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Math Help - Definition of A Non-Linear Reflection?

  1. #1
    Member mfetch22's Avatar
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    Definition of A Non-Linear Reflection?

    I am a recently graduated highschool senior and have been reconsidering a question I had been investigating in the past. A question that none of my teachers seemed to think was possible. I was considering what a reflection over a non-linear base would look like, and its actually taken me awhile to "define" what I think would be considered reflection, since its reversable. Say I want to reflect the line g(x) = (1/3)*x over the parabola f (x) = x^2 + 2 and called the resulting graph R(x). The way I defined it is the following: At any point x on f (x) construct a line normal to f (x), and call this line N(x) . Then determine where N(x) intersects g(x) and call this point P . Construct a line tangent to  f (x) at x and call it T(x). Finnally, reflect evey point  (P , g( P ) ) over every corresponding line T(x) to arrive at the equation of R(x). I used this definition of non-linear reflections and another process using a function for the intersection point to derive the equation of R(x) in terms of x alone. I have digitally generated all my work, and I was wondering if anybody is willing to take a look at my work and give me some feed back. I'm not expecting anybody to read it over word for word, just a quick over-view so I can get some feed back as to weather this could be consider a correct definition. I will attach a few preview pictures of my work to this thread, the first picture being a reflection of the x-axis over f(x) as defined, and the last picture is a reflection of g(x) over f (x) as defined [assuming that I didnt make any simple algebra mistakes] . If you are willing to look at the whole series of my work, email me at mfetch22@yahoo.com and I'll be glad to send it to you. Thanks in advance.

    NOTE: The attached images are not in the proper order and have pages missing between them. If one slide says "on the next page I....." dont expect it to be on the next page. You need to whole set of images for it to work like that, the forum file attachment limit is 5, so I put the most important slides on this thread
    Attached Thumbnails Attached Thumbnails Definition of A Non-Linear Reflection?-06-05-2010-image008.jpg   Definition of A Non-Linear Reflection?-06-05-2010-image009.jpg   Definition of A Non-Linear Reflection?-06-05-2010-image011.jpg   Definition of A Non-Linear Reflection?-06-05-2010-image014.jpg   Definition of A Non-Linear Reflection?-06-05-2010-image017.jpg  

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  2. #2
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    I saw your work, and I will ask you just one question?

    Do you know what is reflection? Could you please define mathematically reflection?

    Regards.
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  3. #3
    Member mfetch22's Avatar
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    Quote Originally Posted by p0oint View Post
    I saw your work, and I will ask you just one question?

    Do you know what is reflection? Could you please define mathematically reflection?

    Regards.
    I'm only fimmilar with what I know to be a reflection of a line over another line, or an equation over the x-axis, y-axis, or y=x . I was trying to move this definition to a curved base as an analog. I'm not certain of the exact definition, so I aplogize if the following definition is incorrect but this is what I understand to be a refelction of a point over a line:

    A point P is reflected over a line L by constructing a line segment N at 90 degrees to L such that N connects P to L, and N is connected to L at a point we shall call A. The distance from P to A will be called D, i.e. the length of N. A line segment is then constructed at 90 degrees to L at point A in the opposite direction of P (realitive to L ) with a length of D. We shall call this segment N'. The point on the end of segment N', which is not point A , is the reflection of point P, which we shall call P'.

    Again, I was trying to move this defintion of reflection [assuming it is correct, but your post has me thinking otherwise] to a different setting in which the base L is actually a curve. If my definition is incorrect, I apologize, and I will remove my post until I research the proper method of reflection for linear situations exstensively, and reapply it to my work in the proper manner.

