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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Lattice Points in x-y plane

What are Lattice Points? Which points in x-y planes are Lattice Points? Is (m,n) a lattice point where m,n are any integers?
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36 views

Two definitions for a smooth curve equal.

I've encountered these two definitions: 1. $\gamma\colon [a,b]\longrightarrow\mathbb{R^3}$ is smooth if all three derivatives exist and $\gamma^{\prime}(t) \neq 0 \;\; \forall t \in [a,b]$ ...
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12 views

Difference between containing point and pass through point?

I do not understand this, What is the difference between the equation of the plane containing the points and the equation of the plane through the point? Is it the same thing or are they different?
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54 views

injective curve inside curve

I am struggling to prove the following intuitive result: Take $\phi:[a,b]\rightarrow \mathbb{R}^{n}$ a continuous mapping with $\phi(a)\neq\phi(b)$. Then there is a continuous injective mapping ...
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3answers
334 views

How to find the length of a curved path.

We have to find a continuous model for a curved path which you then solve. A woman is running in the positive y-direction starting at x=50 (50,0) which is orthogonal to the x axis. At this point a dog ...
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1answer
63 views

Paths followed by Morley triangle vertices as apex moves parallel to base

Let the vertices of a triangle $T$ be $(A,B,C)$, and $(a,b,c)$ the vertices of its Morley triangle $M$. Designate vertex $C$ as the apex of $T$. Now move apex $C$ parallel to $AB$, all the while ...
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328 views

Understanding the Spiro Spline

My name's Wray. This is my first time here. Firstly, I like curves. I've been keeping a pet project for a long time that would implement a delightful new curve-interpolation algorithm named the Spiro ...
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1answer
35 views

convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane: For a curve $r(t)=(x(t),y(t))$ 1.The whole curve lies on only one side of any tangent of the curve 2.Any ...
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1answer
55 views

How to estimate curve length from random points along a (possibly not connected) curve

I'm working with a closed, smooth, planar curve. I don't know the curve exactly, but I do have an effectively-random collection of points along the curve. What's a good way to estimate the curve ...
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35 views

Euclidean space and vector field

Can someone explain me what a Euclidean space is? and more detailed what a vector field is? Or a continuous vector field
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43 views

Why is $\sum^{n}_{i=1} \int_{t_{i-1}}^{t_i} = \int_{a}^{b} $?

This is a part of my proof: $$\begin{align} \left| \sum^{n}_{i=1} V(r(\tau_i)) \cdot \int_{t_{i-1}}^{t_i} r'(t) dt - \int_{a}^{b} V(r(t)) \cdot r'(t) dt \right| &\leq \sum^{n}_{i=1} ...
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30 views

Intersection of planes

A line perpendicular to the plane $ 3x-5y+4z-11=0 $ passes through the origin. At what point does this normal intersects the plane?
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finding the line of intersection

Find the line of intersection between two planes x+y+z=1 and x-2y+3z=1 ? I found r1,r2,n1 and n2 but I don't know what are the other steps
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1answer
70 views

Birrational curves and singularities [closed]

If $C$ and $D$ are two birrational plane curves. Is there some relation between their singular points?
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1answer
125 views

Is this variant of the Jordan Curve Theorem true?

This feels as though it should be falsifiable, but it's not immediately obvious to me. The informal version of the statement is 'for every non-intersecting curve between two opposite corners of a ...
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1answer
118 views

How do I turn a “broken” plot into a smooth curve

I developed and solved a differential equation that predicts fluid temperature along the length of a long pipe with time. Analytical solution is such that it is causing a "discontinuity" in the ...
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0answers
23 views

What is the meaning of “slope of ca”?

I'm reading a paper, when this article refers to the function: $$\beta(v)=\frac{(\frac{v}{I})^k}{1+(\frac{v}{I})^k}$$ It say that "around the $I$, $\beta$ is approximately linear in $I$, and has a ...
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5answers
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Ambiguous Curve: can you follow the bicycle?

Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
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0answers
130 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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76 views

Harris Exercise 5.13 (points in general position)

I have a question about the second part of Exercise 5.13 in Harris' Algebraic Geometry: A First Course: Given $n \leq 2d + 1$ points in $\mathbb{P}^2$, characterize the subset of $(\mathbb{P}^2)^n$ ...
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87 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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0answers
33 views

equiaffine arc length, moving frame, and affine curvature

I am trying to learn affine geometry, and I'm having some trouble getting started with the following problem. Compute (a) the equiaffine arc length, (b) the moving frame, and (c) the affine ...
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1answer
363 views

arc length parameterization of planar curve in Matlab

Let $\gamma (t)$ be a planar curve parameterized by time $t$. For fun, let it be a limacon. In Matlab $\gamma (t)$ looks like this. ...
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2answers
27 views

Lengths of Plane Curves - Calculus 2: $\sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2}$

$$ \sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2} $$ I am having problems setting this up. Taking the derivative of $\sqrt{1-x^2}$. Leaves me with: $$ \frac{1}{2}\left(1-x^2 ...
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3answers
779 views

A hyperbola as a constant difference of distances

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, the distance between the two ...
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1answer
28 views

Curvature of the boundary curve of convex set

I've got a very simple question, but I can't get a rigorous proof. Suppose that $X\subset\mathbb{R}^2$ is a convex set, and suppose further that $\gamma$ is a closed regular curve with image $\partial ...
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2answers
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Find the area of the parallelogram with vertices (4,1), (6, 6), (7, 7), and (9, 12).

I am trying to find the area of the parallelogram with vertices (4,1), (6, 6), (7, 7), and (9, 12). So I believe the way to solve this problem is through the cross product and then taking the ...
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1answer
68 views

Do line integrals of non smooth curves exist?

