent plane?

Determine equation of tangent plane in points $(\frac{1}{2},1,f(\frac{1}{2},1) )$ $f(x,y)=x^{4}-x^{2}+y^{2}$ I know usually how these examples work, but I am confused with these $3$ points. I have ...
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153 views

Blowing up a model for a plane curve

Let $R = \mathbb{C}[[t]]$ and let $\mathcal{X} \hookrightarrow \mathbb{P}_R^2$ be the arithmetic surface defined by the equation $$(X^2 - 2Y^2 + Z^2)(X^2 - Z^2) + tY^3Z = 0.$$ The generic fiber ...
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1answer
76 views

Every point of Ext(C) lies on some tangent line to C

In our differential geometry class our professor gave the following problem as a homework problem: Let $ C $ be a smooth non-singular simple closed curve in plane. Prove that every point of the ...
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52 views

Integers characterizing singularities of algebraic curves

The problem in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field $\...
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404 views

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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610 views

What's so special about involute curves??

An involute curve (specifically, an involute of a circle) is very commonly used to define the shape of the teeth on a gear. Apparently this idea goes back to Euler. Why is this? What special ...
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200 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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424 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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135 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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1answer
48 views

bending an arc to accommodate a constraint

I'm working with piecewise polynomial spirals: curves of the form $z(t) = z_0 + \int_0^t e^{i f(s)} ds$ where $f$ is a quartic polynomial determined by the tangent angles and curvatures at given ...
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2answers
115 views

The shapes of general lemniscates (i.e., Cassinian curves) on the complex plane

On the complex plane, curves given by an equation of the form: $$ |z-z_1|\cdot |z-z_2| \cdots |z-z_n| = C $$ with $ C \gt 0$, are known as general lemniscates, or Cassinian curves with $n$ foci. I ...
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1answer
287 views

How to create animated plot of curve depending on parameter in WolframAlpha?

Is it possible to create in WolframAlpha an animated plot of a curve given by an equation, where some of the coefficients depend on the parameter (=on time)? For example if I would like to have a ...
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84 views

A nonplanar closed curve such that the plane curve with the same curvature as function of the arclength is not closed

Find a non plane, closed curve such that the plane curve with the same curvature as function of the arclength is not closed. Been thinking a lot in this problem and haven't got a clue. Any ...
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57 views

Tractable indefinite integral of the exponentiation of some function

Consider the function $z(s)\in\mathbb{C}$ defined as $z(s)=\int_0^s \exp\left[i(q u+\lambda(u))\right]du$ for some $q\in\mathbb{Q}-\mathbb{Z}$ and $\lambda(s)$ a $2\pi$-periodic real differentiable ...
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1k views

Equation for the Maximum-Area Rotor in a Wankel Engine

I am attempting to construct a specialized Wankel Engine in Autodesk Inventor. For my particular project, the rotor must take up the maximum area, in order to separate the air spaces of the top and ...
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812 views

How to estimate/determine surface normals and tangent planes at points of a depth image (point cloud)?

I have depth image, that I've generated using 3D CAD data. This depth image can also be taken from a depth imaging sensor such as Kinect or a stereo camera. So basically it is depth map of points ...
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70 views

Generalisation of circumscribed circle for polygones instead of triangles.

Given a triangle, there exists a unique circle passing through its $3$ vertices. I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the ...
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1k views

Definition of multiplicity of a point (in a plane curve)

In the book "Basic Agebraic Geometry I (third edition, 2013)" at page 14 Shafarevich says, about plane curves, what it follows: If $P=(0,0)$ and the leading terms (note:by leading terms I suppose ...
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53 views

Solving the algebraic equations .

I am working with an equation to find the singular points in $\mathbb P^2 (\mathbb C)$ . Basically after taking the partial derivatives and doing some manipulations it reduces to $$y^2 + (2-k)xz +(...
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875 views

Reconstructing a curve from its curvature

I'm with a problem in an exercise form Do Carmo's Differential Geometry of Curves and Surfaces; it is number 9 section 1.6. I have a differentiable real function $k(s)$, $s\in I$, and I need to show ...
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47 views

How do I use k-dimensional planes as bounds for generating k-dimensional vectors?

I am an essentially self-trained programmer with little mathematical background. I do not quite know where to start for this problem, and do not know the terminology to help me get conclusions ...
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1answer
143 views

Equivalence of focus-focus and focus-directix definitions of ellipse without leaving the plane

Take a look at the following two definitions of ellipse: For some fixed points $F_1,F_2$ and real number $2a>|F_1F_2|$ an ellipse is the locus of points $P$ such that $|F_1P|+|F_2P|=2a$. ...
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24 views

Computing integral when the derivative factors (separates)

I have a specific smooth function $F(x,y,z) = 0$ which implicitly defines a function $\widehat{z}(x,y)$. In my analysis of this function's properties, the term $$M(x,y) \equiv -\frac{\widehat{z}_y}{\...
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44 views

Formulations of Cauchy's theorem that don't seem consistent

So I am reading through the book Complex Analysis by Lars Ahlfors, and there is one point that is causing some confusion for me: In chapter 4.2 - Cauchy's integral formula, we first encounter the ...
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35 views

How does one find a parameter representation for bounded region?