    Thanks
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by mfetch22 View Post
    I am a recently graduated highschool senior and have been reconsidering a question I had been investigating in the past. A question that none of my teachers seemed to think was possible. I was considering what a reflection over a non-linear base would look like, and its actually taken me awhile to "define" what I think would be considered reflection, since its reversable. Say I want to reflect the line g(x) = (1/3)*x over the parabola f (x) = x^2 + 2 and called the resulting graph R(x). The way I defined it is the following: At any point x on f (x) construct a line normal to f (x), and call this line N(x) . Then determine where N(x) intersects g(x) and call this point P . Construct a line tangent to  f (x) at x and call it T(x). Finnally, reflect evey point  (P , g( P ) ) over every corresponding line T(x) to arrive at the equation of R(x). I used this definition of non-linear reflections and another process using a function for the intersection point to derive the equation of R(x) in terms of x alone. I have digitally generated all my work, and I was wondering if anybody is willing to take a look at my work and give me some feed back. I'm not expecting anybody to read it over word for word, just a quick over-view so I can get some feed back as to weather this could be consider a correct definition. I will attach a few preview pictures of my work to this thread, the first picture being a reflection of the x-axis over f(x) as defined, and the last picture is a reflection of g(x) over f (x) as defined [assuming that I didnt make any simple algebra mistakes] . If you are willing to look at the whole series of my work, email me at mfetch22@yahoo.com and I'll be glad to send it to you. Thanks in advance.

    NOTE: The attached images are not in the proper order and have pages missing between them. If one slide says "on the next page I....." dont expect it to be on the next page. You need to whole set of images for it to work like that, the forum file attachment limit is 5, so I put the most important slides on this thread
    What do you mean by a "correct" definition? We are free to define anything in any way we like, as long as our definitions are consistent with notions that already exist in the literature. But there is no point, generally, to give a name to something which is without great interest or usefulness inside mathematics. I'm not saying this is the case for what you've done; what I'm saying is that you haven't shown your idea to be of interest generally. What can you say about such generalized "reflections"? Probably not much, unless you heavily restrict the kind of curve about which to reflect. What properties of the figure, if any, do they preserve? None in the case of reflection about an arbitrary curve... Your "reflection" shares almost none of the properties of usual reflections; they are generally not invertible, they are not isometries of Euclidean space, etc. There are perhaps, however, some interesting things to be said in specific cases.

    If you've just finished high school, and are interested in mathematics, I would suggest you pick up a book about linear algebra or multivariable calculus; you will not regret it; clearly you will not have any difficulty in learning the material by yourself.
    Last edited by Bruno J.; June 6th 2010 at 06:38 PM.
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  5. #5
    Member mfetch22's Avatar
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    I found an error in my original defintion of the final reflection graph of R(x) . I attached to proper definition to this post. The error was in the definition of the range of values of t in the parametric definition of all points (u, v) . My apologies. The new, proper reflection is below
    Attached Thumbnails Attached Thumbnails Definition of A Non-Linear Reflection?-06-07-2010-image017.jpg  
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    Quote Originally Posted by mfetch22 View Post
    I found an error in my original defintion of the final reflection graph of R(x) . I attached to proper definition to this post. The error was in the definition of the range of values of t in the parametric definition of all points (u, v) . My apologies. The new, proper reflection is below
    Ok, again your graph doesn't share common characteristics of reflection.

    What are the common characteristics of reflection lets say around some line?
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  7. #7
    Member mfetch22's Avatar
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    Reversability? I dont know. You win. I give up.
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    Sorry, I am not expert in this field, but what I can see is that the graphic which is about to be reflected have equal distance from the graphic of reflection (lets say the line), with the reflected graphic.

    So by my logic, the normal distance from every point of R(x) to f(x) should be equal to the distance of g(x) to f(x). I think this is not the case with your graphic. Am I right?
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by mfetch22 View Post
    Reversability? I dont know. You win. I give up.
    It's generally far from reversible. For instance, if you "reflect" a circle of radius 2r about a concentric circle of radius r, everything goes to a single point.
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  10. #10
    Member mfetch22's Avatar
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    Quote Originally Posted by p0oint View Post
    Sorry, I am not expert in this field, but what I can see is that the graphic which is about to be reflected have equal distance from the graphic of reflection (lets say the line), with the reflected graphic.

