---
title: Spherical Mirrors
module: Geometrical Optics
moduleNumber: 11
lessonNumber: 3
order: 1103
summary: >
  Curve a mirror and it stops merely reflecting an image and starts forming one: the
  same $1/s+1/s'=1/f$ that governs lenses reappears, now with $f=R/2$ and reflected
  rays and object sharing one side of the glass. We derive the mirror equation from
  the reflection geometry of a single paraxial ray, then let signed distances do the
  sorting — real inverted images on the near branch, virtual upright ones behind the
  surface — and check the concave, convex, and plane-mirror limits against each other.
  The second half turns to how focal length is actually measured on a bench, by
  finite conjugates, distant targets, return imaging, and sagitta, and to the
  aperture and off-axis aberrations the single paraxial focus cannot capture.
topics: [Geometrical Optics]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 32 — Optical Images; §32-1"
  - book: Tipler & Mosca
    ref: "Ch. 31 — Reflection"
---

## Spherical geometry, reflection side, and signed distances

A spherical mirror is a small reflecting portion of a sphere. The sphere’s center
is the center of curvature $C$; its radius is the signed radius of curvature
$R$. The vertex $V$ is the point at which the optic axis meets the reflecting
surface. A concave mirror facing an object on the left has $C$ left of
$V$, on the side occupied by both incoming and reflected rays. A convex mirror
facing that object has $C$ behind the reflecting surface, on the right of $V$.

The relation $R=2f$ belongs to the paraxial model, not to every reflected ray of
a spherical surface. It defines focal length $f$ in the central, low-angle
region of the mirror. A wide mirror still has the same geometric radius, but outer
rays form a different axial crossing pattern. The distinction becomes central when
the image is measured with a large aperture or a sharp screen criterion.

$$
% caption: Reference geometry of a concave mirror. The center of curvature $C$, focal
% point $F$, and vertex $V$ lie on the optic axis, with $F$ halfway between $V$ and
% $C$, so $f=R/2$.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.3,0)--(9.0,0) node[right] {axis};
\draw[acc,very thick] (7.5,-2.0) .. controls (8.2,-1.0) and (8.2,1.0) .. (7.5,2.0);
\filldraw[draw=black,fill=black!8] (2,0) circle (2pt);
\filldraw[draw=black,fill=black!8] (5,0) circle (2pt);
\filldraw[draw=black,fill=black!8] (8.03,0) circle (2pt);
\draw[black,dashed] (2,0)--(8.03,0);
\node[below] at (2,0) {center};
\node[below] at (5,0) {focus};
\node[below] at (7.35,0) {vertex};
\node[above] at (7.9,2.05) {mirror};
\end{tikzpicture}
$$

The adopted sign convention records the side on which a point or curvature center
lies. Incoming light and reflected light occupy the same physical side of a mirror.
With the object in front of a mirror on the left:

- $s>0$ for a real object in front of the mirror.
- $s'>0$ for a real image in front of the mirror, where physical reflected rays
  meet.
- $s'<0$ for a virtual image behind the mirror, where backward extensions meet.
- $R>0$ and $f>0$ for a concave mirror; $R<0$ and $f<0$ for a convex
  mirror.
- $m=y'/y$ is positive for an upright image and negative for an inverted image.

The terms front and behind refer to the reflecting surface, not to the orientation
of a drawing on a page. A real image lies in front because rays travel there after
reflection. A virtual image lies behind because the rays leave the mirror diverging
and their backward geometric extensions intersect there.
Moving a screen through the physical ray bundle can locate a real image; it cannot
produce a projected image at the virtual point.

Distance origin matters. The equations use the tangent plane at the vertex in the
ideal spherical construction. A physical mirror has a substrate, coating, rim, and
mount. Measuring from a rim or a mechanical back surface adds a common offset to
every reported distance. That offset can be insignificant for a long focal length
and dominant for a compact mirror. State the reference mark used on the bench and
convert it to the vertex plane before applying the mirror equation.

At each surface point, the normal of a spherical mirror passes through $C$. The
law of reflection applies to the angle between a ray and this normal:

$$
\theta_{\rm in}=\theta_{\rm out}.
$$

The radius-normal construction is the reason a ray sent through $C$ returns on
its incident line. It reaches the surface at normal incidence. This radial ray is
one of the three standard construction rays because normal incidence fixes its
direction. A ray that reaches the vertex has a normal parallel to the axis, so its
reflection is obtained by equal slopes about the axis.

$$
% caption: Reflection at a mirror surface. The radius from the center of curvature $C$
% to the hit point is the local surface normal, and the incoming and outgoing rays
% make equal angles with that normal.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.3,0)--(9.0,0) node[right] {axis};
\draw[acc,very thick] (7.5,-2.0) .. controls (8.2,-1.0) and (8.2,1.0) .. (7.5,2.0);
\filldraw[draw=black,fill=black!8] (2,0) circle (2pt);
\filldraw[draw=black,fill=black!8] (7.72,1.5) circle (2pt);
\draw[black,dashed] (2,0)--(8.55,2.09);
\draw[->,black,thick] (2.9,2.81)--(7.66,1.53);
\draw[->,black,thick] (7.66,1.47)--(4.2,-1.98);
\node[below] at (2,0) {center};
\node[above right] at (7.85,1.55) {normal};
\node[above] at (3.7,2.95) {incoming};
\node[right] at (4.5,-1.35) {outgoing};
\end{tikzpicture}
$$

## Paraxial mirror equation and the focal relation

The spherical-mirror equation follows from the reflection geometry of one ray from
an object point. Let the ray strike the mirror at a point a height $h$ from the
axis. The object distance is $s$, the image distance is $s'$, and the center of
curvature is one radius from the vertex. The normal through the hit point runs to
$C$. Equal incident and reflected angles, combined with the triangle exterior
angles, give

$$
\alpha+\gamma=2\beta.
$$

A paraxial ray has angles small enough for each angle to be approximated by height
divided by its corresponding axial distance:

$$
\alpha\simeq\frac{h}{s},
\qquad
\beta\simeq\frac{h}{R},
\qquad
\gamma\simeq\frac{h}{s'}.
$$

Canceling $h$ yields the signed spherical-mirror relation

$$

\frac{1}{s}+\frac{1}{s'}=\frac{2}{R}.

$$

The derivation assumes that the ray height is small relative to the radius and that
the object, image, and axial distances support the same small-angle approximation.
It is not a general exact equation for a wide spherical mirror. Its agreement with
ray tracing becomes worse as the aperture or field angle grows.

