My Math Forum Question about a curve - 1 or 2 directions?
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August 31st, 2012, 07:20 AM   #1
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Question about a curve - 1 or 2 directions?

Hiya!

Newbie here who has been lurking this forum more due to being curious about maths :)

I am not good at maths and I am embarrassed by my little knowledge about it as I had to stop studying it in my early years, but I'm determined to learn slowly as I am an adult now and have a stressful day job. I'm glad to be here!

I have a question which I hope you guys can help me out with. The other day, I was watching some cool geometrical stuff and I had a question, does a curve have one direction or two directions?

Please see the attached diagram of the curve. What I am talking is from the start of the curve (A) to the end of the curve (C), as in the diagram. Furthermore, am I right in using the "term" direction? Would it be another term? Thus direction mean something else in mathematical terms?

The way I thought is that a curve simulating a semi-circle would ultimately end as a circle thus the curve would only have 1 direction as the line continues going on a set path. But then, I thought that perpaps there were 2 directions instead: first direction would be from A to B, then the second direction would be from B to C. I also understand that there are other types of curves and that this one would be a too perfect one in that it is about a half circle.

Can you guys help? I understand this is quite possibly a basic question but I am trying to question stuff which I am unsure about.

Oh so many thanks! :)
Attached Images
 curve directions.jpg (4.3 KB, 309 views)

 August 31st, 2012, 07:26 AM #2 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Question about a curve - 1 or 2 directions? What do you mean by "direction"? Can you explain it? In mathematics, direction means the direction of a vector. If you meant the vector-direction, then the curve in you attachments have two, 1. C to A, the counterclockwise one and 2. A to C, the clockwise one. Any line has two directions. A clockwise one, and an anti-clockwise one. You mentioned A to B is a direction and B to C is another direction but actually it's not true. Both are the same - clockwise direction.
 August 31st, 2012, 07:30 AM #3 Senior Member   Joined: Feb 2012 Posts: 628 Thanks: 1 Re: Question about a curve - 1 or 2 directions? I would say that the curve itself does not have the property of direction, but it can be traced in two directions. It may be useful to consider a curve which is described parametrically. For example, the circle $x^2 + y^2= 1$ can be described parametrically by the parameter t as follows: $x= \cos t$ $y= \sin t$ Now, we can trace the curve in the counterclockwise direction by moving in the direction of increasing t, or in the clockwise direction by moving in the direction of decreasing t. You could also consider a curve which is a function: $y= f(x)$ You can move along this curve in the direction of increasing x, or in the direction of decreasing x.
August 31st, 2012, 07:57 AM   #4
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Re: Question about a curve - 1 or 2 directions?

Quote:
 Originally Posted by mathbalarka What do you mean by "direction"? Can you explain it? In mathematics, direction means the direction of a vector. If you meant the vector-direction, then the curve in you attachments have two, 1. C to A, the counterclockwise one and 2. A to C, the clockwise one. Any line has two directions. A clockwise one, and an anti-clockwise one. You mentioned A to B is a direction and B to C is another direction but actually its not true. Both are the same - clockwise direction.
Great! I somewhat hinted that using the term direction was wrong so I'm glad to know that direction would be going up and down. As I say, I'm a total beginner so I'm trying to learn.

I guess what I had in mind is that, as a non-mathematical person, I would have said that there is only one "bend" in the curve previously posted, or that there is only one curve (in the same previously attached image). However, I am not sure what these somewhat loose terms (bend, curve) would equate to in mathematical terms or if I'm using them correctly. Please see the attached new image.

What I have now done (excuse my Paint abilities!) is that I have joined a new curve going in the opposite "_____" (this is what I don't know). This 2nd curve is the same as the previous one in that it is meant to be a semi-circle but it is the reverse mirror reflection of the original curve and has been now joined at Point C of the previous curve.

The way I see it is as if the newly attached image was the aerial view of a road and I start with my car at Point A. When does the 1st curve start/occur/end, the 2nd curve, the 3rd curve etc? And what is the actual term for the aforementioned "_____"? Am I still going in the same "direction" when I enter the second semi circle?

Please let me know if I am not being clear and excuse my minimal math knowledge. Thanks!
Attached Images
 curve directions 2.jpg (7.0 KB, 300 views)

August 31st, 2012, 08:04 AM   #5
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Re: Question about a curve - 1 or 2 directions?

Quote:
 Originally Posted by icemanfan I would say that the curve itself does not have the property of direction, but it can be traced in two directions. It may be useful to consider a curve which is described parametrically. For example, the circle $x^2 + y^2= 1$ can be described parametrically by the parameter t as follows: $x= \cos t$ $y= \sin t$ Now, we can trace the curve in the counterclockwise direction by moving in the direction of increasing t, or in the clockwise direction by moving in the direction of decreasing t. You could also consider a curve which is a function: $y= f(x)$ You can move along this curve in the direction of increasing x, or in the direction of decreasing x.
Thank you for offering your help. Unfortunately my math knowledge is really basic so I cannot understand what you said although I'm sure others who may have a similar enquiry will do. I hope to eventually get to a level to feel comfortable with relatively advanced math jargon. Regardless, thanks!

