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August 16th, 2011, 04:22 PM   #1
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Spherical mirrors, spherical?

Let us prove that there exists no spherical mirror. Suppose an object is lying on the principal axis of a spherical mirror (concave, in this example), specifically on its focus, as shown in the image below.

1) Light has perfect reflection, so ? OVF is congruent to the angle formed by reflected ray 1 and the axis (let's call this $\displaystyle \theta$)

2) Since the object is placed on the focus, no image is formed, therefore reflected rays 1 and 2 are parallel.

3) We get from (2) that ? OAF = $\displaystyle \theta$.

4) We get from (3) that $\displaystyle \triangle OVF \equiv \triangle OAF$ since both share a side and they have equal internal angles.

5) We get from (4) that OA = VF, then FVA is right and the mirror is plane and not spherical. Therefore spherical mirrors do not exist.



We know that spherical mirrors do exist; so there is obviously something wrong above. From (3) on, no new lemmas are added, so we have the following possibilities:

i) Light does not suffer perfect reflection.
ii) Images placed on the focus actually generate images.
iii) There is no focus.
iv) This statement is false.

All of the above hypotheses sound absolutely false and would destroy all I claim to know about optics; which of them must be true?

Last edited by skipjack; December 6th, 2018 at 06:26 AM.
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August 16th, 2011, 11:23 PM   #2
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Re: Spherical mirrors, spherical?

Interesting thinking! Congratulations - you will get far - yes QUESTION everything you are told!

Even on your diagram, you can see what you are saying in not true! Hence the paradox.

The fact is the images we are TOLD ABOUT (at school) are idealised dreams.
For a start, they are infinitely small! Also infinitely sharp and NOT (school teachers conveniently forget) caused by waves!
In fact, the wave theory of images is very interesting and far more enlightening than the "light ray" theory.
The light goes by the shortest distance, thus all the wave components ARRIVE IN PHASE (and thus add).
(You will need the wave theory to explain holograms too.)

Your object (source of light or image) has finite size.
Most of it is NOT on the axis of the mirror.
This causes distortion. They call this aberration (they like big words and there are dozens of mirror aberrations)
Even for points on axis there is distortion and to avoid this distortion the mirror should be an ellipse (an ellipsoid of revolution)
Even then, there will be aberrations for all off-axis points
Your spherical mirror only APPROXIMATES the shape of part of an ellipse (the bit near the axis).

Your angles are only "almost equal" and the error depends on some big power of distance off-axis divided by distance from mirror. The theory blithely ASSUMES this ratio is so small that sin(theta) = theta and cos(theta) = 1

The assumptions made by theory are justified by a need for the results to be USEFUL . For this they must be easy enough to calculate (and "explain"). Hence ray diagrams.
You can work out this by calculating the angles (sines and cosines) and writing them as series as they are small.

Last edited by skipjack; December 6th, 2018 at 06:32 AM.
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August 17th, 2011, 08:58 AM   #3
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Re: Spherical mirrors, spherical?

Quote:
Originally Posted by gin palace View Post
Even for points on axis there is distortion and to avoid this distortion the mirror should be an ellipse (an ellipsoid of revolution)
Even then, there will be aberrations for all off-axis points
Your spherical mirror only APPROXIMATES the shape of part of an ellipse (the bit near the axis).
But an analogue proof may be done for any kind of curved mirror, an elliptic not being an exception in any sense.

Last edited by skipjack; December 6th, 2018 at 06:33 AM.
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August 17th, 2011, 10:13 AM   #4
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Re: Spherical mirrors, spherical?

I'm really not sure what you are getting at with your original argument. What, precisely, do you mean by the focus? If you're talking about the optical phenomenon, then a focus needs a mirror (or lens) AND an object to define it by, and even then it doesn't necessarily exist.
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August 17th, 2011, 10:45 AM   #5
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Re: Spherical mirrors, spherical?

