3
votes
1answer
39 views

Calculating curvature of a curve on a the surface $x^2+y^2=1$. [closed]

Find a curve on the cylinder surface $x^2+y^2=1$ in $\mathbb R^3$ such that its curvature is equal to $\frac1{100}$ at each point of this curve. Does this easily generalize to different surfaces?
3
votes
1answer
54 views

Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
1
vote
2answers
86 views

Extend a vector field of normal vectors beyond the surface

I am not terribly well-versed in differential geometry, so please keep that in mind when answering the question. We are given a surface in ${R}^3$ defined parametrically by $\vec{r}(u,v)$ where ...
1
vote
1answer
118 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: $$ ...
2
votes
2answers
342 views

Curve arc length parametrization definition

I did some assignments related to curve arc length parametrization. But what I can't seem to find online is a formal definition of it. I've found procedures and ways to find a curve's equation by ...