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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Unique solution of a simple functional equation

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in ...
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System of parametric equation [on hold]

Given : point $A(1,2,3)$ and line $\ell: x=m-1 ; y=-m ; z=2m-2 $ find system of parametric equation of line passing through $A$ and perpendicular to $\ell$.
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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

Planes And Lines

Given :Point $A(1,2,4)$ and plane $P: x-y+z+2=0$ How to find coordinates of point $A'$ the symmetric of point $A$ with respect to plane $P$
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Reparametrization with non-vanishing lateral derivatives

Let $\mathbf{r}:I\subseteq\mathbb{R}\to\mathbb{R}^2$ be a $C^{1}(I)$ function, non-constant on any subinterval, such that $\forall\ t_0\in I$, the following limits exist: $\lim\limits_{t\nearrow ...
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1answer
63 views

Curvature in $\mathbb{R}^2$

Let $f(t) = (x(t),y(t))$, not necessarily parametrized by arclength. We define the unit tangent vector, $T(t) = (1/|f'(t)|)(x',y')$. Also the normal vector, $N(t) = (1/|f'(t)|)(-y',x')$, which is ...
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25 views

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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Image of any curve can be parametrized without zero derivative [closed]

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a $C^{1} ([a,b])$ injective application. Prove that there is another continuous parametrization $\rho:[c,d]\to\mathbb{R}^2$ such that the following two sets are ...
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2answers
46 views

What's the name of this wavy curve?

What's the name of the curve you get from changing the x or y frequency on what was previously a path around an ellipse? The equation would be: f(t) = (Acos(ut), Bsin(vt)) And it looks like a wavy ...
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20 views

Intersection of point normal to plane

Hi if I have a point p (red dot) and I have a plane P (yellow) how do I find the intersection of the point normal to the plane? thanks! enter image description here
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94 views

Fourier series for convex plane curves.

The following problem is from Stein's Fourier analysis. This problem explores another relationship between the geometry of a curve and Fourier series. The diameter of a closed curve $\Gamma$ ...
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28 views

Showing a limit is equivalent to the curvature of a planar curve

I've been working on this limit for a long time and just don't know how to show this. I tried representing h and d as vectors, but I cannot seem to accurately describe them. What would be a good ...
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20 views

A 4th grade curve meets a line in one point with multiplicity 4

Suppose a 4th grade curve meets a line in one point with multiplicity 4. Example: the lemniscate $(x^2 + y^2)^2 = y^2 - x^2$ meets the line $x=y$ when the condition $x^4=0$ holds. This shows that ...
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38 views

Finding curves with special constraints. How do i verify the answer?

Let $O$ be a fixed point on the plane. Find all curves $\gamma : \mathbb{R} \to \mathbb{R}^2$ such that if $C(t_0)$ is the centre of the curvature of $ \gamma$ at $t_0$ then the angle $\widehat{ ...
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2answers
12 views

Finding the directional derivative parallel to an intersection of planes

We must find the directional derivative of $ f(x,y,z) = x^2 + 2xyz -y^2 $ at $ (1,1,2) $ in a direction parallel to the straight line $ \frac{x-1}{2} = y-1 = \frac{z-2}{-3} $ The straight line is an ...
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27 views

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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4answers
30 views

length of path travelled on $(t, \cos t, \sin t)$ from times $t = 0$ and $t = 2\pi$

Let the position of a particle in three dimensional space at time t be $(t, \cos t, \sin t)$. Then the length of the path traversed by the particle between the times $t = 0$ and $t = 2\pi$ is (A) ...
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2answers
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Equation of a plane that crosses the axes at points equidistant from the origin.

