In our applications of lenses and mirrors, we will be very interested in light rays coming in from "infinity" which are parallel to the optical axis.

What is a flat mirrorfor kids

From an object point P, drop a perpendicular to the mirror, meeting at B. Construct a ray originating from P that strikes the mirror at A. Then, the Law of Reflection and elementary angle geometry shows us that the marked angles are equal. Using the common side and the right angles, the ASA theorem of congruent triangles shows that PBA and P'BA are congruent; the object is as far in front of the mirror as the image is behind. The ray reflected at A thus seems to originate from P'. Repeating the argument for other rays shows that all the reflected rays due to a point P on an object trace back to the same image point P'.

Simple plane geometry also shows us that all reflected rays from a point on an object can be traced back to the same image point.

What is a flat mirrorused for

We can think of these light rays' source as being so far away that for all intents and purposes, the rays interacting with the optical system are parallel. In the study of optical systems, we will be mainly looking at the focus and the magnification of the systems. Also of importance is whether the image formed by the system is upright or inverted.

[S] Sénéor, R.: In: Renormalization of quantum field theories with non-linear field transformations, Breitenlohner, P., Maison, D., Sibold, K. (eds.), Lect. Notes Physics, Vol. 303. Berlin, Heidelberg, New York: Springer 1988

Flat mirrorreflection

[F] Federbush, P.: A phase cell approach to Yang-Mills theory. I–VII. Commun. Math. Phys.107, 319 (1986);110, 293 (1987);114, 317 (1988); Ann. Inst. Henri Poincaré47, 17 (1987)

What is a flat mirrorcalled

[L] Lüscher, M.: Construction of a self-adjoint, strictly positive transfer matrix for the Euclidean lattice gauge theories. Commun. Math. Phys.54, 283 (1977)

Many of the optical systems that we shall consider will consist of spherical arcs1. We first note that we can find the centre of any circular arc using elementary geometry.

In this section and the next, we will discuss mirrors, flat and curved. Many of the ideas explored in our discussion of mirrors will also be applicable in the realm of lenses.

Planemirrorimage

Planemirrorexamples

[FMRS2] Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: A renormalizable field theory; the massive Gross-Neveu model in two dimensions. Commun. Math. Phys.103, 67 (1986)

Any chord's perpendicular bisector passes through the arc's centre. Thus we can find the centre of an arc by constructing chords AC and CE and construct their perpendicular bisectors through B and D respectively. The intersection of the perpendicular bisectors is necessarily the centre of the arc, as such a centre is unique.

Spherical mirrors are classified according to their curvature. Concave mirrors bulge away from the objects they are reflecting, whereas convex mirrors bulge toward them. The optical axis of a spherical mirror is the line that joins its centre with the middle of the reflecting surface.

[B] Balaban, T.: Commun. Path. Phys.95, 17 (1984);96, 223 (1984);98, 17 (1985);99, 75 (1985);99, 389 (1985);102, 277 (1985);109, 249 (1987);116, 1 (1988);119, 243 (1988);122, 175 (1989);122, 355 (1989)

[FMRS1] Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Construction of infraredΦ 44 by a phase space expansion. Commun. Math. Phys.109, 437 (1987)

What is a flat mirrorin physics

We provide the basis for a rigorous construction of the Schwinger functions of the pure SU(2) Yang-Mills field theory in four dimensions (in the trivial topological sector) with a fixed infrared cutoff but no ultraviolet cutoff, in a regularized axial gauge. The construction exploits the positivity of the axial gauge at large field. For small fields, a different gauge, more suited to perturbative computations is used; this gauge and the corresponding propagator depends on large background fields of lower momenta. The crucial point is to control (in a non-perturbative way) the combined effect of the functional integrals over small field regions associated to a large background field and of the counterterms which restore the gauge invariance broken by the cutoff. We prove that this combined effect is stabilizing if we use cutoffs of a certain type in momentum space. We check the validity of the construction by showing that Slavnov identities (which express infinitesimal gauge invariance) do hold non-perturbatively.

[MS] Magnen, J., Sénéor, R.: In Third International Conference on Collective Phenomena. New York: The New York Academy of Sciences 1980

Magnen, J., Rivasseau, V. & Sénéor, R. Construction ofYM 4 with an infrared cutoff. Commun.Math. Phys. 155, 325–383 (1993). https://doi.org/10.1007/BF02097397

The simplest of mirrors are the flat ones; much of their function is explained using the Law of Reflection. Flat mirrors produce virtual images, images that cannot be caught and displayed in a screen. This is because there is no actual light coming from the image; what we see is the light from an object, reflected by the mirror.

The object is reflected in the mirror according to the Law of Reflection. The eye traces the reflected ray back to the image and sees it "behind" the mirror. The image is as far behind the mirror as the object is in front.