The small-angle step uses $\sin\phi\simeq\tan\phi\simeq\phi$ for each
ray-normal angle $\phi$, with angles measured in radians. The relevant parameter
is the ray height relative to the local radius and to the axial distances. A mirror
can have a long radius yet still violate the paraxial condition if a wide beam uses
an appreciable fraction of its aperture. Conversely, a narrow central beam can give
an accurate focal measurement on a mirror whose full clear aperture has visible
spherical aberration. The formula therefore predicts an aperture-dependent
experimental result whenever the focus criterion admits nonparaxial rays.

The vertex plane is also part of the derivation. Object and image distances are
measured from the tangent plane at $V$, while the radius extends from $V$ to
$C$. Replacing the vertex plane by a rim, clamp, or mirror back surface changes
$s$ and $s'$ without changing the physical rays. A systematic shift of that
kind can preserve an apparently good ray diagram yet alter the inferred focal
length, especially for short-radius mirrors.

$$
% caption: Paraxial construction for the mirror equation. A ray from the axial object
% strikes the mirror near the vertex, the radius to the hit point is the normal, and
% equal reflection angles $\alpha+\gamma=2\beta$ relate $s$, $s'$, and $R$.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.3,0)--(9.0,0) node[right] {axis};
\draw[acc,very thick] (7.5,-2.0) .. controls (8.2,-1.0) and (8.2,1.0) .. (7.5,2.0);
\filldraw[draw=black,fill=black!8] (0.6,0) circle (2pt);
\filldraw[draw=black,fill=black!8] (3,0) circle (2pt);
\filldraw[draw=black,fill=black!8] (2,0) circle (2pt);
\filldraw[draw=black,fill=black!8] (7.75,1.3) circle (2pt);
\draw[->,black,thick] (0.64,0.05)--(7.72,1.28);
\draw[->,black,thick] (7.72,1.32)--(3.05,0.06);
\draw[black,dashed] (2,0)--(8.5,1.47);
\node[below] at (0.6,0) {object};
\node[below right] at (3,0) {image};
\node[below] at (2,0) {center};
\node[above] at (7.9,1.42) {hit point};
\end{tikzpicture}
$$

At object distances where $1/s$ is negligible, the image distance approaches $R/2$.
That limiting distance is the paraxial focal length:

$$

f=\frac{R}{2},
\qquad
\frac{1}{s}+\frac{1}{s'}=\frac{1}{f}.

$$

The focus is a point only for axial paraxial rays. Parallel rays entering at a
small nonzero angle still intersect in the focal plane, displaced from the axis.
The same field-angle dependence later contributes to off-axis image errors. A
mirror calibration that quotes one focal length therefore also needs an aperture and
wavelength-independent geometric condition: small ray angles about the optic axis.

$$
% caption: Parallel paraxial rays at a concave mirror. Rays parallel to the axis cross
% at the focal point, halfway between the vertex and center of curvature, defining the
% focal length $f=R/2$.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.3,0)--(9.0,0) node[right] {axis};
\filldraw[draw=black,fill=black!8] (2,0) circle (1.8pt);
\filldraw[draw=black,fill=black!8] (5,0) circle (2pt);
\draw[acc,very thick] (7.5,-2.0) .. controls (8.2,-1.0) and (8.2,1.0) .. (7.5,2.0);
\draw[->,black,thick] (0.7,1.2)--(7.72,1.2);
\draw[->,black,thick] (0.7,0.6)--(7.86,0.6);
\draw[->,black,thick] (0.7,-1.2)--(7.72,-1.2);
\draw[->,black,thick] (7.72,1.2)--(5,0);
\draw[->,black,thick] (7.86,0.6)--(5,0);
\draw[->,black,thick] (7.72,-1.2)--(5,0);
\node[below] at (2,0) {center};
\node[above left] at (5.05,0.12) {focus};
\end{tikzpicture}
$$

Reflection reversibility gives the companion construction. A point source at the
focal point of a concave mirror sends a reflected paraxial bundle parallel to the
axis. The ray paths retrace when their directions are reversed because the local
reflection angles remain equal. This is the physical basis for using a concave
mirror as a collimator and for testing focal position with an autocollimation
arrangement.

A convex mirror has negative radius and negative focal length under the same sign
convention. Parallel incoming rays reflect outward. Their backward extensions meet
behind the mirror at a virtual focus. The focal distance still has magnitude
$\lvert R\rvert/2$; its negative sign denotes that no physical
reflected rays cross at that location.

$$
% caption: Parallel paraxial rays at a convex mirror. Reflected rays diverge in front
% of the surface, and their dashed backward extensions meet at the virtual focus a
% distance $\lvert f\rvert=\lvert R\rvert/2$ behind the mirror.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.3,0)--(9.0,0) node[right] {axis};
\draw[acc,very thick] (3.5,-2.0) .. controls (2.8,-1.0) and (2.8,1.0) .. (3.5,2.0);
\filldraw[draw=black,fill=black!8] (5,0) circle (2pt);
\draw[->,black,thick] (0.7,1.0)--(3.05,1.0);
\draw[->,black,thick] (0.7,-1.0)--(3.05,-1.0);
\draw[->,black,thick] (3.05,1.0)--(1.1,2.0);
\draw[->,black,thick] (3.05,-1.0)--(1.1,-2.0);
\draw[black,dashed] (3.05,1.0)--(5,0);
\draw[black,dashed] (3.05,-1.0)--(5,0);
\node[below right] at (5,0) {focus};
\node[above] at (3.25,2.05) {mirror};
\end{tikzpicture}
$$

The relation between radius and focal distance is linear within the paraxial model.
Doubling the radius doubles the focal length and reduces the mirror’s optical power.
The sign changes together: a concave mirror has positive $R$ and $f$, while the
corresponding convex surface has negative $R$ and $f$. This pairing provides a
quick dimensional and sign check before any image-distance calculation.

