Lens Field of View Calculator

$M$ is the magnification, the size of the image on the sensor divided by the size of the scene, $f$ the focal length and $s$ the distance from the sensor to the scene. Of the two roots, the one below 1 applies: the image is smaller than the scene.

$u$ is the distance from the lens to the scene.

$\mathrm{FOV}_x$ and $\mathrm{FOV}_y$ are the width and height of the field of view on the scene, and $H_x$ and $H_y$ the sensor’s width and height; in portrait they swap.

$R_w$ is the length of the scene one pixel covers on the plane in focus, and $R_{sx}$ the number of pixels across the sensor.

$\theta_x$ and $\theta_y$ are the angles off the optical axis at which the lens sees the left and right, and the top and bottom edges of the scene; the angles of view are $2\theta_x$ and $2\theta_y$. With the lens focused at infinity, as on lens data sheets, they are $2\arctan(H_x / 2f)$ and $2\arctan(H_y / 2f)$.

$a$ and $a'$ are the distances from the front principal plane $H$ to the scene and from the rear principal plane $H'$ to the sensor, and $\overline{HH'}$ the distance between the two planes. The whole lens moves to focus.

$\theta_x$ is the angle, at the entrance pupil, of the scene point whose chief ray (the ray through the centre of the aperture stop) reaches the edge of the sensor, found by tracing; $u_E$ is the distance from the entrance pupil to the scene. Likewise for $\mathrm{FOV}_y$ and the diagonal.

$D$ is the distortion: $r$ is where the chief ray from a scene point at height $h$ lands on the sensor, and $M h$ where the ideal thin lens would put it.

$c$ is the crop factor: the diagonal of the 36 × 24 mm frame, 43.27 mm, divided by the sensor’s diagonal. $f_{35}$ is the 35 mm equivalent focal length, the focal length that gives the same angle of view across the diagonal on that frame.

• The main results come from a typical published lens design, scaled to the focal length and traced ray by ray, so the field, the angles and the distortion follow from its optics; the ideal thin lens’s values are in the line under each result.
• The design is chosen by how far off the optical axis the image corner lies: up to 23°, a double Gauss; up to 46°, a retrofocus wide-angle; beyond, a very wide-angle (fisheye) design.
• Real lenses of the same field can distort more or less than the design: of the eight lenses whose data sheets were read, two with the image corner about 37° and 38° off the axis list −6.84 % and −18.4 %, where the retrofocus design gives −2.4 % and −2.6 %.
• Rays are traced at one wavelength, 587.6 nm; colour is not included.
• The angles are those at the set distance. Lens data sheets give them with the lens focused at infinity, slightly wider: by up to about 1 % at a distance of 100 focal lengths, and more at closer distances.
• Pixels are taken as square.
• Their size is the sensor maker’s pixel size or, where the maker gives the sensor’s size, its width divided by the number of pixels across it; every other dimension follows from those two. For Other sensor, they are the values entered.
• The pixel counts are those a camera delivers by default.
• The lens must also cover the sensor’s format, and the lens maker’s data decides the final setup.
• The crop factor is taken across the diagonal, against the 36 × 24 mm frame.
• The neighbouring stock lenses are the standard focal lengths of two makers’ current C-mount machine vision series and, for the security-camera sensors, one maker’s current fixed lenses (decision R2).

$\mathbf{d}$ is the unit vector from the scene centre $\mathbf{C}$ to the camera, from the yaw $\psi$ and the pitch $\varphi$.

$\mathbf{S}$ is the sensor’s place and $\mathbf{E}$ the entrance pupil’s, the point the camera sees from; $u_E$ is the distance from the entrance pupil to the scene, as in Field of View.

$\mathbf{r}$ and $\mathbf{u}$ are the image’s right and up in the model’s coordinates, and $\mathbf{r}_0$ and $\mathbf{u}_0$ the same before the roll $\rho$. Straight above or below the scene centre the image’s right is $\mathbf{r}_0$, set by the yaw. A pixel at a distance $r$ from the image’s centre shows the direction $\theta(r)$ off the optical axis, from the traced lens, as in Field of View.

• The camera points at the scene centre, which is the centre of the image.
• The model’s coordinates are taken in the unit chosen, with z up.
• Only the model’s geometry is read: materials, colours and textures in the file are not, and an OBJ file’s groups are drawn as one part.
• The lens design and the other assumptions of Field of View apply.

• The lens is a typical published design scaled to the focal length, not your lens. Real lenses of the same field distort more or less (the assumption of Field of View, with the eight data sheets’ figures).
• The image shows geometry only: where each point of the part lands on the sensor. It shows no blur (focus, depth of field, diffraction or aberrations other than distortion), no vignetting or fall-off in brightness towards the corners, no colour fringes, no noise and no motion blur. Every point is drawn sharp.
• The shading is a fixed light that shows the part’s shape, not the station’s lighting; surfaces are one grey, without their colour, gloss or texture.
• Black areas lie beyond the design’s field. A real lens may cover more or less of the sensor, and a camera’s housing, mount or hood may also cut the image.
• The camera in the isometric view is a symbol: the camera body, the lens barrel and the rest of a station are not modelled, so they cannot block the view here as they may in place.
• The image is drawn at the screen’s resolution, not the sensor’s pixels; detail finer than a screen pixel is not shown as the sensor would record it.
• The lens is focused on the plane through the scene centre, square to its axis; parts of the model nearer or farther are drawn sharp all the same.
• The model is read as given: its units as chosen and its z up; the page does not repair or check its geometry.