Field of View
Introduction
In photography, field of view (FOV) describes the angular extent of a given scene that is imaged by a camera. It is used interchangeably with the more general term angle of view (AOV).
In this blog post, I would like to derive the equation for field of view.
Thin Lens Equation
The thin lens equation is the fundamental equation that is used for deriving the FOV equation.
Assuming the thickness of the lens is infinitely small, the thin lens equation
$$
\frac{1}{f} = \frac{1}{u} + \frac{1}{v}
$$
where $f$ is the focal length, $u$ is the object distance to the lens, $v$ is the image distance to lens.
The proof of the thin lens equation can be found in my previous blog post “How Camera Lens Works”.
Field of View
Suppose the focus plane is at the object $AB$. The object distance to the lens is $u$, the image distance to the lens is $v$, the focal length of the lens is $f$, and the sensor size is $d$. Based on the simple geometry, we have
$$
\tan \left( \frac{\theta}{2} \right) = \frac{\frac{d}{2}}{v} = \frac{d}{2v}
$$
where $\theta$ is the field of view angle.
Solving this equation, we have
$$
\theta = 2 \arctan \left( \frac{d}{2v} \right)
$$
With the thin lens equation, we have
$$
v = \frac{uf}{u - f}
$$
Therefore, the field of view angle $\theta$ can be expressed as
$$
\theta = 2 \arctan \left( \frac{d (u - f)}{2uf} \right)
$$
When $u = \infty$ or $u \gg f$, we have $v \approx f$, and the field of view angle $\theta$ can be approximated as
$$
\theta \approx 2 \arctan \left( \frac{d}{2f} \right)
$$
FAQs
Does Camera with Prime Lens Have a Fixed Field of View?
Strictly speaking, the field of view of a camera with a prime lens is not fixed.
By definition, prime lens has a fixed focal length but the user can adjust the focus plane so that different objects in the field of view can be in focus. The field of view will remain roughly the same if $u \gg f$. However, according to the field of view precise equation we have just derived, numerically the field of view will change slightly due to the change of the focus plane. It is just our eyes may not be able to tell the difference in most of the scenarios.
In addition, prime lens should not to be confused with fixed focus lens. The fixed focus lens not only has a fixed focal length but also has a fixed focus plane, which means the field of view is always fixed.
How Does Field of View Change for Camera with Zoom Lens?
For a camera with a zoom lens, the focal length of the lens can be adjusted. Therefore, the field of view can be adjusted as well.
If the user zooms in, the focal length of the lens increases, and the field of view angle decreases, and the object in the field of view appears larger. If the user zooms out, the focal length of the lens decreases, and the field of view angle increases, and the object in the field of view appears smaller.
References
Field of View