## Principal rays and image construction

Ray diagrams provide a geometric check on signed algebra. Draw rays from one
well-defined object point, usually the top of an arrow. The crossing of the
reflected rays gives the corresponding image point. Two independent paraxial rays
are sufficient; a third ray tests the construction. The mirror can be drawn as its
curved surface or as a tangent line at the vertex, provided the center, focus, and
direction of the reflecting side remain unambiguous.

A concave mirror has the standard principal rays:

- A ray parallel to the axis reflects through the focal point.
- A ray directed through the focal point reflects parallel to the axis.
- A ray directed through the center of curvature reflects back on itself.

The first two rely on the focal definition and reversibility. The radial ray relies
on normal incidence. At a finite aperture the three lines are paraxial
approximations; their intersection is the ideal image location used by the mirror
equation. Nonparaxial physical rays can cross at other longitudinal positions.

Start a construction at one object point. Draw two rays with unambiguous rules, then
locate the image point from the intersection
of physical reflected rays or dashed backward extensions. Repeat the construction
for a second object point only when image height or shape is needed. A line leaving
the mirror must carry an arrow in the physical reflected direction; a dashed
extension is geometrical information about a virtual point and carries no light
through the apparatus. This convention prevents a common error in which a virtual
image is placed on a screen behind a mirror.

The radial ray should be directed to the actual center of curvature. At an off-axis
object point, a vertex-directed line is generally non-normal to the spherical
surface and therefore does not retrace. A construction that uses the vertex as a
shortcut for every ray can appear orderly while producing a wrong image position.
The center-directed rule remains valid for concave and convex mirrors; only the
location of $C$ relative to the reflecting surface changes.

$$
% caption: Three principal rays for a concave mirror, object beyond the center of
% curvature. The parallel, focal, and center-directed rays meet at one real, inverted,
% reduced image with magnification $m=-s'/s$.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.2,0)--(9.0,0) node[right] {axis};
\draw[acc,very thick] (7.5,-2.4) .. controls (8.25,-1.2) and (8.25,1.2) .. (7.5,2.4);
\filldraw[draw=black,fill=black!8] (2,0) circle (1.8pt);
\filldraw[draw=black,fill=black!8] (5,0) circle (1.8pt);
\draw[acc,very thick,->] (0.5,0)--(0.5,1.4);
\draw[black,very thick,->] (3,0)--(3,-0.93);
\draw[->,acc,thick] (0.5,1.4)--(7.95,1.4);
\draw[->,acc,thick] (7.95,1.4)--(2.5,-1.19);
\draw[->,acc!70!black,thick] (0.5,1.4)--(8.0,-0.93);
\draw[->,acc!70!black,thick] (8.0,-0.93)--(2.5,-0.93);
\draw[->,black,thick] (0.5,1.4)--(3.4,-1.31);
\node[left] at (0.5,1.4) {object};
\node[below left] at (2,0) {center};
\node[above] at (5,0.05) {focus};
\node[below] at (3.0,-1.5) {image};
\end{tikzpicture}
$$

The tangent-line version of a diagram reduces drawing clutter. Put a vertical line
at the vertex, mark the focal point and center on the reflecting side, and preserve
the direction rules. The line provides a distance reference and a convenient
paraxial-ray intercept. Curvature continues to determine the focal point and the
center-directed ray.

An object inside the focal distance of a concave mirror produces diverging reflected
rays. The radial and focal constructions remain valid, but their backward extensions meet
behind the mirror. The image has negative $s'$, positive magnification, and no
screen plane. This configuration is the geometrical basis of a concave shaving or
inspection mirror: the observer receives an enlarged upright virtual image.

A convex mirror uses principal-ray rules with backward extensions. A parallel
incoming ray reflects as though it originated at the virtual focus behind the
surface. A ray directed toward that focus reflects parallel to the axis. A ray
directed toward the center of curvature reaches the surface normally and returns on
its line. With a real object, every construction gives a virtual, upright, reduced
image between the vertex and the virtual focus.

$$
% caption: Principal rays at a convex mirror with a real object. A parallel ray
% reflects as if from the virtual focus and a center-directed ray retraces; the dashed
% extensions locate an upright, reduced, virtual image behind the surface.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.3,0)--(9.0,0) node[right] {axis};
\draw[acc,very thick] (3.5,-2.2) .. controls (2.75,-1.1) and (2.75,1.1) .. (3.5,2.2);
\filldraw[draw=black,fill=black!8] (7,0) circle (1.8pt);
\filldraw[draw=black,fill=black!8] (5,0) circle (1.8pt);
\draw[acc,very thick,->] (0.5,0)--(0.5,1.4);
\draw[black,very thick,->] (4.11,0)--(4.11,0.62);
\draw[->,acc,thick] (0.5,1.4)--(3.0,1.4);
\draw[->,acc,thick] (3.0,1.4)--(1.0,2.8);
\draw[black,dashed] (3.0,1.4)--(4.5,0.35);
\draw[->,acc!70!black,thick] (0.5,1.4)--(3.0,0.86);
\draw[black,dashed] (3.0,0.86)--(4.6,0.52);
\node[left] at (0.5,1.4) {object};
\node[below] at (7,0) {center};
\node[below] at (5,0) {focus};
\node[above right] at (4.11,0.62) {virtual image};
\end{tikzpicture}
$$

Off-axis object points use the same local rules, but the radial ray must be drawn
to the actual center of curvature. A vertex-directed line is usually non-normal to
the surface. At a small field angle the corresponding image
point follows the paraxial equation. At larger field angles, tangential and sagittal
focus differ and a single sharp image plane can fail even when the axial focus has
been calibrated.

