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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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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1answer
77 views

Collinearity of triangle vertices on circular paths

Suppose as a function of time, the vertices of a triangle move with constant speed along circular trajectories: $$ \vec p_i(t) = a_i \left( \begin{array}{c} \cos(b_i t+ c_i)\\ \sin(b_i t+ c_i) \end{...
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1answer
44 views

The plane curve of maximum average speed under constant gravitational force

A line gives us the minimum distance from $A$ to $B$. A cycloid gives us the minimum traveling time of a point mass from $A$ to $B$ (under constant gravitational acceleration $g$). What about the ...
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12 views

Measure a pyramid or prism [closed]

Is it more challenging to measure a pyramid or prism? Discuss, making clear what attributes you are measuring, and whether you are determining measurements physically or through calculation. My ...
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33 views

Convex, closed plane curve is a Jordan curve

The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it'...
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122 views

3-D Geometry Problem. Find a curve which touches the straight line.

If two perpendicular tangent planes to paraboloid $x^{2}+y^{2}=2z$ internsects in a straight line in the plane $x=0$, obtain the curve to which the straight line touches. I don't know how to ...
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2answers
16 views

What is the definition of a path along a multivariable function?

I'm taking a class equivalent of Calculus III, and we saw how to prove continuity of a multivariable function. Recently we looked at the following example: \begin{align} f(x,y) = \begin{cases} \frac{...
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1answer
21 views

Find volume of cube with the help of eqn of plane

The volume of cube whose two faces lie on the plane 6x-3y+2z+1=0 and 6x-3y+2z+4=0?
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36 views

Level curve of the function $f(x,y)=\min\{x^2+y^2,xy\}$ [closed]

How I can find the level curve of the function $$f(x,y)=\min\{x^2+y^2,xy\}$$ From where I need to start to solve this problem? Thank you!
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44 views

Injective curve

How can i show that the curve $\gamma:\mathbb{R}\to\mathbb{R}^2$ defined by \begin{equation} \gamma(t)=\left(\frac{t}{1+t^2},\frac{t}{1+t^4}\right) \end{equation} is injective (without using algebraic ...
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1answer
38 views

Does there exist a curve with non zero area?

Does there exist a curve with a bounded infinitely diffrentiable derivative (i.e. has a minimum |curvature|) of hausdorf demension 2? Or even a diffrentiable curve of hausdorf demension 2?
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1answer
22 views

Path from a start point at a certain heading to an end point at a certain heading while obeying a minimum turn radius

So not sure if my title is clear (also no idea what to tag, because you need 1000 rep to add tags) so I will try my best to explain the problem. I'm working in 2D space and to simplify the problem, I'...
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2answers
71 views

Index of a Jordan curve

Winding number theorem: If $J\subset \mathbb{C}$ is a Jordan curve and a point $z$ lies in its interior domain, then the winding number $n(J,z)=\pm 1$. Now suppose that $J$ is smooth and we have the ...
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1answer
44 views

How to prove this inequality about the arc-lenght of convex functions?

Let be $f,F:[0,1] \to \mathbb{R}$ with $f,F \in C^2([0,1])$ two convex functions such that $f \le F$ in all points and $f(0)=F(0)$, $f(1)=F(1)$. Considering the plane-curves $\gamma(t)=(t,f(t))$ and $...
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36 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 ...
6
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1answer
127 views

converse to the jordan curve theorem

Suppose $K\subset \mathbb{R}^2$ is compact and locally connected, and does not contain 0. Let $A$ be the component of $\mathbb{R}^2-K$ containing 0, and let $B$ be the unbounded component. Assume ...
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1answer
32 views

find a closed curve with given winding numbers

Is there a closed (piecewise) $C^1$ curve $\gamma: [a,b]\rightarrow\mathbb{C}$, so that $\mathbb{C}\setminus\gamma([a,b])$ has four components with winding numbers -1, 0, 2, 3? Thanks!
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34 views

Which curve (surface) is this?