Give the equation of a plane that crosses the axes at points equidistant from the origin. Explain your reasoning. I know the equation should be on a 45 degree angle looking towards the axis. I have ...
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2answers
32 views

Finding a local parameterization of a plane curve

I'm attempting to find a parameterization of $\frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} = 1$. I find a tangent vector field: $X = \left( \frac{2x_2}{b^2}, -\frac{2x_1}{a^2} \right)$ (by taking the ...
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Determine the Winding Numbers of the Chinese Unicom Symbol

I'm practicing with Winding Numbers, and encountered an interesting example. You might be familiar with this liantong symbol, the logo of China Unicom: Suppose we make this into a fully closed and ...
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30 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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1answer
62 views

Find the condition for a center of a circle with exactly one lattice point on its circumference

Statement Find the condition for a center of a circle with exactly one lattice point on its circumference (this lattice point must not be the only one lattice point of the disk) What I have ...
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1answer
18 views

Is it possible to join a set of points using multiple circular arcs to get a smooth curve?

I have a set of points on a plane, and I want to join these points using a circular arc between consecutive points such that the final curve I get is smooth (no sharp edges). Is this possible? If so, ...
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26 views

Finding $y$ In Calculus(Area) Problem? [duplicate]

Find the number b such that the line $y=b$ divides the region bounded by the curves $y = x^2$ and $y = 4$ into two regions with equal area.
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find travel time given path and velocity field

As I was studying refraction, I began wondering what path would light take when entering a non-homogeneous transparent medium, i.e. a certain material in which the refraction index $n$ varies ...
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2answers
45 views

If $0 < \theta < \frac{\pi}{2}$, then $\gamma$ is a logarithmic spiral

Let $\gamma$ be a plane curve parametrized by the arc length, having the property that its tangent vector $T(t)$ forms a fixed angle $\theta$ with $\gamma(t)$. Before explaining where I am, let ...
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51 views

Can we classify plane cubics, What are they?

There are four qualitatively distinct pictures of the plane cubics. What are the polynomials corresponding to them? I know two of them have special names: nodal cubic and cuspidal cubic with ...
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2answers
32 views

Find equation of plane

I have to find the equation of the plane that is perpendicular to the line $\overline{l}(t)=(10, 0, 4)t+(6, -2, 2)$ and passes through $(10, -2, 0)$. We know that a plane that has a perpendicular ...
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1answer
44 views

Direction of outward normal differential $\hat n\ ds$

In vector calculus, why is the outward normal differential $\hat n\ ds$ around a closed curve $dy\ \hat i - dx\ \hat j$? Why isn't it $-dy\ \hat i + dx\ \hat j$ or something else?
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84 views

Prove $\int_{[a,b]}f=\int_{[a,c]}f+\int_{[c,b]}f$

Let $a,b\in\mathbb C$ and $c\in[a,b]$. Let $f$ be continuous on $[a,b]$. Use the definition to show that \begin{equation} \int_{[a,b]}f=\int_{[a,c]}f+\int_{[c,b]}f \end{equation} Note: You should ...
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1answer
17 views

Multivariable Calculus: Intersection of surface and planes

I have this math problem: Let $S$ be the surface that consists of all points $(x,y,z)$ that satisfy the equation $x^2+y^2=z^2$. 1) What are the intersections of $S$ with horizontal planes ...
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1answer
38 views

Do Carmo DG Q. 1.7.2 finding the arc with a given length which bounds the largest area

I'm struggling with the following question for long. I tried to apply isoperimetric inequality $4\pi A\leq L^2$, but my attempt has been unsuccessful. Could anyone give me a hint? Let $AB$ be a ...
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37 views

Alebgraic curve and Riemann surfaces

How do we prove that any smooth complex algebraic curve $C\subset\mathbb{P}^2$ is a Riemann surface? Does there exist a complex version of the implicit function theorem?
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1answer
13 views

Deriving tangent plane equation from scalar equation of plane

There's one step in the derivation that I don't understand. I'll explain some of the derivation and then explain which step I don't understand. Begin with scalar equation of plane: A(x-x0) + ...
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1answer
34 views

Arc length of curve (regular condition)

I have a question regarding the defintion of arc length of a curve in $\mathbb{R}^n$. If $\gamma$ is a regular curve, the define arc length as $S(t)=\int_{t_0}^t|\gamma'(t)|dt$. Since $\gamma$ is ...
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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 to show that unit normal vector is $n = \pm(ai+bj+ck)/(a^2+b^2+c^2)^{0.5}$
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1answer
35 views

(hyper) elliptic curve in characteristic two and the Jacobian criterion

Let $k$ be a field of characteristic two and let $E$ be a curve given by $$ y^2=x*(x+1)*(x^2+x+1)*(x^3+x+1)\quad\text{or}\quad y^2=f(x) $$ Now we have $dy^2/dy=2y=0$ and consider the Jacobian ...
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80 views

Is simple closed curve homeomorphic to a circle?