    So by my logic, the normal distance from every point of R(x) to f(x) should be equal to the distance of g(x) to f(x). I think this is not the case with your graphic. Am I right?
    I'm not sure if your right. I could be completely wrong and you could be right, but consider the followoing process before you deteremine weather the distances are not equal:

    Pick any point x on f(x) and construct a line normal to f(x) at that point. (weather you do it mentally with your mouse pointer, or want to actually crunch through the short algebra) Observe where this normal intesects the graph of g(x) and R(x) , the distances along this line normal to f(x) at x [ the distances being along the normal from f(x) to R(x) and from f(x) to g(x) ] are the same, are they not?

    Make sure you are looking at the corrected reflection graph I posted. As for this whole question and the work I did, it seems to be a flop. But as for your specific question on the distances I beleive it the are the same as far as the definition a gave of reflection goes.

    But I could be completely wrong, I am no expert either.
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    Take for example x=4 and draw the line y=-x+4.

    Another thing you should notice. Wheter you take line from y=(1/3)x normal to the parabola, also R(x) should be normal to the line. It is not the case with your graphic.

    Just look at the example that gave you Bruno J.

    It perfectly matches with what I am saying. For ex. if the reflection of circle with radius 2r around circle with radius r is the center of the concentric circles, then the distance from the center to the inner circle is r. If you put r+r (the diametar) you'll see that the distance between the inner and outer circle is r in every point of the circle.

    Also if you take normal line of the circle 2r in some point which passes to the centre of the concetric circles, it would also be normal to the circle r in the same point, which is amazing.

    Just try to improve your graphic with these characteristics that I mentioned.
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  12. #12
    Member mfetch22's Avatar
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    Quote Originally Posted by p0oint View Post
    Take for example x=4 and draw the line y=-x+4.

    Another thing you should notice. Wheter you take line from y=(1/3)x normal to the parabola, also R(x) should be normal to the line. It is not the case with your graphic.

    Just look at the example that gave you Bruno J.

    It perfectly matches with what I am saying. For ex. if the reflection of circle with radius 2r around circle with radius r is the center of the concentric circles, then the distance from the center to the inner circle is r. If you put r+r (the diametar) you'll see that the distance between the inner and outer circle is r in every point of the circle.

    Also if you take normal line of the circle 2r in some point which passes to the centre of the concetric circles, it would also be normal to the circle r in the same point, which is amazing.

    Just try to improve your graphic with these characteristics that I mentioned.
    Okay, I gotcha. My reflection does not hold any interesting properties as other reflection do, and thats what I need to work on, I get that. I figured this reflection might resemble what a wall angled to a parabolic mirror might look like in the mirror, you know? Thats the interesting property that I was hoping to generate.

    Anyway, I understand all the problems, thank you both for pointing them out. My question is, how would I go about making these properties arise? I mean I defined reflection as I did in the first post and simple followed it where it led. I'm not sure how else to define reflection. And biggest of all, how can I go from a definition and expect it to led to a specific property? I mean I thought by the definition of my reflection that the distances along the normal from the refelcted graph to f (x) , and from the graph to be reflected to f (x) , would be the same. But as you have stated and show this isnt so. I simply dont know where to go next. I worked on defining a non-linear reflection like this 2 months ago and it came out that the reflection had a nasty cusp, so I gave up. I was hoping this new definition would led me to more productive work. I guess this is more of a pure mathematical question, having no real application even within mathematics, and so I should let it go.
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    @mfetch22 I see that you're interested in studying mathematics and exploring new things.
    Reflection of non-linear figure has no real application (as I know). Maybe you need to take the common characteristics of reflection over line. Read here. Also read more about reflection over a plane. Take as much characteristics as possible if you want to produce some reflection.

    For example. Start by reflection over line. Then "bend" the line a little bit so it would look like some kind of parabola. Then reflect the same figure over the parabola so that you will start to have sense how should non-linear reflection look like.

    Regards and good luck.
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