## Mirror equation, magnification, and calculation checks

Replacing $2/R$ with $1/f$ gives the signed mirror equation

$$

\frac{1}{s}+\frac{1}{s'}=\frac{1}{f}.

$$

It has the same algebraic form as the thin-lens equation, but the physical geometry
is different. Object and real-image points for a mirror lie on the same side of the
reflecting surface. A positive image distance means reflected rays meet in front of
the mirror. A negative image distance means that their backward extensions meet
behind it. Substitution into a familiar-looking formula is safe only after the
reflection sign convention has been assigned.

Solving the equation for image distance and combining it with similar triangles
gives

$$

s'=\frac{fs}{s-f},
\qquad
m=\frac{y'}{y}=-\frac{s'}{s}=-\frac{f}{s-f}.

$$

The sign of $m$ classifies orientation. A real image from a concave mirror has
$s'>0$ and $m<0$, so it is inverted. A virtual image has $s'<0$ and $m>0$,
so it is upright. Magnification magnitude classifies size. These results are tied
to signed distances; taking absolute values before the final classification erases
the information that distinguishes virtual and real images.

$$
% caption: Similar-triangle geometry for magnification. The ray from the object tip to
% the vertex reflects symmetrically about the axis, so object and image heights obey
% $m=y'/y=-s'/s$, negative for the inverted real image.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.3,0)--(9.0,0) node[right] {axis};
\draw[acc,very thick] (7.5,-1.6) .. controls (8.2,-0.8) and (8.2,0.8) .. (7.5,1.6);
\draw[acc,very thick,->] (2,0)--(2,1.4);
\draw[black,very thick,->] (5,0)--(5,-0.7);
\draw[->,black,thick] (2,1.4)--(8.03,0);
\draw[->,black,thick] (8.03,0)--(4.7,-0.77);
\draw[black,dashed] (2,0)--(8.03,0);
\node[above] at (2,1.4) {object};
\node[below] at (5,-0.7) {image};
\node[below right] at (8.03,0) {vertex};
\end{tikzpicture}
$$

The denominator $s-f$ exposes the major concave-mirror boundary. At $s=f$, the
image distance has no finite value because the reflected rays are parallel. Just
outside the focus, $s'$ is large and positive; just inside it, $s'$ is large and
negative. A small uncertainty in object position near the focal point therefore
causes a large change in image position. The large derivative is a physical
sensitivity set by the image geometry.

$$
% caption: Image distance versus object distance for a concave mirror. Objects beyond
% the focus give real images ($s'>0$); objects inside it give virtual images ($s'<0$);
% both diverge as $s\to f$.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.6,1.5)--(6.5,1.5) node[right] {object distance};
\draw[->,black] (0.6,0.2)--(0.6,3.3) node[above] {image distance};
\draw[black,dashed] (3.0,0.3)--(3.0,3.15);
\draw[black,dashed] (0.62,1.5)--(6.3,1.5);
\draw[acc,very thick] plot coordinates {(3.25,3.05)(3.5,2.55)(3.9,2.15)(4.5,1.9)(5.4,1.72)(6.2,1.66)};
\draw[black,very thick] plot coordinates {(0.85,1.15)(1.4,0.95)(2.0,0.68)(2.55,0.38)(2.85,0.15)};
\node[below] at (3.0,0.3) {focus};
\node[acc,above] at (4.9,1.95) {real};
\node[black,below] at (1.7,0.75) {virtual};
\end{tikzpicture}
$$

Magnification has the same singular boundary. It is negative on the real-image
branch of a concave mirror and positive on the virtual branch. Large
$\lvert m\rvert$ near the focus is accompanied by a large sensitivity to
object placement and a narrow depth of focus. A reported enlarged image needs both
its height ratio and its orientation. Projectability follows from the sign of image
distance and the intersection of physical reflected rays.

Four checks expose most sign errors before numerical work is accepted. A distant
object at a concave mirror must give an image near the positive focal point. An
object at twice the focal distance must give a same-size inverted image at twice the
focal distance. An object inside the focus must give a negative image distance. A
real object at a convex mirror must give a negative image distance and positive
magnification. These checks test geometric limits alongside the algebraic result.

## Concave-mirror image regimes

A real object in front of a concave mirror has image side, size, and orientation set
by its position relative to the focus and center of curvature.
The center lies at $2f$. The five limiting regimes form one continuous sequence
under the mirror equation and principal-ray geometry.

When $s>2f$, the image forms between $f$ and $2f$. It is real, inverted, and
reduced. This is the distant-scene regime: a large object maps to a small image near
the focus, where a sensor or screen can intercept the reflected rays.

At $s=2f$, the image also lies at $2f$. The magnification is $-1$: real,
inverted, and equal in height to the object. Equal object and screen distances,
both measured from the vertex plane, provide a direct bench check. Accurate
vertex-plane location remains necessary.