Wolfram says that the theorem of conservative fields is : The following conditions are equivalent for a conservative vector field on a particular domain $D$: For any oriented simple closed ...
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2answers
65 views

Finding singularities of a projective curve

For $w \in \mathbb{C}$ we define the projective curve $$p(x,y,z):= x^3+y^3+z^3+wxyz.$$ Now I have to find all $w \in \mathbb{C}$ for which the projective curve $p(x,y,z)$ is singular and show that ...
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1answer
60 views

How can I find a curve based on its tangent lines?

Let's say for some curve its tangent lines at every point have a property that the length of a segment within the first quarter $[0;+\infty)^2$ is exactly $C>0$. How can such a curve be defined ...
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36 views

$r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$

What does the following equation represent? $r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$ My approach: I factorized the equation and it became $(a+r\cos\theta)(a-r)=0$ I feel that ...
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51 views

Green's Theorem and Divergence (2D)

I am reading the book Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson. In Chapter 1 he talks about the Possion Equation, and to prove that FEM ...
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1answer
33 views

Determing if a parametric curve is smooth

I have to determine whether the following curves are smooth or not and I'm having trouble with the following two functions: Consider $f(t) = (t^{2}-1,t^{2}+1)^{T}$ The solution states: $f'(t) = ...
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1answer
159 views

How can a simple closed curve not look locally like the rotated graph of a continuous function?

A simple closed curve is a continuous closed curve without self-intersections. The question of whether you can inscribe a square in every simple closed curve is currently an open problem, but this ...
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1answer
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Manipulating An Equation into A Workable Form

The question asks me to find the arc length of $$y= (x-x^2)^{1/2} + \sin^{-1}(x^{1/2})$$ I know I need to take the derivative: $$\frac{1-2x}{2(x-x^2)^{1/2}} + \frac{1}{(1-x)^{1/2}}$$ I've tried ...
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1answer
77 views

GNU Octave draw figure of 2 planes

How can I draw two planes in same figure in GNU Octave? $$ x + y + z = 1\\ 2x - y + 3z = 4$$ Thanks!
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2answers
161 views

Curves with “constant speed”?

I am new to the concept of curves. Let us a assume we have a simple function such as $f:\mathbb{R}^+\rightarrow \mathbb R^+\quad f(x) := \sqrt{x}$. (Or $f(x)=\exp(x)$ or a polynomial etc.). We can ...
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1answer
52 views

The area of a region around a curve

If we are given a simple closed curve with length $L$ in the plane, and we have a fixed number $r$ such that for each point $x$ on the curve there is a related disc $D(x,r)$ with radius $r$, how can ...
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35 views

A function to fit a certain S-shaped curve

I am looking for a function to fit a certain type of S-shaped curve. Here are my criteria: The curve always pass three points (0,0), (0.5,0.5) and (1,1). For 0 < x < 0.5, f(x) < x; for ...
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24 views

Could a hyper elliptic curve of degree 5 admit only three ordinary double points?

The definition for hyper elliptic curves is those curves which admit a ramified double cover of $P^1$. The given degree five is for some homogeneous equations of degree five that defines the curve, ...
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1answer
156 views

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 ...
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148 views

s-shaped reverse logistic curve

Is there any curve that grows very slow at the beginning then growth picks up exponentially before hitting the wall. I need sort of reverse behavior of the logistic curve ...
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2answers
54 views

Strange property of parametrization of a class of plane curves

My studies lead me to the following parametrization of perhaps a new class of plane curves ( which are similar in shape to the classical sinusoidal spirals but not identical ). If the curves are not ...
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61 views

Creating a $C^2$ smooth convex curve which joins circular arcs

Suppose that I have many points $p_1$,..., $p_n$, all very close to origin 0. For R very, very large, I have circular arcs of radius R, $\alpha_i$, about $p_i$. I do not speak about their ...
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Limit of an Expression on a Plane Curve

Say we have a curve $\mathcal{C}$ like this: $$ x^5 + y^5 = 5xy.$$ Say then that we want to find (and prove) the limit of the quantity $x/y$ (if it exists) as $x \to + \infty$ on the curve ...
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Limit of an expression as $x$ tends to a particular quantity on a curve

Say we have a curve like this: $$ xy + y^2x^2 = x.$$ Let's call it the curve $\mathcal{C}$. Say then that we want to find the limit of the quantity $x^2/y$ (if it exists) as $x \to - \infty$ on the ...
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71 views

Finding an arc-length between 2 points in 3 dimensions

I know how to find an arc-length between two points with coordinates, say $X=(a,b)$ and $Y=(c,d)$. But how do I find the same thing but for, say $X=(a,b,c)$ and $Y=(d,e,f)$? Thanks!
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Degree of ramification divisor and number of fixed points under a group action.

Let $C$ be a projective plane nonsingular complex curve and a finite group $G$ acts on $C$. Consider the quotient $f: C\rightarrow C/G=:C'$. Then, by Riemann–Hurwitz $2g_C-2=|G|(2g_{C'}-2)+\deg R,$ ...
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Parameterization of simple closed curve

A curve $Z$ in a two dimensional space is parametrized by $0\leq t < 1$ , and satisfies $Z(t) = Z(t+1)$. If it is sufficiently well behaved, it can be represented using a Fourier series with a ...
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AP Calculus Integral Problem

A solid is formed by revolving the curve $y=x^{2/3}+1$, for $0 \leq x \leq 2.5$, about the $X$-axis Estimate the volume of the solid by partitioning $[0,2.5]$ into five sub-intervals of equal length, ...