I need help with this question. I have been stuck at it for a few days. My main problem is how I use the curve $K_r$ to find the parametric representation. I have a curve $K_r$ in the $(x,y)$-plane ...
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29 views

Prove that the curve $\alpha(t)$ is tangent to the $x$ axis.

I have the curve $\alpha(t):(-1,\infty) \rightarrow R^2$ given by $\alpha(t)= ((\frac{nat}{1+t^3}), (\frac{nat^2}{1+t^3}))$ with $n$ a natural an $a$ a constant both of them fixed. I need to prove ...
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26 views

Convex curve has convex interior

Let $c: \mathbb{R} \rightarrow \mathbb{R}^2$ be a simple closed curve with curvature $\kappa \geq 0$. Then the interior of $c$ is convex. I know that in this case $$ \langle N(t_0), c(t) - c(t_0)...
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1answer
82 views

When to parameterize to find equation of tangent plane and normal line?

I'm working through some problems asking to find the equation of a tangent plane and the normal line to the surface. I notice that some example questions parameterize the curve before solving and ...
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41 views

Show that $P(x,y)=0$ is a hyperbola if $b^2−4ac>0$ .

The following exercise is from Thomas Garrity's Algebraic Geometry: A Problem Solving Approach. I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax^2 + Bx + C$ where $A$...
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36 views

How to deduce Jordan Curve Theorem from Schönflies Theorem

Recently I started reading Ethan Bloch's "A First Course in Geometric Topology and Differential Geometry" and I came upon this exercise to deduce the Jordan Curve Theorem from the Schönflies Theorem: ...
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66 views

Finite Unions of Dendrites

I will ask the main question first, and then give the motivation for this one! The question is a bit specific, but seems to be the most general question to ask after handling some obvious ...
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47 views

Examples of smooth implicit curves and surfaces

I am currently constructing a method to approximate implicitly given plane curves and surfaces, which are smooth and single-sheeted. Now I have finished writing a Matlab function doing all the ...
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33 views

Expression of a form as a sum of powers.

I have a small question about one short sentence appearing in page : $376$ of the following electronic textbook : https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf The short ...
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49 views

Transform vectors from cartesian coordinates to curve coordinate system [2d]

I have an object moving in a two-dimensional space and its position is given by cartesian coordinates $(x_i, y_i)$. This object also has a velocity vector $({v_x}_i,{v_y}_i)$ and an acceleration ...
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35 views

How to slice a cone to get a given hyperbola?

Hyperbolas are made whenever a plane is normal to the radius of rotation. Which hyperbola is formed is dependent on the radius and a scaling factor. What radius R and scaling factor would be needed ...
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1answer
36 views

Function of curve length s in term of x

I am looking into a certain problem and decide to formulate it in a way that use the equation s=F(x) to describe a planar curve. Normally, we express the equation of a planar curve in the form y=f(x). ...
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1answer
38 views

On points with same $x$-co-ordinate on certain cubic curve

Consider the cubic curve $y=x^3+ax^2y+bxy^2+cy^3$. If $(t,r)$ and $(t,s)$ are two distinct points on the curve, then is it necessarily true that $t=0$ ?
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1answer
30 views

Question on the span of a tangent plane

I was reading this answer regarding the span of a tangent plane here. The answer says the graph of $f$ is also the graph of the map $F(x,y) = (x,y,f(x,y))$. The tangent plane is spanned by $(1,0,f_x)...
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92 views

Proving regularity and orthogonality of two curves

Let $\alpha(s) = (x(s)),y(s))$ be a regular plane curve that is parameterized by arclength, and let $n(s)$ be the normal vector and $k(s)$ be the curvature of $\alpha$. Consider the family of curves: ...
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1answer
116 views

image of parametric quadratic curve with three components contained in a plane

I am studying Differential geometry I tried to prove this by taking all three components as quadratic with $t$ as a parameter but could not be successful. If all three component functions of a ...
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43 views

Proving that strictly monotonic curvature implies no self intersections (more specifically, using the following inequalities)

Let $a(s)$ be a regular curve that is parametrized by arclength. Prove that, if the curvature $k(s)$ is a strictly monotonic function, then $a(s)$ has no self intersections. Suggestions: a) [will be ...