For sure every curve that is homeomorphic to a circle is a simple closed curve, but is every simple closed curve homeomorphic to a circle? Is there a proof for that, or is there some topological ...
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Determining the curvature of $y=Asin(bx)$

I am asked to determine the curvature of $y=Asin(bx)$. Unfortunately, I don't think I am on the right track. So the curvature of a curve is: $\kappa = \frac{1}{\lvert \vec{V}\rvert}\lvert ...
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1answer
39 views

Pull-back of regular map and rational function field

I don't understand what I'm missing in this example. Let $X=V(X_1^2+X_2^2-X_0^2)$ the circle in $\mathbf{P}^2_k$, being $k$ an algebraically closed field. Let be also $f:X\longrightarrow ...
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Embedding a $deg \geq 4$ curve in $\mathbb{P}^2$

Reading the paper by Kollar "The structure of algebraic threefolds", among some examples he does talking about intrinsic and extrinsic geometry over $\mathbb{C}$, he mentions that "an irreducible ...
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How to explain why a curve is on a cylindrical surface?

The question may be a bit general but I'm unsure about how to define it. I have a curve: $\vec r(t) = (2\cos(t),2\sin(t), 2t)$, for $0\le t \le 2\pi$, The problem I'm trying to pose : "Show that ...
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For which functions $r$ is the curve $r(t)(\cos t,\sin t)$ regular or unit speed?

Decide conditions on smooth function $r$ such that $\nu$ is a regular curve and decide functions $r$ for which $\nu$ has speed $1$. Let $r: \mathbb R \rightarrow \mathbb R$ be a smooth function ...
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Laplacian discretization for parametric curves

I know how to compute the discrete Laplacian of a graph and of a mesh (the Laplace-Beltrami operator). Is there an analogous definition for the computation of the Laplacian of a parametric curve ? ...
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26 views

Isomorphism from $\mathbb{R}^2$ to $\mathbb{D}$ such that lines become circular arcs

I'm currently working on a hobbyist math project that require taking lines on an infinite plane, and projecting them onto a finite (euclidean) surface such that intersections are preserved. Does ...
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1answer
38 views

critical point for the curvature does not correspond to a local maximum/minimum.

Draw an example where a critical point for the curvature does not correspond to a local maximum/minimum Does the curve for infinity sign satisfy this? I am having trouble seeing why it's true, if of ...
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1answer
49 views

Parameterizing the path of a point on a circle rolling on another circle

Problem: A wheel of radius $a$ rolls on the outside of a circle with radius $b$ (see figure). Find the parameterization for the curve a point on the wheel follows. You may choose freely how you ...
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32 views

The Point of Tangency Between a Sphere and a Tangent Plane

Find the equation of the sphere centered at (2,0,-3) that is tangent to the plane x=y. What is the point of tangency? Describe the interior of the sphere with an inequality. What I have thus far: ...
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Points at infinity correspond to asymptotic slopes

Let $ P^2\mathbb{C} = \{ [a, b, c] | a,b,c \in \mathbb{C}^* \} $ the complex projective plane. So $ [a,b,c] \sim [x,y,z] $ iff $ \exists \lambda \in \mathbb{C}^* \colon \lambda(a,b,c) = (x,y,z) $. In ...
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The Cusp $w^2 + p(z,w)=0$ is desingularizable in the origin $O \in \mathbb{C}$

I have just studied a method in projective geometry over complex numbers on how to desingularize a curve in a point but i'm a little bit confused. I don't know the name of this classical method in ...