# 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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### 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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### 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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### 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 \det\begin{pmatrix} 0 & 1 & 1 & 1 & 1\\ 1 & 0 & ...
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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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### 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 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 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. $\frac{\partial}{dx}=4x$ $\Rightarrow$ $f_x(1)=4$ $\frac{\... 1answer 34 views ### For all unit vectors$\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}$; they form a surface ... 1answer 68 views ### Convexity criterion for piecewise regular planar curves A theorem of classical differential geometry states, that a simple, closed and regular$\mathcal{C}^2$-curve$\gamma:[0;1]\to\mathbb{R}^2$is convex, iff its signed curvature doesn't change sign. ... 4answers 161 views ### What is the circumference (arc length) of$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 ... 1answer 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 ... 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$\gamma=\alpha([a,b])$. ... 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 ... 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?... 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 ... 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 $$x(u,v) = \cos^2 (u) \cos (v) + \cos (u) \sin (u) \... 1answer 50 views ### How can I plot curves relative to other curves? Suppose I have a curve – say a sine curve y=sin(x) Now I want to draw a second curve, y = ½sin(x), but relative to the first – so it has the first one as a “baseline”. I don’t mean just adding the ... 1answer 70 views ### If three points on a quadric surface then the line going through them is contained in the quadric I am having trouble understanding a step in my Professor's Lecture notes She shows that Lemma 2.2.4 Let P_1,\ldots,P_5 be distinct points in \mathbb{P}_k^2. There exists a conic in \... 1answer 43 views ### Are there computational short cuts to calculating the distance from a large number of points to 3 different planes? I have three planes, and i want to calculate distance of my point to each of them. However, there are 68000 points in the space, so it does not make sense( computationally) to calculate the distance ... 0answers 79 views ### Logarithmic spiral characterized by signed curvature and arc length parameter. This is a homework problem I am having trouble with: Show that if a planar unit speed curve q(s) satisfies$$\kappa_s = \frac{1}{es+f}$$for constants e, f >0, then the curve is a logarithmic ... 0answers 40 views ### Involute of a Curve A string of length ℓ is attached to the point γ(0) of a unit-speed plane curve γ(s). Show that when the string is wound onto the curve while being kept taught, its endpoint traces out the curve ι(s) = ... 0answers 22 views ### Describe a curve by other than a fomula, fitting or interpolation I have a curve defined by a set of (x,y) given points. After ploting these points I get : My aim is to study the sensitivity of the operation that results these points. In order to do that, the ... 1answer 37 views ### Finding curves whose tangents intersect with the x-axis at (\frac{x}{2},0) I have to find the family of curves in \mathbb{R}^2 with this property: The tangent in an arbitrary point on the curve does intersect with the x-axis in (\frac{x}{2}, 0). I think I have to make ... 0answers 39 views ### Looking for rounded corner plane curve with certain properties (SIDESTEPPED) For a project involving simulating traffic lights, I am currently looking for a formula to get a rounded 90-degree corner (to describe the path of a turning car) with certain properties: Defined in ... 0answers 26 views ### advice for curve fitting I have numerically obtained some curves, corresponding with it I have also obtained some roots. I strongly believed these curves can be fitted with some (elliptic) functions taken the roots as ... 1answer 48 views ### What is my line integral answer incorrect? EDIT: Is my computation not correct, possibly because the parametrization that I used requires x,y to be on the xy-plane? If so, can I adjust from here, and not start over? I.e., is there some ... 1answer 65 views ### Prove it is not a closed Curve I wanna prove that (cos(t^3+t),sin(t^3+t))=γ(t) which is a reparametrization of a circle .Is not a closed curve like the circle. WHat i did is take this problem to the Complex plane so$$γ(t)--->... 0answers 23 views ### 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 ... 0answers 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 ... 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. ... 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 ... 2answers 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, ... 0answers 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) ... 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: $$\langle \alpha''(t),n(t)\... 2answers 175 views ### Help in showing that an evolute is the envelope of the normals to a curve Let \alpha:I\to R^2 be a regular parametrised plane curve (arbitrary parameter), and define n=n(s) and k=k(t) to be the normal and curvature respectively. Assume k(t)\neq0, t\in I. In this ... 1answer 78 views ### Find a vector function represented by the curve of intersection? I'm struggling with the following problem: Given \, z = \sqrt{x^2 + y^2}\, and \, z = y+1\, find the vector function represented by the curve of intersection of the surfaces using the ... 3answers 69 views ### Parametrization of Hyperbola I "know" that a parametrization of an Hyperbola (x^2-y^2=1) is given by:$$\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 ... 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 ... 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 ... 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,$(x,y)$, ... 0answers 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 ... 0answers 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 ... 0answers 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}}}+\...
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 ...