1
vote
0answers
30 views

Finding curve that minimizes an integral due to constraints

In the euclidean plane I want a smooth curve $\gamma (t)$ which satisfy:$$\gamma (0)=(0,0)\quad\quad \gamma '(0)=(1,0)\quad\quad n\in \mathbb{N}\land n\neq 1\Rightarrow \gamma ^{(n)}(0)=(0,0)\\\gamma ...
0
votes
0answers
20 views

Given a numerical curvature, compute the shape of the function

Given a numerical curvature, I want to accurately compute the shape of the function. Let $s$ be the arc-length along a curve of total length $L$ which is divided into $N-1$ segments. I am given 1) ...
6
votes
0answers
262 views

Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
1
vote
1answer
117 views

Find the Frenet frame

Consider the following space curve: $$ \gamma(x)=(e^x\cos(x), e^x\sin(x), e^x). $$ My main goal is to find the Frenet Frame T,N,B. So far I have found the arc-length using the following formula: $$ ...
3
votes
1answer
233 views

Integral of a curve with respect to its curvature?

I've been struggling with this one for about $3$ weeks: What is the integral of a $\mathbb{R}^3$ curve with respect to its curvature? I though about approaching it with the Ferret-S formulas, and ...
4
votes
1answer
483 views

The volume and surface area of pipe?

A line segment turns around a curve with right angle from point A to point B. I would like to find the closed region volume and surface area that figured out in the picture. Could you please give ...