Quote:
Originally Posted by mattpi
I'm really not sure what you are getting at with your original argument. What, precisely, do you mean by the focus? If you're talking about the optical phenomenon, then a focus needs a mirror (or lens) AND an object to define it by, and even then it doesn't necessarily exist.
That's weird; the definition of focus I had been given is: the point to which rays parallel to the principal axis converge after being reflected.. So it doesn't exist? What happens to the famous formula "$\displaystyle \frac{1}{f} = \frac{1}{p} + \frac{1}{p'}$", where f is the focal length and p and p' are the x-axis coordinate of the object and image, respectively?

How can we find the coordinates of the image of an object then?

Last edited by skipjack; December 6th, 2018 at 06:36 AM.
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August 17th, 2011, 11:22 AM   #6
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Re: Spherical mirrors, spherical?

Ah. Parallel rays only converge to a point if you have a parabolic mirror. Spheroidal mirrors have the property that rays emanating outwards from one focus converge to the other focus. Since a spherical mirror is a special case of a spheroidal mirror where the two foci are at the same point, the 'focus' of a spherical mirror could only be said to be the centre; all rays originating here will reflect back to the same point.
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August 17th, 2011, 12:45 PM   #7
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Re: Spherical mirrors, spherical?

Quote:
Originally Posted by mattpi
Ah. Parallel rays only converge to a point if you have a parabolic mirror.
If you place an object in such focus, is an image formed? If not, then the same arguments used in the original post hold for the parabolic mirror..

Quote:
Originally Posted by mattpi
Spheroidal mirrors have the property that rays emanating outwards from one focus converge to the other focus. Since a spherical mirror is a special case of a spheroidal mirror where the two foci are at the same point, the 'focus' of a spherical mirror could only be said to be the centre; all rays originating here will reflect back to the same point.
What's the difference between a spheroidal and a spherical mirror? If spherical mirrors don't have foci (I don't think that the centre should be considered a focus) then what does the focal length mean?

Last edited by skipjack; December 6th, 2018 at 06:40 AM.
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August 18th, 2011, 12:29 AM   #8
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Re: Spherical mirrors, spherical?

The point is points have ZERO size.
There is never more than ONE PAIR of points of focus.
For example an elliptical mirror has 2 foci.
There are ONLY TWO and to be at a point an object has to be zero size
The points of the object not at the focus of the mirror are imaged as a BLUR close to the other focus.

Ray diagrams are approximate!
That is why they are so USEFUL.

Last edited by skipjack; December 6th, 2018 at 06:42 AM.
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December 6th, 2018, 06:08 AM   #9
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Holy thread resurrection!

This is a question that has to do with both mathematics and physics. It is true that not all parallell rays will be reflected to the same point, which would be the definition of a focal point.

However in physics models that describe reality well enough are accepted. And the focal point is usually said to be at a distance of half the radius.

Here comes in the mathematical part. This can be analyzed in the unit circle. Take a ray parallel to the horizontal axis, travelling to the right that will reflect from the point (cos x, sin x), where x is a small (for example) positive angle. Then formulate where the reflected ray will cross the horizontal axis. You will notice that for small angles you are very near the point ((0,5),0). Take the limit and see you will get that.

A mathematician will say there is no focal point and will change the topic to limits and definitions. A physician will say who cares, it works and think about how accurately people can tell the difference between a parabola and a circle arc, when most mirrors have even bigger flaws in them.

Last edited by Lasse; December 6th, 2018 at 06:37 AM. Reason: Fixed the coordinates.
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December 10th, 2018, 07:20 AM   #10
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Just did the geometry again. When the ray parallell to the horizontal axis hits the circle at P = (cos x, sin x) both the angle of incidence and reflection are also x.

If you then take the triangle OAP, where O = (0,0) and A the point where the reflected ray intersects with the horizontal axis, A = (a,0) you will get a triangle with two angles x, a side with the length 1, two sides with the length a and the angle π - 2x. Then the sine rule will give you

a = sin x / sin 2x. The assumption is that for small angles this will be near 0,5.

For even ten degrees it is as good as 0,5077, but for example for five degrees it is already 0,5019. For one degree it's 0,500076.



So a paraboloidal mirror approximates in these cases very well to a spherical mirror and then the usual assumption that the focal point is half the radius holds true.
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