For $f<s<2f$, the real image moves beyond the center of curvature. It is inverted
and enlarged. Projection requires this regime when a screen image larger than the
object is needed, but its long image distance consumes bench length and increases
the effect of a screen-position uncertainty.

$$
% caption: Enlarged real image from a concave mirror, object between focus and center.
% The reflected rays meet beyond the center of curvature, forming an inverted image
% taller than the object.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.2,0)--(9.0,0) node[right] {axis};
\draw[acc,very thick] (7.5,-2.4) .. controls (8.25,-1.2) and (8.25,1.2) .. (7.5,2.4);
\filldraw[draw=black,fill=black!8] (2,0) circle (1.8pt);
\filldraw[draw=black,fill=black!8] (5,0) circle (1.8pt);
\draw[acc,very thick,->] (2.8,0)--(2.8,1.1);
\draw[black,very thick,->] (0.9,0)--(0.9,-1.5);
\draw[->,acc,thick] (2.8,1.1)--(8.0,1.1);
\draw[->,acc,thick] (8.0,1.1)--(0.6,-1.61);
\draw[->,acc!70!black,thick] (2.8,1.1)--(8.0,-1.5);
\draw[->,acc!70!black,thick] (8.0,-1.5)--(0.6,-1.5);
\draw[->,black,thick] (2.8,1.1)--(0.75,-1.72);
\node[above] at (2.8,1.1) {object};
\node[below right] at (2,0) {center};
\node[above] at (5,0.05) {focus};
\node[below] at (0.9,-1.55) {image};
\end{tikzpicture}
$$

At $s=f$, rays from the object point leave the mirror parallel. The image is at
infinity in the paraxial model, so a finite screen cannot be positioned at the
image plane. The arrangement is valuable for a collimated-beam test but unstable
for focal-length work based on a screen distance.

For $0<s<f$, the reflected rays diverge and the concave mirror forms an upright
enlarged virtual image behind the surface. The image distance is negative and
$m>1$. The eye can focus on this apparent image because it receives the same
diverging bundle that would have come from a physical object at the virtual point.

$$
% caption: Upright virtual image from a concave mirror, object inside the focal
% distance. The reflected rays diverge, and their dashed backward extensions meet
% behind the mirror at an enlarged upright virtual image.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.3,0)--(9.0,0) node[right] {axis};
\draw[acc,very thick] (5.5,-2.2) .. controls (6.2,-1.1) and (6.2,1.1) .. (5.5,2.2);
\filldraw[draw=black,fill=black!8] (1,0) circle (1.8pt);
\filldraw[draw=black,fill=black!8] (3.5,0) circle (1.8pt);
\draw[acc,very thick,->] (5,0)--(5,1.0);
\draw[black,very thick,->] (7.67,0)--(7.67,1.67);
\draw[->,acc,thick] (5,1.0)--(6.0,1.0);
\draw[->,acc,thick] (6.0,1.0)--(3.0,-0.2);
\draw[black,dashed] (6.0,1.0)--(7.9,1.79);
\draw[->,acc!70!black,thick] (5,1.0)--(6.0,1.667);
\draw[->,acc!70!black,thick] (6.0,1.667)--(3.0,1.667);
\draw[black,dashed] (6.0,1.667)--(7.9,1.667);
\node[left] at (5,1.0) {object};
\node[below] at (1,0) {center};
\node[below] at (3.5,0) {focus};
\node[above right] at (7.67,1.67) {virtual image};
\end{tikzpicture}
$$

The regimes connect continuously. Moving an object inward from far away shifts a
reduced real image from the focus toward the center, then sends an enlarged real
image away from the mirror. Crossing the focal point transfers the image to the
virtual branch behind the mirror. The corresponding ray intersections move through
the same sequence; the
equation gives signed location and magnification for a chosen object distance.

## Convex-mirror imaging and the plane-mirror limit

A real object and convex mirror have $s>0$ and $f<0$. The mirror equation gives

$$
\frac{1}{s'}=\frac{1}{f}-\frac{1}{s}<0.
$$

The image distance is therefore negative. Its magnitude lies between zero and
$\lvert f\rvert$:

$$
\lvert s'\rvert
=\frac{\lvert f\rvert s}{s+\lvert f\rvert}
<\lvert f\rvert.
$$

The virtual image is between the vertex and virtual focus. Since
$m=-s'/s$, its magnification is positive and less than one. Convex mirrors
therefore give upright reduced virtual images for all real-object distances. Their
wide field of view comes with image-size reduction and increasing off-axis
distortion toward the rim.

Object distance changes the convex image position smoothly. A very distant object
has a virtual image close to the virtual focus. Bringing the object close to the
mirror moves the virtual image toward the vertex and enlarges it slightly, while
retaining $0<m<1$. A convex rear-view or safety mirror therefore cannot supply
metric object distance from apparent size alone without a calibrated model of the
surface and viewing geometry.

Curving a mirror outward allows it to collect rays from a larger angular region than
a plane mirror of the same physical width. The price is a smaller, distorted image
and a field-dependent mapping between viewing angle and image position. Geometrical
optics treats the surface as a spherical reflector; a practical device also needs a
specified usable field, mounting angle, and distance range.

The plane-mirror result is a limiting check. If
$\lvert R\rvert$ tends to infinity, then $\lvert f\rvert$ tends to infinity and
the mirror equation becomes $s'=-s$. The magnification becomes $m=+1$. This
limit confirms the signed convention: plane-mirror images are upright, virtual,
and located behind the surface at equal distance.

## Measurement, alignment, and uncertainty

### Focal-length and radius measurement

Focal length can be measured in several ways, each with a different dominant
uncertainty. A distant target gives a fast estimate. Parallel input is approximated
when $s$ is much larger than the mirror radius, so the screen is translated until
the reflected image is sharp and the vertex-to-screen distance is recorded as
$f_{\rm est}$. The finite-distance correction follows directly from the mirror
equation:

$$
s'=\frac{fs}{s-f}
=f\left(1+\frac{f}{s}+\frac{f^2}{s^2}+\cdots\right).
$$

Using a target that is merely far by eye can bias the result high. The correction is
small only when $f/s$ is small compared with the stated relative uncertainty. A
distant roofline, collimator, or laboratory target at a recorded distance gives a
more defensible estimate than an unspecified distant object.

