Questions tagged [curves]

For questions about or involving curves.

50
votes
5answers
11k views

Is it possible to plot a graph of any shape?

In school, I have learnt to plot simple graphs such as $y=x^2$ followed by $y=x^3$. A grade or two later, I learnt to plot other interesting graphs such as $y=1/x$, $y=\ln x$, $y=e^x$. I have also ...
6
votes
3answers
4k views

Curves with constant curvature and constant torsion

Describe all curves in $\mathbb{R}^3$ which have constant curvature $κ > 0$ and constant torsion $τ$. Any ideas what we can do to describe all such curves? Do we have to use the formulas of the ...
0
votes
1answer
964 views

Proving that second derivative is perpendicular to curve

How can I prove the following? $\gamma (t)$ is unit speed, $\dot \gamma(t) \not= 0 \Rightarrow \ddot \gamma(t)$ is perpendicular to $\gamma(t)$ I don't really see where a problem would arise when $\...
23
votes
4answers
826 views

Prove that the boy cannot escape the teacher

I'm struggling with the following problem from Terence Tao's "Solving Mathematical Problems": Suppose the teacher can run six times as fast as the boy can swim. Now show that the boy cannot ...
16
votes
4answers
529 views

Can the boy escape the teacher for a regular $n$-gon?

This is related to Prove that the boy cannot escape the teacher Suppose there is a boy in the center of a regular $n$-gon. The teacher is on the edge of the $n$-gon (but cannot leave the edge) and ...
7
votes
3answers
191 views

A continuous curve intersects its 90 degrees rotated copy?

This is almost the same problem as in this question. However, the OP there was looking for a solution where we could assume any number of things, while I want to stick with just the given assumption (...
6
votes
3answers
240 views

Will a path between $(x, y)$ and $(-x, -y)$ always intersect a 90 degree rotated copy?

Suppose we have a path between two points $(x, y)$ and $(-x, -y)$. If we rotate it by 90 degrees around the origin, will the copy intersect the original? (You can add any number of assumptions to ...
24
votes
2answers
38k views

What is parameterization?

I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up. On Wikipedia it says: Parametrization is... the process of ...
14
votes
1answer
303 views

Fitting a closed curve on the roots of ${x \choose k}-c$

Let $${n \choose k} = \frac1{(n+1) \operatorname{B}(n-k+1, k+1)}.$$ be the generalized binomial coefficient, and here $\operatorname{B}$ is the Beta function. Let $f_{k,c}(x)$ be the following ...
3
votes
2answers
2k views

What equation will create a 3D rose curve?

The parametric equation $x=a\cos(bt)\cos(t)$, $y=a\cos(bt)\sin(t)$ where $a$ & $b$ are constants and $t$ is parameter gives a rose curve which looks like, On a similar basis, is there a equation ...
4
votes
2answers
2k views

deriving the formula of the torsion of a curve

in our class we defined the torsion $τ(s)$ of a curve $γ$ parameterized by arc length this way $τ(s) = B'(s) \cdot N(s) $ where $B(s)$ is the binormal vector and $N(s)$ the normal vector in many ...
4
votes
3answers
329 views

Parametrization of the intersection between a sphere and a plane

I can't find a way to get the parametric equation $\gamma(t)=(x(t),y(t),z(t))$ of a curve that is the intersection of a sphere and a plane (not parallel to any coordinate planes). That is $$\begin{...
1
vote
1answer
127 views

Tangent to Cubic curve has positive slope.

This is in continuation to my earlier post at : How many times tangent to a cubic curve $y = x^3$ from a point on it, meets again at another point. A brief summary of the previous post : If take a ...
1
vote
1answer
103 views

Space filling curve

Let $C$ be the Cantor set. Then the Cantor function $f:C \to [0,1]$ can be extended to $F:[0,1]\to [0,1]$ linearly as the end points of an removed interval takes the same value. For example $f(1/3)=f(...
1
vote
0answers
181 views

Derivative of B-spline basis functions for degree 2

Lets first derivative of $N_{i,p}(t)$ (i-th Bspline basis function) is as follow: $N'_{i,p}(t)=\frac{p}{t_{i+p}-t_{i}}N_{i,p-1}(t)+ \frac{p}{t_{i+p+1}-t_{i+1}}N_{i+1,p-1}(t)$ Now Let's consider a ...
1
vote
1answer
134 views

Nonsingular projective curve model corresponding to $y^2 = x^4+1$

Consider the affine curve $C_1 = V(y^2 - (x^4+1)) \subset \Bbb A^2_k$. In the answers to this question, they claim that there is a (unique?) nonsingular projective curve $C_2$ corresponding to $C_1$ (...
4
votes
3answers
593 views

Why is the Koch curve homeomorphic to $[0,1]$?