August 31st, 2012, 08:09 AM   #6
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Re: Question about a curve - 1 or 2 directions?

Quote:
 Originally Posted by Alamos Am I still going in the same "direction" when I enter the second semi circle?
Yes. Here is an attachment with an explanation :

[attachment=0:2iafynzj]Alamos.png[/attachment:2iafynzj]

ABC is the ant-clockwise direction.

CBA is the clockwise direction.

Feel free to ask for more clarification.

Balarka

.
Attached Images
 Alamos.png (8.2 KB, 298 views)

August 31st, 2012, 08:17 AM   #7
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Re: Question about a curve - 1 or 2 directions?

Quote:
Originally Posted by mathbalarka
Quote:
 Originally Posted by Alamos Am I still going in the same "direction" when I enter the second semi circle?
Yes. Here is an attachment with an explanation :

[attachment=0:2ykg28yb]Alamos.png[/attachment:2ykg28yb]

ABC is the ant-clockwise direction.

CBA is the clockwise direction.

Feel free to ask for more clarification.

Balarka

.
Thanks!

Allow me to ask more questions with your newly revised S shape image.

What does it change when you have covered the distance A to B, and are now at B to cover the distance to C? This is what initially I had erroneously called "changing direction" but have learnt that I am effectively going in the same direction. However, something does change, right? What would be the name for this? What has changed?

Also, what do you call the distance between A and B, and the distance between B and C? Would it be a curve? So we'd have 2 curves in the image (A-B & B-C)? Could you also call it a "bend"? Is there another term for these 2 distances?

Thanks again guys.

August 31st, 2012, 08:23 AM   #8
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Re: Question about a curve - 1 or 2 directions?

Quote:
 Originally Posted by Alamos What does it change when you have covered the distance A to B, and are now at B to cover the distance to C?
There are a plenty of thing that changes. However, they depends on the structure of the curve. The main thing that changes is the position of your "car"
Also, the curvature changes. etc etc.

Quote:
 Originally Posted by Alamos Also, what do you call the distance between A and B, and the distance between B and C?
It will be a piece of the curve AC.

Quote:
 Originally Posted by Alamos Could you also call it a "bend"?
The "bend" will be the change of curvature, I guess.

Balarka

.

August 31st, 2012, 08:50 AM   #9
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Re: Question about a curve - 1 or 2 directions?

Thanks. I am still a bit confused though.

Quote:
 Originally Posted by mathbalarka What does it change when you have covered the distance A to B, and are now at B to cover the distance to C? There are a plenty of thing that changes. However, they depends on the structure of the curve. The main thing that changes is the position of your "car" Also, the curvature changes. etc etc.
I would like to know in terms of the distance that makes A to C and which looks like a pseudo S: what changes at Point B then? The curvature?

Perhaps, I am wrong in what I understand as a curve. I would have thought that A to B, and B to C are two distinct curves but what I seem to understand from your following response is that the distance from A to C (which resembles an S and is made up of to pseudo semi-circles joined) is ONE whole curve. This to me would be new as I thought a curve was a distance which has one "_____" (again, I initially called this "direction" yet I don't know what to call this) as is the case of a semi circle. In the wase of a wave (such as the S shape portrayed), I would have thought that there are 2 curves.

Quote:
 Originally Posted by Alamos Also, what do you call the distance between A and B, and the distance between B and C? It will be a piece of the curve AC.
Referring to my previous inserted comment above, the 2 semi-circles would be 2 pieces of the distance A to C (this distance resembling an S or wave)?

Quote:
 Originally Posted by Alamos Could you also call it a "bend"? The "bend" will be the change of curvature, I guess. Balarka
I am trying to look for the dictionary definitions of "curve" and "bend" yet they seem to be similar. Are these 2 terms similar in mathematics and in real life? And by curvature, we could say then that the distance from A to C changes curvature ONCE (at Point B)?

Jeez, I am sounding like a big ignoramus. Thanks for helping guys, I'd like to be 100% clear on this issue bearing in mind my limited knowledge.

August 31st, 2012, 09:02 AM   #10
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Re: Question about a curve - 1 or 2 directions?

Quote:
 Originally Posted by Alamos what changes at Point B then? The curvature?
Perhaps. Actually, it doesn't changes at B. The curvature changes at the neighborhood of B.

Quote:
 Originally Posted by Alamos what I seem to understand from your following response is that the distance from A to C
Not the distance. A curve is the geometrical figure, not the length!

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