Finite-conjugate measurement avoids the distant-target approximation. For each
object–screen pair that forms a sharp real image, measure $s$ and $s'$ from the
vertex plane and calculate

$$

f=\frac{ss'}{s+s'}.

$$

Several pairs detect systematic errors more effectively than one pair. Near
$s=2f$, object and image distances are similar and their contributions to
focal-length uncertainty are balanced. Near $s=f$, image distance becomes large
and screen-position uncertainty is amplified. At very large $s$, the image lies
close to the focal plane and the correction to a distant-target estimate becomes
hard to resolve. A moderate real-image geometry generally gives the clearest
measurement.

Center-of-curvature testing measures radius more directly. A small illuminated mark
placed at $C$ sends rays toward the mirror along radii. Every paraxial ray returns
to the mark after reflection. The vertex-to-mark distance is $R$, and
$f=R/2$. A practical bench may place the source and viewing aperture close
together or separate outgoing and returning light with a partially reflecting
plate. Measure the vertex-plane distance; a holder’s front edge belongs to the
mechanical mount geometry. The vertex plane is the optical reference.

Surface measurements can give an independent radius estimate. A spherical surface
with sagitta $q$ over a chord of half-width $a$ has

$$

R=\frac{a^2+q^2}{2q}.

$$

This geometric result becomes sensitive when $q$ is small: a shallow curve
requires resolving a small sagitta against a much larger chord. It is best used as a
cross-check on an optical measurement, because coating thickness, local surface
figure, and a poorly defined edge can make a mechanical chord differ from the
effective reflecting region.

The reciprocal plot is a diagnostic for repeated finite-conjugate data:

$$
\frac{1}{s'}=-\frac{1}{s}+\frac{1}{f}.
$$

A graph of $1/s'$ against $1/s$ should have slope $-1$ and intercept
$1/f$ for an aligned paraxial mirror. A slope discrepancy can indicate a reversed
distance sign, a reference offset that changes between runs, a screen tilt, or a
focus criterion that changes with magnification. A nonlinear fit in the original
distance variables is preferred when both distances have substantial uncertainty;
the reciprocal graph remains an effective error detector.

A complete measurement record separates random scatter from common offsets. Store
raw object, screen, and mirror-stage coordinates together with their reduced
distances. A calibrated shift of the mirror vertex changes every derived
finite-conjugate pair in a correlated way. Repeating the same uncorrected
measurement reduces random focus scatter but leaves the common reference error in
the mean. Center-of-curvature return imaging, a known calibration mirror, or an
independent sagitta measurement can constrain that offset.

Focal length also depends on the active surface zone. A value obtained with a
central stop is a paraxial result for that aperture. Opening the stop may change the
selected best plane even when the radius measurement is unchanged. State the
illuminated diameter, target wavelength range, and focus metric with the final
number. These conditions allow another measurement to distinguish a true surface
or mounting difference from a change in the operating definition of focus.

### Alignment, focus selection, and uncertainty

Mirror measurements are alignment problems as well as distance problems. The object,
the vertex normal, and the screen center need one common bench axis. A mirror tilt
changes outgoing ray angles; a screen tilt makes one side of the image sharp at a
different axial coordinate from the other. Decentering clips part of the beam and
can change the apparent best focus through aberration. These effects are systematic
until measured and modeled; repetition leaves the systematic terms in every result.

A practical alignment sequence is:

1. Place object center, mirror vertex, and screen center on one bench line.
2. Limit the aperture to the paraxial region required by the claimed uncertainty.
3. Translate the screen through the sharp region from both directions and record
   repeated focus coordinates.
4. Record the vertex reference, target geometry, aperture, source band, and focus
   metric for every data point.
5. Compare signed ray construction with observed image side and magnification before
   combining repeated values of $f$.

Screen focus estimates a coordinate. A broad target, detector pixels, source
bandwidth, aberration, and screen grain create a range of positions that can appear
sharp. Define one selection rule, such as maximum edge contrast or minimum
line-pair width, and use it across the run. A single rounded screen coordinate
without a stated focus rule omits a significant uncertainty component.

Near the focal boundary, image-position sensitivity follows from

$$
\frac{\partial s'}{\partial s}
=-\frac{f^2}{(s-f)^2}.
$$

Its magnitude rises sharply as $s$ approaches $f$. A small target-stage motion
then produces a much larger screen shift. This behavior explains why an enlarged
real image near the focus is a poor casual choice for a precision focal-length
measurement. It also provides a physical criterion for selecting a less sensitive
object–screen geometry before data are taken.

For independent object and image distance uncertainties $u_s$ and $u_{s'}$,
propagation through $f=ss'/(s+s')$ gives

$$
u_f^2=
\left[
\frac{s'^2}{(s+s')^2}u_s
\right]^2+
\left[
\frac{s^2}{(s+s')^2}u_{s'}
\right]^2.
$$

The coefficients show which distance controls the result. A reduced image has large
$s$ and image distance near $f$, so screen-position uncertainty has the larger
coefficient. An enlarged image shifts weight toward object distance. At unit
magnification, equal distance uncertainties contribute equally. A common
vertex-reference shift is correlated across measurements and needs a covariance term
or calibration parameter.

Tilt is readily visible with a two-sided target. If upper and lower target marks
reach their sharpest screen images at different longitudinal positions, the optical
axis, screen plane, or mirror normal is misaligned. A central mark alone can conceal
the error. Record a focus interval or a target-dependent focus map when the
discrepancy exceeds the stage resolution.

