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why does a ray, parallel to the principle axis, intersect the principal axis at half the radius of curvature, i.e. focus? enter image description here

EDIT

I was taught that in a spherical concave mirror, the rays parallel to the principle axis always converge at $f$, which is at $\frac{r}{2}$, but I found something like this in GeoGebra :

enter image description here

I think to be converging at a single point, the mirror has to be parabolic, as Olin and Floris said. Or rather it should be called paraboloid mirror. Now, why is the focus of paraboloid mirror half of radius of curvature?

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    $\begingroup$ This is a most excellent question, presenting the OP's own careful and well placed research that questions a widely taught assertion in high school. Moreover, that assertion, although an approximation, has its valid uses- to wit the location of the cardinal points and principal planes of an optical system. To call this question "failed mathematics" IMO bespeaks a complete misunderstanding of the relationship between physics and mathematics (by this reasoning we shouldn't teach Newton's gravitation, nor indeed GTR as many of us think the latter will be superseded). Anyhow Tu Papi, well done. $\endgroup$ – WetSavannaAnimal Oct 20 '14 at 1:49
  • $\begingroup$ I think Olin thought that I should have known this. But what can be done? I was taught that all spherical concave mirrors focus at $\frac{R}{2}$, and what's sadder is that our teacher still maintains this. $\endgroup$ – user49111 Oct 20 '14 at 11:08
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As Olin said, for a parabolic mirror lines that start out parallel to the principal axis all converge to a point. This can easily be confirmed with a bit of simple math. For a mirror, angle of incidence = angle of reflection. I show that this results in a parabolic shape in my answer to an earlier question (see the last part of the answer).

Now a spherical surface is a reasonable approximation for a parabolic surface when the angles are small - the error gives rise to spherical aberration which limits the performance of (especially large) spherical optical elements, but it's a small error for small lenses. You can convince yourself by writing down the difference:

$$y = a x^2\tag{parabolic}$$

$$(r - y)^2 + x^2 = r^2 \tag{spherical}$$

We can rewrite the spherical equation as

$$r^2 - 2ry + y^2 + x^2 = r^2\\ y = \frac{x^2}{2r}+\frac{y^2}{2r}$$

And when $y$ is very small, the first term dominates. However, at larger distances off axis, the second term becomes significant and the spherical surface is no longer a good approximation of the parabolic one.

As for the last part of your question - your own sketch effectively shows the answer. The tangent to the mirror surface is perpendicular to the ray from center of curvature. The laws of reflection tell you that incident =reflected angle, so for small angles the ray goes throught the point that is midway between the mirror and the center of curvature.

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It doesn't. You need a parabolic surface, not a spherical one, to have rays parallel to the axis converge at a focal point.

I have no idea where you got your failed math from, but it's just plain wrong.

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    $\begingroup$ Olin, that's too blunt! Please sugar-coat it! $\endgroup$ – 299792458 Oct 17 '14 at 16:42
  • $\begingroup$ We've been taught in school that all concave spherical mirrors focus at $\frac{r}{2}$ $\endgroup$ – user49111 Oct 18 '14 at 0:49
  • $\begingroup$ @TuP: Then you were taught badly. Or more likely, you misunderstood something. $\endgroup$ – Olin Lathrop Oct 18 '14 at 12:27
  • $\begingroup$ Our teacher plainly said that in a concave mirror, all rays parallel to principal axis converge at $f$, which is $\frac{r}{2}$. Confusing information was found here too wikipedia - Focal Length. It says For a spherically curved mirror in air, the magnitude of the focal length is equal to the radius of curvature of the mirror divided by two $\endgroup$ – user49111 Oct 18 '14 at 13:20
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    $\begingroup$ @TuP: What you are talking about is a approximation, useful only for spherical mirrors over small angles. A small piece of a sphere approximates a parabaloid well enough for some purposes, like a shaving mirror for example. For such small patches of a sphere, rays parallel to the axis get reflected so that they pass close to r/2. The bigger the angle of the spherical section, the more the error. To eliminate this error, the shape has to be parabaloid. $\endgroup$ – Olin Lathrop Oct 18 '14 at 20:56

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