Curved mirrors form images by reflection. In geometric optics, the most important quantitative tool for a single spherical mirror is the
mirror equation,
\[
\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}.
\]
Here \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance. Once the image distance is known,
the size and orientation of the image are described by the magnification
\[
m=-\frac{d_i}{d_o}.
\]
If the object height is \(h_o\), then the image height is
\[
h_i=m\,h_o.
\]
These three relations are enough to solve most introductory mirror problems. The only tricky part is the
sign convention. Different textbooks use different coordinate conventions, and that is why mirror problems often look inconsistent even when the physics is the same.
This calculator supports two common choices.
In the classroom / real-object-positive convention, a concave mirror has \(f>0\), a convex mirror has \(f<0\),
and a real object in front of the mirror has \(d_o>0\). In that same setup, a positive \(d_i\) corresponds to a real image in front of the mirror,
while a negative \(d_i\) corresponds to a virtual image behind the mirror.
In the Cartesian signed convention, the mirror vertex is placed at the origin and positions in front of the mirror are negative,
while positions behind the mirror are positive. Then a real object in front of the mirror has \(d_o<0\). For a concave mirror, the focus is also in front,
so \(f<0\). For a convex mirror, the focus is behind the mirror, so \(f>0\). In this convention, a real image has \(d_i<0\) and a virtual image has \(d_i>0\).
The formulas themselves stay the same; only the meaning of the signs changes.
The magnification contains two different pieces of information. First, the sign of \(m\) tells you the image orientation.
If \(m<0\), the image is inverted. If \(m>0\), the image is upright. Second, the magnitude \(\lvert m\rvert\) tells you the size comparison.
If \(\lvert m\rvert>1\), the image is larger than the object. If \(\lvert m\rvert<1\), it is smaller. If \(\lvert m\rvert=1\), it has the same size.
This matters because some informal examples quote only the size factor and ignore the sign, which can make a result such as \(m=-1\) get described loosely as “magnification 1.”
The calculator therefore shows both \(m\) and \(\lvert m\rvert\).
For example, in a Cartesian-sign setup with a concave mirror, \(f=-15\ \text{cm}\) and \(d_o=-30\ \text{cm}\), the mirror equation gives
\[
\begin{aligned}
\frac{1}{d_i} &= \frac{1}{f}-\frac{1}{d_o} \\
&= \frac{1}{-15}-\frac{1}{-30} \\
&= -\frac{1}{15}+\frac{1}{30} \\
&= -\frac{1}{30},
\end{aligned}
\]
so \(d_i=-30\ \text{cm}\). Then
\[
m=-\frac{d_i}{d_o}=-\frac{-30}{-30}=-1.
\]
The negative sign means the image is inverted, while the magnitude \(\lvert m\rvert=1\) means it is the same size as the object.
This is exactly the familiar case where the object is placed at the center of curvature of a concave mirror.
The ray diagram in this calculator uses two standard construction rays from the top of the object. One ray travels parallel to the optical axis and reflects
through the focus for a concave mirror, or reflects so that its backward extension passes through the focus for a convex mirror.
The second ray goes to the mirror vertex and reflects symmetrically. Where the reflected rays meet, or where their dashed backward extensions meet,
the top of the image is found.
A makeup mirror is a classic concave-mirror example. When your face is inside the focal length, the image is virtual, upright, and enlarged.
A security mirror is usually convex, because convex mirrors keep images upright and reduced, increasing the field of view.
At more advanced levels, real mirrors depart from the ideal paraxial model because of spherical aberration, but the mirror equation and magnification formula
remain the standard foundation for basic geometric optics.