Repeated data should be retained before averaging. For every object–screen pair,
store raw stage positions, direction of approach, aperture setting, screen criterion,
and any correction applied to the vertex reference. Plot focal-length residuals
against object distance, image distance, and acquisition order. A trend with
distance suggests a reference or alignment error; a drift with time suggests source,
temperature, or stage-settling change. Random scatter about the uncertainty model
is consistent with the stated measurement process.

## Spherical aberration, aperture, and model limits

The mirror equation describes rays close to the optic axis. A true spherical surface
brings marginal and paraxial rays from an axial object to different crossing points.
In a concave mirror, outer rays generally cross the axis closer to the mirror than
central rays. The sequence of axial crossings is longitudinal spherical aberration;
a screen at any one plane receives a finite blur patch from the axial object.

$$
% caption: Spherical aberration of a concave mirror. Marginal (edge) rays cross the
% axis closer to the mirror than paraxial rays, so no single plane images every
% reflected ray from an axial object.
\begin{tikzpicture}[>=stealth,font=\footnotesize,scale=0.9]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,->] (0.3,0)--(9.0,0) node[right] {axis};
\draw[acc,very thick] (7.4,-2.4) .. controls (8.25,-1.2) and (8.25,1.2) .. (7.4,2.4);
\draw[->,black,thick] (0.7,0.5)--(7.95,0.5);
\draw[->,black,thick] (0.7,-0.5)--(7.95,-0.5);
\draw[->,black,thick] (7.95,0.5)--(4.6,-0.07);
\draw[->,black,thick] (7.95,-0.5)--(4.6,0.07);
\draw[->,acc,thick] (0.7,1.9)--(7.6,1.9);
\draw[->,acc,thick] (0.7,-1.9)--(7.6,-1.9);
\draw[->,acc,thick] (7.6,1.9)--(5.4,-0.56);
\draw[->,acc,thick] (7.6,-1.9)--(5.4,0.56);
\filldraw[draw=black,fill=black!8] (5,0) circle (1.8pt);
\filldraw[draw=acc,fill=acc!25] (5.9,0) circle (1.8pt);
\node[below] at (3.8,-0.05) {paraxial};
\node[above] at (5.85,1.0) {marginal};
\end{tikzpicture}
$$

Reducing the aperture blocks marginal rays and narrows the geometrical blur. It also
reduces the reflected flux at the image, making the focus metric noisier. A measurement
aperture balances these effects: wide enough to give a repeatable
screen signal, narrow enough that the selected focal length represents the
paraxial model. At very small apertures, wave spreading becomes another limit and
requires a wave-optics description beyond the present ray model.

The best screen plane depends on the focus criterion. Minimum spot diameter,
maximum line contrast, and least-confusion circle can select slightly different
longitudinal positions in an aberrated system. Reporting a focal length without
aperture, focus criterion, and axial field position can omit a systematic shift
larger than the repeatability of the translation stage. A parabolic reflector has a
different axial parallel-ray behavior; the equations developed here remain specific
to spherical surfaces and their paraxial region.

Off-axis points add coma, astigmatic separation of meridional and sagittal focus,
and distortion. Their full calculation requires a more complete optical model, but
the experimental signature is direct: a centered axial target can give a sharp
paraxial image while a target near the usable field edge gives an asymmetric or
stretched spot. A focal-length claim should state whether it is axial or a
field-averaged performance measure.

Surface figure error, roughness, coating stress, and thermal deformation can create
blur or scattered light beyond the ideal spherical model. A mirror can therefore
have a repeatable paraxial focal distance yet poor contrast. Separate the reported
quantity—paraxial focal length, on-axis spot size, field performance, or reflected
throughput—and match the apparatus to that quantity.

An aperture sweep is a direct model check. Measure the selected focus plane at a
sequence of centered aperture diameters while holding target, vertex reference, and
focus metric constant. A stable result over the intended aperture supports the
paraxial approximation at the requested resolution. A systematic focus shift with
diameter identifies spherical aberration or a surface-zone error. Repeat the sweep
at a second target height to separate an axial aperture effect from off-axis
aberration or decenter.

Surface reflectance primarily affects signal level, while the ideal ray-direction
law remains geometric. A weak or uneven coating changes the practical measurement.
Lower contrast broadens a screen-focus maximum, and scatter can obscure the edge of a
test target. Represent the resulting focus uncertainty in the experimental record.

## Worked reductions, reporting, and model audit

> **Worked example (Concave-mirror image).** A concave mirror has
> $R=+0.400\ \mathrm{m}$, so $f=+0.200\ \mathrm{m}$. A $0.0500\ \mathrm{m}$ object
> lies $s=0.600\ \mathrm{m}$ in front of the vertex. The mirror equation gives
>
> $$
> \frac{1}{s'}
> =\frac{1}{0.200}-\frac{1}{0.600}
> =3.33\ \mathrm{m^{-1}},
> \qquad
> s'=+0.300\ \mathrm{m}.
> $$
>
> The positive image distance places the image in front of the mirror. The
> magnification and image height are
>
> $$
> m=-\frac{0.300}{0.600}=-0.500,
> \qquad
> y'=(-0.500)(0.0500\ \mathrm{m})
> =-0.0250\ \mathrm{m}.
> $$
>
> The image is real, inverted, and half the object height. The regime check agrees:
> $s=3f>2f$, so the image lies between $f$ and $2f$ and is reduced. A screen
> $0.300\ \mathrm{m}$ from the vertex intercepts it; a screen behind the mirror would
> contradict both the sign of $s'$ and the reflected-ray direction.