Henning Makholm has provided a nice proof that the limiting curve is a continuous function from $[0,1]$ to the plane. I was curios if the function is homeomorphism. A quick search gave me many sources ...
3
votes
3answers
1k views

Minimum distance between the curves $f(x) =e^x$ and $g(x) =\ln x$ [closed]

What is the minimum distance between the curves $f(x) =e^x$ and $g(x) = \ln x$? I didn't understand how to solve the problem. Please help me.
2
votes
1answer
93 views

Why do we need that $\alpha$ is for regular to existence of arc length as integral?

Take $\alpha:I\rightarrow R^n$ a regular curve. For $t_0\in I$ Define $$s(t)=\int_{t_0}^t|\alpha'(x)|dx$$ as the arc lenth. In Differential Geometry of Curves and sufaces of Do Carmo is written ...
1
vote
1answer
153 views

Problem using Parametric Equation of Semicubical Parabola

I've been working my way through an old A'Level maths book and am having a lot of difficulty with a problem given in the Chapter on Loci & Parametric Equations: "Find the equation of the tangent ...
0
votes
0answers
27 views

Unclear equation with partial and total derivatives

Suppose $M$ is a manifold and $f:M\to \mathbb{R}$ a smooth function. Also let $\alpha :I\to \mathbb{R}^n$be a representation of a curve and $(\phi,U)$ be a chart. Now why it holds that $$\frac{\...
-1
votes
3answers
9k views

Create a formula that creates a curve between two points

We have two points, $A$ and $B$. The difference in $x$ is 1 unit, and the difference in $y$ is arbitrary. For each point we also know the gradient. First, I want to draw a smooth line that connects ...
20
votes
5answers
557 views

Area enclosed by an equipotential curve for an electric dipole on the plane

I am currently teaching Physics in an Italian junior high school. Today, while talking about the electric dipole generated by two equal charges in the plane, I was wondering about the following ...
17
votes
1answer
132 views

Tracing a curve along itself - can the result have holes?

Let $\varphi:[0,1]\to\Bbb R^2$ be a continuous curve (not necessarily injective) with $\varphi(0)=(0,0)$. Let $f:[0,1]^2\to\Bbb R^2$ be defined as $f(s,t)=\varphi(s)-\varphi(t)$. Question: Is the ...
37
votes
5answers
4k views

Helix with a helix as its axis

Does anyone know if there is a name for the curve which is a helix, which itself has a helical axis? I tried to draw what I mean:
15
votes
4answers
242 views

Can every curve be subdivided equichordally?

This question build on top of this other question: Dividing a curve into chords of equal length, for which I wrote an (incomplete) answer. I got the feeling we might need some help from a real ...
7
votes
1answer
640 views

Orthogonal differentiable family of curves

This problem is out of section 4-4 in M. do Carmos' Differential Geometry of Curves and Surfaces: We say that a set of regular curves on a surface $S$ is a differentiable family of curves on $S$ if ...
1
vote
1answer
545 views

What type of curve does Photoshop's Curves use?

I'm trying to figure out what kind of curve Photoshop's Curves interface uses. Here are some examples: I'm trying to recreate the effect using D3.js and SVG. None of the curves available in D3 (cubic ...
5
votes
1answer
124 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\...
8
votes
6answers
1k views

What equation can produce these curves?

Does an equation exist that can produce the curves shown in the attached image, by varying a single variable?
2
votes
0answers
91 views

a function $f$ is differentiable in $\vec{0}$ if $f \circ \gamma $ is differentiable in 0

Please help me solve this question: let $f:R^n \rightarrow R$ be a function. for all differentiable curve $\gamma: [-1,1] \rightarrow R^n$ such that $\gamma(0) = \vec{0} \space$, $f \circ \gamma :[-...
2
votes
1answer
685 views

Proving a few properties of Bertrand curves

Here's what I've got so far (and I'm assuming $\alpha$ is a unit speed curve): a) The fact that $\beta(s) = \alpha(s) + r(s)N(s)$ for some scalar function $r$ follows trivially because of the fact ...
2
votes
4answers
768 views