We're having trouble fitting our data... well, we don't even know which function we should fit onto. Anybody knows if this function is well defined mathematically?
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35 views

A curve with Lebesgue measure non zero

In this Continuously Differentiable Curves in $\mathbb{R}^{d}$ and their Lebesgue Measure the domain of the curve is a compact set. I want know if the same answer holds for curves with non-compact ...
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1answer
24 views

Log power rule problem

According to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian ...
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35 views

Roots of the Taylor approximation of the exponential

While answering another question, I looked at the roots of the $n^{th}$ degree Taylor approximation of the exponential. $$e^x\approx E_n(x):=\sum_{k=0}^n\frac{x^k}{k!}.$$ Apparently, these root are ...
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Parametric version of a simple equation

I have a simple relation that I need to plot in a plane. I could do it, but I believe that I don't get the best way. A plane curve is defined implicitely by the following equation : \begin{equation}\...
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25 views

A space curve with non-vanishing curvature is planar iff its torsion is 0

Intuitively this is simple and to prove the backwards direction: $\tau = 0 \Rightarrow \mathbf{b'}=0 \Rightarrow \mathbf{b} $ is constant. Then letting $\gamma$ be the parametrisation of the curve ...
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11 views

Exercise concerning areas inside closed curves

Let $\alpha (s)$, $s\in[0,l]$, be a closed, convex, plane curve with $\kappa >0$. Let $r$ be a positive constant and define $\beta (s)=\alpha (s)-rn(s)$, where $n(s)$ is the normal vector of $\...
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1answer
61 views

Find the intersection of plane and sphere

If the equation of the sphere is $x^2+y^2+z^2=1$ and the plane is $x+y+z=1$, then how can the equation of a circle be determined from the equations of a sphere and a plane? and what is the parametric ...
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1answer
49 views

Intersection of sphere and ellipsoid

Ellipsoid: $$ x^2+\frac{y^2}{4}+\frac{z^2}{2}=1 $$ Sphere: $$ x^2+(y-1)^2+(z-d)^2=1 $$ For what values of $d$, there is a common tangent plane to both curves? Part of my resolution: Consider $f(x,y,...
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17 views

Intersection of sphere and plane is a point, find c, such that (0,0,c) is center of sphere

Sphere: $$ x^2+y^2+(z-c)^2=1 $$ Plane: $$ x+2y+3z=0 $$ Find the values of $c$, for which the intersection of the sphere and the plane is a point. Well, I know that the sphere has the center (0,0,c) ...
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68 views

Graph theory: creating surfaces

If we draw a circle in the plane , we separate the plane into 2 regions , inside and outside. On the following surfaces, determine the maximal number of circles that you can draw on the surface ...
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1answer
18 views

Finding the radii of an ellipse from the intersection of a plane and a sphere

I'm trying to solve the following problem, regarding Stokes Theorem: $F = z i + xj + yk $; C the curve of intersection of the plane $x + y + z = 0$ and the sphere $x^2 + y^2 + z^2 = 1$ [Hint: ...
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46 views

Dual curve of an algebraic curve in affine coordinates

$F$ is an irreductible algebraic curve and we consider the application that sends a non-singular point $(a : b : c)$ in homogenius coordinates, to $\phi(a, b, c) = \bigl(f_x(a, b, c), f_y(a, b, c), ...
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2answers
51 views

Closed subset of $\mathbb R^2$ disconnecting $\mathbb R^2$ must contain a curve?

Edit: Per Alex S's answer, my first statement was false, so I've moved it to the bottom. Here is the weaker statement that I am trying to prove (this was included in my original question): Let $C \...
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1answer
29 views

How to draw cubic plane curve?

In Python, using MatPlotLib, given [vector] parameters $a$ and $b$ and [scalar] parameter $c$, I want to draw a general cubic plane curve in 2-dimensional space (regular plane with $x$ and $y$ axes): $...
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75 views

Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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12 views

Curve fitting: How to identify the appropriate function for a beat-like phenomena?

I have a time series data which shows some beat like behaviour. The envelope does not look exponentially decreasing, as it is impossible from a physics point of view. The envelope is likely to be a ...
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1answer
70 views

convex curve as boundary of a convex set

In short, my question is to know if the following statement is true, and how to show it : Theorem Let $\gamma$ be a closed simple $C^2$ convex curve in $\mathbb{R}^2$. We denote $\Gamma$ its image. ...
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1answer
39 views

What is the rotation index of a figure 8?

