Hakop
2004-10-31 06:29:55 |
Boundary of parameterized surface
Hey guys, I'm new here and i'd like to pose an interesting problem for everyone:
Let's suppose you have a surface parameterized by the following:
r(s,t) = <(1-s)*t,s*(1-t)> ; where 0 <= t <= 1 and 0 <= s <= 1
when you try to draw this surface, you will see that as you use more and more s and t surface lines, the boundary of the surface converges to want looks like a parabola. My question is can anyone derive a parameterization of the boundary curve, call it c(z)?
It seems like the best way to approach this problem is to break the boundary curve into a finite amount of linear divisions and develop an expression for that curve. Then take a limit as those small divisions approach infinity to come up with an express for the continuous curve. I have not come close to solving the problem this way, so I may be looking at the problem in the wrong. Regardless, can anyone help? |
owl
2004-10-31 13:06:40 |
Re: Boundary of parameterized surface
Hi Hakop! How are you generating a surface with the 2-d vector r? |
Hakop
2004-10-31 16:37:29 |
Re: Boundary of parameterized surface
Yes, r is a 2-D vector. You think of the surface like this:
you have two curves: c1 = <t,0> and c2 = <0,1-t> t goes from [0,1]
c1 is just a line on the x-axis and c2 is a line on the y-axis.
Then the surface joining these two curves is r(s,t) = (1-s)*c1(t)+s*c2(t) where s goes from [0,1]
<a href='http://www.simionavich.net/apalyan/surface.bmp'>here's a visual from mathematica:
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Hakop
2004-10-31 16:38:52 |
Re: Boundary of parameterized surface
Sorry, let me try that again
<img src='http://www.simionavich.net/apalyan/surface.bmp'> |
Hakop
2004-10-31 16:44:35 |
Re: Boundary of parameterized surface
I guess images don't work so here's a link:
<a href='http://www.simionavich.net/apalyan/surface.bmp'>here's a visual from mathematica:
|
Hakop
2004-10-31 16:45:46 |
Re: Boundary of parameterized surface
heres a visual from mathematica. |
owl
2004-10-31 18:01:02 |
Re: Boundary of parameterized surface
So this surface sits in the x-y plane, yes? |
owl
2004-10-31 18:30:13 |
Re: Boundary of parameterized surface
Okay, I am pretty sure I understand. Consider the curve C(t)=<t^2, (1-t)^2>; you may be pleased with its attributes :-)
Rewriting r as a linear equation of (x,y) with parameter t, led me to this result. |
Hakop
2004-10-31 23:23:19 |
Re: Boundary of parameterized surface
Can you show me how you did this? |
owl
2004-10-31 23:50:51 |
Re: Boundary of parameterized surface
x=(1-s)t y=s(1-t)
solving for s ...
y=((t-1)/t)(x-t)
These are the lines parameterized by t.
Note that these are all tangent lines to the boundary. If we can parameterize the boundary by t, we know the intercepts of the curve, we know it probably is some kind of conic section, and we know the slope at t is (t-1)/t. A little guessing took me home :-)
What do you think? |
Hakop
2004-11-01 06:35:53 |
Re: Boundary of parameterized surface
That's very clever. So there is no real systematic way of doing this? |
Juggler
2004-11-01 06:42:45 |
Re: Boundary of parameterized surface
this looks to be a good problem, so why waste it by putting it in the forum?
check out the rules of the site, problems do not live here, there are a number of good reasons why not.
if you had been paying attention when you clicked on the link to create new thread, you would have read that you shouldn't put this stuff in here. |
owl
2004-11-01 13:43:17 |
Re: Boundary of parameterized surface
I plead ignorance; I apologize to Perplexus for the offense, obvious annoyance, and waste. Thank you, Juggler, for the polite correction.
Hakop; all is not lost. This can be generalized to a whole class of popular "string art" questions. Juggler is right, why not submit it? Definitely reword carefully :-) And the answers will certainly be without the educational ambiguity I tend to insert. If you don't want to submit, email me and I will; we could call it "Hakop's Strings" :-) |
Hakop
2004-11-01 22:02:55 |
Re: Boundary of parameterized surface
Juggler,
I put this problem in the forums becuase I had just recently registered, so I believe that I don't have the ability to post problems yet. With regards to generalizing the problem, I am not sure to what extent I can generalize this such that the problem will remain interesting and non-trivial. Perhaps I can say something like: there are two curves c1(t) and c2(t) that generate a surface r(s,t) by (1-s)*c1(t)+s*c2(t). Express the boundaries (in parameterized form) that result from generating this surface.
Is that reasonable? It seems like I will need to get more specific with the curves c1 and c2 in order for this problem to be solvable. For starters, maybe we can let c1 and c2 be any two lines in a 2-D plane and build up from there. |
Juggler
2004-11-01 22:22:14 |
Re: Boundary of parameterized surface
The reason you can't post a puzzle is that you are still a novice, this is because you haven't posted a comment on an existing puzzle yet. 24hrs after you do this, you will be able to post this problem.
When this problem gets to the front of the queue, the journeymen and scholars will comment on it, if it gets the Thumbs Up, it will be posted to the site.
I suggest you read the FAQ's on the main page, they will tell you all you need to know about this site for night. |
Juggler
2004-11-01 22:48:42 |
Re: Boundary of parameterized surface
ooops, made a mistake, you need to post 3 comments and rate one puzzle before you get to post puzzles |