Intuitive definition for curvature

I was reading about curvature of a curve but I didn't understand it. I'm looking for an easy definition and also I want to know is there any general formula for finding curvature of the all curves ? ...
2
votes
1answer
830 views

A curve of constant curvature and zero torsion must be a circle

From Elementary Differential Geometry by Pressley I don't understand the last paragraph. Why does it show $\gamma$ lies on the sphere $\mathcal S$ with center $\mathbf a$ and radius $1/\kappa$?
1
vote
1answer
347 views

Torsion of an asymptotic curve with nonzero curvature

I wish to solve the following problem using the matrix of the shape operator $S_{P}$. Suppose $K= \text{det} S_{P} <0$, and $C$ is an asymptotic curve with curvature $\kappa (P)$ nonzero. I want to ...
-4
votes
1answer
769 views

How can I get smooth curve at the sigmoid function?

I'm trying to implement the sigmoid curve by using the following function. A is 3.2505508013 B is 1.5223545069 and K is 0.56. $\left(\frac{\left(\frac{\left(\left(2\left(\sqrt[A]{xx}\right)-1\right)...
5
votes
2answers
2k views

Gradient vector of parametric curve

I have ellipse $$(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$$ Gradient is $$(\frac{2x}{a^2}, \frac{2y}{b^2})$$ How I can obtain this vector from parametrization of my curve? Let I know only $$(x, y) = (a ...
4
votes
2answers
135 views

Does intrinsic mean existing regardless of some bigger space?

How is the arc-length of a regular parametrized curve in a surface $S\subset\mathbb{R}^3$ intrinsic? Let $\bf{x}\rm(u,v)$ be a parametrization of $S$. Letting $E,F,G$ denote the coefficients of the ...
4
votes
1answer
442 views

Find the parametric equation of the curve to be the intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $y=z$

A space-curve $C$ is defined to be the intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $y=z$. How should one try to find the parametric equation of the curve? It seems natural to let $x=(...
3
votes
1answer
115 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 $\...
3
votes
1answer
142 views

Is this a polygon

Suppose we take a rectangle and cut out a smaller quadrilateral within it, as shown below: Would the resulting figure be a polygon? If it is, would it be a concave octagon? According to what I'm ...
2
votes
0answers
402 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 ...
2
votes
2answers
891 views

infinite length of a curve

I need to show that $$f(x)= \begin{cases} \sin(1/x),& x \neq 0 \\ 0,& x = 0\end{cases}$$ on $[0,1]$ has an infinite arc length. I've tried to prove that $(f'(x))^2$ is unbounded on $[0,1]$....
1
vote
2answers
68 views

cylinder shape around an arbitary curve

I am trying to find an analytic way to describe a cable (like a cylinder) in order to calculate the exact point of intersection of the cable wall with a given ray (straight line) in $\mathbb{R}^3$. ...
1
vote
2answers
300 views

how to find the maximum height perpendicular to arc without knowing radius of the arc

I am doing the curvature correction of the object boundary, I need to calcluate the maximum height perpendicular to the arc shown in the figure. I know the chord length and arc length, But i don't ...
1
vote
2answers
142 views

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 ...
1
vote
0answers
34 views

Show that these curve families are orthogonal: $f(xy) = C$ and $y^2 - x^2 = D$ [duplicate]

Let the function $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable and its derivative is never zero. Show that these curves are orthogonal: $$f(xy) = C$$ $$ y^2 -x^2 = D$$ $C$ and $D$ are ...
1
vote
2answers
1k views

Physical significance of knot vector in B-spline.

A B-spline blending curve formulation is: $P(u)=\sum_{k=0}^np_k B_{k,d}(u)$ Given $n+1$ control points, B-spline blending functions are polynomials of degree $d-1$, $(1<d<=n+1)$. ...
1
vote
1answer
132 views

n-ellipse parametric equation

I am looking for n-ellipse parametric equation. n-ellipse is an equidistant curve from n foci. https://en.wikipedia.org/wiki/N-ellipse The implicit equation is provided in the document: http://...
1
vote
1answer
167 views

Minimizing error in Bézier circle approximation.

$$ y = a^2+b^2+1 $$ $$ a = 3cx(1-x)^2 + 3x^2(1-x) + x^3 $$ $$ b = 3cx^2(1-x) + 3x(1-x)^2 + (1-x)^3 $$ For x between 0 and 1, for what value of c is the area under the curve the smallest? For ...