My Math Forum Spherical mirrors, spherical?

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 August 16th, 2011, 03:22 PM #1 Senior Member   Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0 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 $\theta$) 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 = $\theta$. 4) We get from (3) that $\triangle OVF \equiv \triangle OAF$ 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 hypothesis sound absolutely false and would destroy all I claim to know about optics, which of them must be true?
 August 16th, 2011, 10:23 PM #2 Member   Joined: May 2011 Posts: 90 Thanks: 1 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 sclool) 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 iteresting and far more enlightening than the "light ray" theory. The light goes by the shortest distance thus all the wave componebts 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 is aberration (they like big words and there are dosens 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 revolotion) Even then there will be aberrations for all off-axis points Your sherical 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 theoty 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 sieries as they are small.
August 17th, 2011, 07:58 AM   #3
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Re: Spherical mirrors, spherical?

Quote:
 Originally Posted by gin palace Even for points on axis there is distortion and to avoid this distortion the mirror should be an ellipse (an ellipsoid of revolotion) Even then there will be aberrations for all off-axis points Your sherical 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.

 August 17th, 2011, 09:13 AM #4 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 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.
August 17th, 2011, 09:45 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 "$\frac{1}{f}= \frac{1}{p} + \frac{1}{p#39;}$", 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?

 August 17th, 2011, 10:22 AM #6 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 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.
August 17th, 2011, 11:45 AM   #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 an 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?

 August 17th, 2011, 11:29 PM #8 Member   Joined: May 2011 Posts: 90 Thanks: 1 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 focii 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 mirrer are imaged as a BLURR close to the other focus. Ray diagrams are approximate! That is why they are so USEFUL

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