Tagged Questions
18
votes
0answers
316 views
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 ...
9
votes
2answers
253 views
A question about curves in $\mathbb{R}^2$
I need to show this result:
Let $\alpha :I\rightarrow \mathbb{R}^2$ a smooth curve, where $I$ is a compact interval of the real line. If $\lVert \alpha (s) - \alpha (t) \rVert$ depends only on ...
6
votes
3answers
223 views
Can closed curves have small curvature?
Let $\gamma$ be a smooth curve in Euclidean space of length $2\pi$ whose curvature function satisfies $-1 < k(t) < 1$. Can $\gamma$ be closed?
This seems like it should be an easy exercise, at ...
6
votes
2answers
335 views
The Dido problem with an arclength constraint
It is well known that the solution to the classical Dido problem is a semicircle, and that the solution to the classical isoperimetric problem is a circle. It's also reasonably obvious that the ...
5
votes
4answers
261 views
Curvature of planar implicit curves
I am trying to understand how the curvature equation
$$\kappa = -\frac{f_{xx} f_y^2-2f_{xy} f_x f_y + f_x^2 f_{yy}}{(f_x^2+f_y^2)^{3/2}}$$
for implicit curves is derived. These curves arise from ...
3
votes
3answers
694 views
The function that draws a figure eight
I'm trying to describe a counterexample for a theorem which includes the figure eight or "infinity" symbol, but I'm having trouble finding a good piecewise function to draw it. I need it to be the ...
3
votes
1answer
131 views
What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid?
What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid?
If possible, show some reference please?
3
votes
0answers
185 views
Reconstructing a curve from its curvature
I'm with a problem in an exercise form Do Carmo's Differential Geometry of Curves and Surfaces; it is number 9 section 1.6.
I have a differentiable real function $k(s)$, $s\in I$, and I need to show ...
3
votes
0answers
85 views
Showing that the cardioid is or isn't the caustic of the truncated cone.
Today I had a nice breakfast, but instead of using the usual cylindrical cup and admiring the nephroid inside it, I chose a cup in the form of a truncated cone. It seemed that this cup produced a nice ...
2
votes
1answer
254 views
Does anyone know the name of this curve?
I have come upon
the curve with the following parametric equations:
$$x(t)=\log(2+2\cos(t))/2$$
$$y(t)=t/2$$
for $-\pi<t<\pi$. It gives the image in the complex plane under $\log(1+z)$ of the ...
2
votes
2answers
367 views
Direction of the second derivative of an arclength parametrized curve
I have a question, it's so simple and stupid ._. If I have a planar curve parametrized by arc length, it's easy to show that the second derivate is orthogonal to the first derivate vector (tangent ...
2
votes
2answers
80 views
What is the limit distance to the base function if offset curve is a function too?
I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that ...
2
votes
1answer
86 views
What does the definition of curvature mean?
First question, am I right in saying that curvature measure how quickly the direction of a cruve changes?
Also, we have been given the "definition":
$$T'(t) = \kappa(t) | \gamma'(t) | U(t)$$
where ...
2
votes
2answers
325 views
Ways to define a curve
I'm trying to give shapes in my physics engine roundness/ curvature.
I am aware of various methods for mathematically defining curvature such as bezier-curves, ellipses, etc; but I'm not sure which ...
2
votes
1answer
453 views
The signed curvature of the Catenary
Now I want to show that the signed curvature of the catenary, with parameterization
$$(t,\cosh(t))$$
is $k(t)=\frac{1}{\cosh^2(t)}$
Now what I have done (and presumably went astray), is first ...
