0
votes
0answers
32 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 ...
1
vote
1answer
24 views

To prove a relation for a smooth, asymptotic plane curve, in arc length parametrization.

Given a smooth plane curve, parametrized in arc length as $\alpha(s) \equiv (x(s),y(s))$ and given that $$\lim_{s \to \infty} \frac{y(s)}{s} = k,$$ $k$ a constant, and $$\lim_{s \to \infty}x(s) = 0,$$ ...
10
votes
4answers
305 views

When is the moment of inertia of a smooth plane curve is maximum?

Given a smooth plane curve $(x(s),y(s))$, parameterized in arc length $s$, of fixed finite length $L$, its moment of inertia about its center of mass (axis perpendicular to the plane) is given as $$MI ...
0
votes
2answers
53 views

Curve is a submanifold

In our class today we said today that an 1-times continuously differentiable immersion $\phi:[0,1] \rightarrow \mathbb{R}^n$ is a submanifold of dimension 1 if it is closed such that $\phi(0)=\phi(1)$ ...
0
votes
1answer
68 views

Hausdorff dimensions of smooth but non-rectifiable curves

Smooth curves of finite length have a Hausdorff dimension of 1. How about smooth but non-rectifiable (i.e. infinite-length) curves? Are they also of Hausdorff dimension 1, or does it depend on the ...
6
votes
1answer
217 views

Does every continuous everywhere but differentiable nowhere curve have an infinite length?

Given a curve $\!\,\gamma : [a, b] \rightarrow ℝ^2$ that is continuous everywhere but differentiable nowhere (or almost nowhere), is its length: $$\text{length} (\gamma)=\sup \left\{ \sum_{i=1}^n ...
3
votes
2answers
287 views

Do simply connected open sets in $\Bbb R^2$ always have continuous boundaries?

A Jordan curve is a continuous closed curve in $\Bbb R^2$ which is simple, i.e. has no self-intersections. The Jordan curve theorem states that the complement of any Jordan curve has two connected ...
2
votes
2answers
42 views

Parameterized curve describing trajectory of thrown object

We describe the trajectory of a thrown object (neglecting friction and similiar effects) with the curve $$k(t) = \left(v_0\cos(\beta)t,\,v_0\sin(\beta)t-\frac{g}{2}t^2\right)$$ with ...
3
votes
1answer
134 views

For any curve of unit length there exists a closed rectangle with area at most $A$ that covers the curve.

Find the minimum possible value of $A$ such that for any curve of unit length there exists a closed rectangle with area at most $A$ that covers the curve.
5
votes
3answers
321 views

Level sets of convex functions

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a convex function. For $t\in\mathbb{R}$, consider the corresponding level set $$f^{-1}\{t\}=\{(x,y)\in\mathbb{R}^2: f(x,y)=t\}.$$ For the application I ...
1
vote
2answers
703 views

definition of length of non-rectifiable curves

Following the usual definition of length of a curve, the curve have to be rectifiable to have finite length. However for example the curve $ \gamma \colon [0,1] \to \mathbb{R}, t \to t \cos^2(\pi/t) ...