Is it 0 since the total turning angle covers one clockwise circle and one counterclockwise circle thus making the total 0 and the rotation index 0?
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74 views

Proof of the formula that computes the genus of smooth projective plane curve

I was searching for a proof of the formula that computes the genus of a smooth projective plane curve of degree $d$: $$g = \frac{(d-1)(d-2)}{2}$$ which do not make use neither of triangulation or ...
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2answers
24 views

Dichotomy in the number of regions on a plane formed by an infinite number of lines

I'm reading Knuth's Concrete Mathematics and we are dealing with recurrence relations. He proves that the number of regions $L_n$ formed by $n$ lines on a plane is $L_n=\frac{n(n+1)}{2}$. I don't ...
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1answer
24 views

Existence and uniqueness of a point with horizontal tangent in a convex curve

I had a look on the proof by E. Schmidt of the Schur's Theorem about arcs of convex curves. It states the following: Let $C$ and $C'$ be two arcs of the same length with the endpoints $a,b,a',b'$ ...
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28 views

Normal and tangent vectors to a curve

Assume we have a displacement field $u=u(x,z)=(u_x(x,z), u_z(x,z)))$ which takes the point $R=(x,z)$ to $r=R+u=(x+u_x(x,z), z+u_z(x,z))$. We want to find the tangent and normal to the curve which was $...
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21 views

Can someone explain the detail of the proof (rectifiable curve)

Curve $C$ in the plain, \begin{align} C:= \begin{cases} x=\phi(t) \\ y=\psi(t) \end{cases} \quad, \quad a \le t \le b \end{align} The graph of C is $\{(x, y): x=\phi(t), y=\psi(t), a\le t\le b\}$ $\...
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65 views

Curve equidistant to sine and cosine.

If I have the sine and cosine curves plotted, what would be the formula of the curve that is equidistant to both curves? Here's a picture of how it looks like. The original question comes from a ...
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47 views

Understand if a curve is parametrized by arc length or not

Show that the curve $$\alpha(t)=(t,1+\frac{1}{t},\frac{1}{t}-t), \quad t\in(0,\infty)$$ is a plane curve. I know $\tau$ must be zero for curve being plane. However, I want to determine the ...
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1answer
20 views

3D Denjoy–Riesz theorem

The Denjoy–Riesz theorem states that every totally disconnected subset of $\Bbb R^2$ is the subset of a Jordan arc. Is this true in $\Bbb R^3$? Originally I thought Antoine's necklace would be a ...
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Find the highest point on curve defined by intersection of the graph of $g(x,y) = \sqrt{xy}$ and plane $x+y-1=0$

So far this is what I have done: $$F(x,y) = \sqrt{xy} + λ(x+y-1) =0$$ $$F_x = \frac12(xy)^\left(\frac{-1}{2}\right).y + λ=0$$ $$F_y = \frac12(xy)^\left(\frac{-1}{2}\right).x + λ=0$$ $$Fλ = x+y-1=0$...
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38 views

use of wolfram in determining area between two curves

I am new to the use of Wolfram (that for the limited cases I have used is very impressive). However I wonder if anyone can tell me what I am doing wrong. I wanted to calculate the area between the two ...
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Can every possible curve be expressed mathematically [closed]

Can every possible curve/parabola shown on a graph (for example $x^2$ or somthing much more complicated) be expressed in an equation like $y=x^2$. Or are there some lines you can't express?
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96 views

Prove this : $\left(a\cos\alpha\right)^n + \left(b\sin\alpha\right)^n = p^n$

I have this question: If the line $x\cos\alpha + y\sin\alpha = p$ touches the curve $\left(\frac{x}{a}\right)^\frac{n}{n - 1} + \left(\frac{y}{b}\right)^\frac{n}{n - 1} = 1$ then prove that $\left(a\...
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32 views

On a formulation in Hilberts original paper about the space-filling Hilbert curve

I have a question on the famous paper Über die stetige Abbildung einer Linie auf ein Flächenstück (which translates roughly as On the continuous mapping of a line onto a square) by D. Hilbert. Let the ...
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1answer
55 views

Is there a plane filling function calculator online?

I recently read about the "Hilbert Curve" and found it very interesting. Does anyone know of a place online where I could extrapolate different shapes and explore this field of mathematics?