> **Worked example (Convex-mirror image).** A convex mirror has
> $R=-0.300\ \mathrm{m}$, so $f=-0.150\ \mathrm{m}$. For a real object at
> $s=0.600\ \mathrm{m}$,
>
> $$
> \frac{1}{s'}
> =\frac{1}{-0.150}-\frac{1}{0.600}
> =-8.33\ \mathrm{m^{-1}},
> \qquad
> s'=-0.120\ \mathrm{m}.
> $$
>
> With object height $y=0.0500\ \mathrm{m}$,
>
> $$
> m=-\frac{-0.120}{0.600}=+0.200,
> \qquad
> y'=+0.0100\ \mathrm{m}.
> $$
>
> The negative image distance and positive reduced magnification identify an upright
> virtual image. Its distance magnitude $0.120\ \mathrm{m}$ is below
> $\lvert f\rvert=0.150\ \mathrm{m}$, so it sits between the vertex and virtual
> focus — no screen can catch it.

Verification uses ray geometry, a calibrated viewing system, or a second optical
element that converts the diverging bundle to a real image.

> **Worked example (Finite-conjugate calibration).** A concave mirror produces a
> sharp real image with $s=0.750\ \mathrm{m}$ and $s'=0.300\ \mathrm{m}$. Both
> distances are measured from the vertex plane, so
>
> $$
> f=\frac{ss'}{s+s'}=\frac{(0.750)(0.300)}{0.750+0.300}
> =0.214\ \mathrm{m},
> \qquad
> R=2f=0.429\ \mathrm{m}.
> $$

The finite-conjugate result can be compared with a return-image radius measurement
only after both distances have been reduced to the same vertex reference. An
agreement between numbers measured from different rim marks can be accidental. If
the optical and sagitta results disagree beyond their combined uncertainty, inspect
aperture, surface zone, vertex offset, screen tilt, and the assumption that the
surface is spherical over the measured region.

The reported quantity should state its domain. A suitable report gives concave or
convex sign, vertex reference, focal length and uncertainty, aperture, source band,
axial or off-axis target condition, focus metric, and method of reduction. For
example: “Concave spherical mirror; paraxial on-axis focal length
$0.214\pm0.003\ \mathrm{m}$; $20\ \mathrm{mm}$ aperture; finite-conjugate
screen measurement; vertex-plane reference.” That statement identifies a
reproducible optical result with its physical conditions stated.

Several failure patterns have distinct causes:

- A real-image prediction with a negative screen distance usually indicates a
  reversed sign for the focal length, image distance, or distance reference.
- A calculated real image that cannot be found on the screen can arise from an
  object inside the focus, a screen placed on the wrong side of the vertex, or a
  hidden aperture/tilt constraint.
- A focal length that changes with aperture is evidence of spherical aberration or
  a changing focus criterion; it calls for an aperture-dependent model check.
- A focal length that changes with target height indicates field dependence,
  decentering, or a tilted setup.
- A return-image radius that disagrees with finite-conjugate imaging can expose a
  non-spherical zone or an uncorrected vertex offset.

The final audit combines limiting cases, signs, and model conditions:

1. Distant concave object gives $s'\to +f$ and $m\to 0^-$.
2. Object at the center gives $s'=s=2f$ and $m=-1$.
3. Object at the focus gives a parallel reflected bundle.
4. Concave object inside focus gives $s'<0$ and $m>0$.
5. Real object at a convex mirror gives $s'<0$ and $0<m<1$.
6. Aperture, field angle, vertex reference, and focus criterion must match the
   conditions under which focal length is reported.

The mirror equation is therefore a compact paraxial model with a precise domain:
one spherical reflecting surface, stated sign convention, distances from the vertex
plane or calibrated principal reference, small ray angles, stated aperture, and a
defined focus criterion. Within that domain, ray construction, algebra, and
measurement give mutually checkable descriptions of the same image geometry.

Reciprocity gives a further bench check for real concave images. If an object at
$s$ forms a real image at $s'$, then a source placed at the former image plane
forms a real image at the former object plane under the same aperture and alignment.
The mirror equation is symmetric under interchange of $s$ and $s'$. The
magnification magnitude changes to its reciprocal, while the focal length remains
unchanged. This test is sensitive to an incorrectly marked vertex plane because
the two distance reductions then fail to exchange cleanly.

The comparison must retain the same physical conditions. Changing aperture changes
the relative contribution of marginal rays; changing screen material or target
contrast changes the focus metric; moving the mirror in its mount changes the
vertex reference. A reciprocal pair that differs beyond its uncertainty therefore
points to a concrete alignment, reference, or model issue. Average only comparable
measurements into a single focal-length value.

Dimension checks complete the reduction. Radius, focal length, object distance, and
image distance use one length unit. The mirror equation has units of inverse length,
and magnification is dimensionless. A reported $R=2f$ result with inconsistent
units can preserve plausible numerical ratios while representing the wrong physical
mirror. Keeping units until the final rounding step also makes a millimeter-scale
vertex offset visible against a meter-scale focal length.

Finite-conjugate measurements deserve a separate uncertainty treatment. The focal
length obtained from object and image distances is most sensitive to the shorter of
the two distances. A fixed scale-reading error therefore has a larger relative effect
when either the object or the screen lies near the focal plane. Moving the object to a
more balanced conjugate pair reduces that sensitivity, but the image must remain far
enough from the mirror to locate the vertex plane reliably. A bench position recorded
from the front rim of a thick mount is not an object distance until the rim-to-vertex
offset has been measured and applied consistently to both trials.

Focus selection also contributes physical uncertainty. A coarse screen, a wide target,
or a bright halo can make several screen
positions appear equally sharp. Record the interval over which the selected focus
metric remains acceptable, then propagate half that interval with the scale reading
and vertex-reference uncertainty. Repeating the measurement at several object
distances exposes a systematic aperture or alignment effect: a random spread narrows
with repeated observations, whereas a focal length that drifts in one direction as
the aperture opens is evidence that the paraxial model is losing accuracy. The final
uncertainty should retain that distinction instead of averaging incompatible focus
conditions into a single optimistic number.