# The parabolic mirror and its remarkable reflection property

A parabolic mirror is a mirror that has the shape of a rotating paraboloid. The latter one is a space surface of equation \(z=a\left(x^2+y^2\right)\) (see example in the figure below for \(a=1\)), obtained by rotating the parabola of equation \(z=ax^2\) about the z-axis. The projection of this surface onto the x–z plane yields the original curve \(z(x)=ax^2\).

A rotating paraboloid has a remarkable property known as the **reflection property**, which finds application in car headlights and satellite dishes. In fact, the latter ones are designed with the shape of a paraboloid for the purpose of having the *focus* as the point of collection or emission of the electromagnetic waves (e.g. the light). To be precise, the reflection property states that **“an electromagnetic beam parallel to the axis of a paraboloid and incident on it converges on the focus after being reflected”**.

The reflection property is based on very simple physical principles. When light (or any electromagnetic ray) is reflected off a surface:

- The incident ray, the normal to the reflecting surface at the point of incidence and the reflected ray lie on the same plane;
- The incidence angle is equal to the reflection angle.

Due to the symmetry of the three-dimensional surface of the paraboloid, to prove the reflection property it is sufficient to consider the geometry of the propagating rays in the x–z plane. Let us consider a parabola of equation \(z=ax^2\) characterized by symmetry axis coincident with the x-axis (\(z=0\)) and *focus* \(F\) placed at the coordinates

\(\displaystyle \left(0; \frac{1}{4a}\right)\).

For sake of simplicity, let us imagine to isolate, from the electromagnetic beam parallel to the axis of the parabola, a geometric ray of equation \(x=x_0\) incident at the point \(P\left(x_0; z_0\right)\), where \(z_0=ax_0^{2}\).

Once that the parabola, the incident ray and consequently the incidence point have been fixed, it is necessary to individuate the normal line to the parabola at that point. In general, the normal line to a curve at a fixed point is the line through that point and *perpendicular* to the tangent line.

For a parabola of equation \(z=ax^2\), it is easy to prove that the tangent line at the point \(P(x_0; ax_0^2)\) has the slope equal to \(m=2ax_0\) (see Appendix A), whereas as a result the normal has slope equal to

\(\displaystyle m’=-\frac{1}{2ax_0}\).

Now, with the aim to determine the Cartesian equation of the reflected ray, we have to both remember that the slope of a straight line is equal to \(\tan{\theta}\), where \(\theta\) is the angle that it forms with the horizontal x-axis, and to relate the angle \(\alpha\) that the tangent line forms with the x-axis (see the figure above for \(a=1\) and \(x_0=2\)) with the angle \(\beta\) that instead the reflected ray forms with the x-axis.

Using the perpendicularity conditions, the relation between complementary angles, it is rather easy to find

\(\displaystyle \beta=2\alpha-\frac{\pi}{2}\).

Then, using the complementary angle identity

\(\displaystyle \tan{\frac{\pi}{2}-\theta}=\cot{\theta}\)

and the double-angle trigonometric formula

\(\displaystyle \cot{2\alpha}=\frac{1-\tan^2{\alpha}}{2\tan{\alpha}}\),

it is straightforward to write down that the slope of the reflected ray is equal to

\(\displaystyle m_r=\frac{\tan^2{\alpha}-1}{2\tan{\alpha}}\).

Now, knowing that the slope \(m\) of the tangent line is equal \(\tan{\alpha}=2ax_0\) and substituting it in the expression of \(m_r\), we can obtain

\(\displaystyle m_r=\frac{4a^2x_0^2-1}{4ax_0}\).

Utilizing the Cartesian equation of a straight line \(z=m_r(x-x_0)+z_0\) crossing the given point \(P(x_0; z_0)\), where \(z_0=ax_0^2\), and slope \(m_r\), it is possible to find the following equation for the reflected ray

\(\displaystyle z=\frac{4a^2x_0^2-1}{4ax_0}(x-x_0)+ax_0^2\).

In the end, with the purpose to find the intersection between the reflected ray and the z-axis, setting \(x\) to zero and simplifying, we finally gets

\(\displaystyle z=\frac{1}{4a}\),

which is the z-coordinate of the focus \(F\) of the parabola of equation \(z=ax^2\).

*Quod Erat Demonstrandum.*

### APPENDIX A

There exist two methods to determine the slope of the tangent line to a curve at a fixed point:

- Considering the intersection of the curve with the tangent line; this is given by the system of equations \(\begin{cases} z=z(x) \\ z-z(x_0)=m(x-x_0)\end{cases}\). Since, the tangent line meets the curve in exactly one point, at the aim to find the desired value of the slope \(m\), it is necessary that the system has a unique solution. In the case of the parabola of equation \(z=ax_0^2\), the system of equations \(\begin{cases} z=ax^2, \\ z-z(x_0)=m(x-x_0)\end{cases}\) leads to the quadratic equation \(ax^2-mx+mx_0-ax_0^2\), which to have coincidental solutions implies that the discriminant \(m^2-4a(mx_0-ax_0^2)\) must vanishes. Observing that the polynomial \(m^2-4ax_0m+4a^2x_0^2\) is equal to the square of the binomial \(m-2ax_0\), it is outspoken to see that \(m\) must be equal to \(2ax_0\);
- Calculating, at the point of interest of coordinates \(P\left(x_0; z_0\right)\), the derivative \(z'(x_0)\) of the function \(z(x)\) representing the curve. In our case, it is straightforward to find that \(m=z'(x_0)=2ax_0\).