focal length

Focal length controls how much a camera magnifies and how wide a slice of the world it captures. Intuitively, a long focal length (a telephoto lens) acts like a telescope: it zooms in, fills the frame with a small far-away region, and flattens depth. A short focal length (a wide-angle lens) takes in a sweeping view and exaggerates near-far size differences. In the pinhole model it is simply the distance from the pinhole to the image plane.

There are two units to keep straight. In physical optics, focal length is a property of the lens measured in millimeters (e.g. a 50 mm lens), defined as the distance behind the lens at which parallel incoming rays converge. In computer vision, focal length appears in the intrinsic matrix as fx and fy measured in pixels — the same distance, but expressed in units of pixel pitch. The conversion is f_pixels = f_mm x (image_width_pixels / sensor_width_mm). This is why fx and fy can differ: if the pixels are not square, the same physical focal length corresponds to different pixel counts horizontally and vertically.

In the projection equations x = fx X / Z + cx, larger f means a scene point at fixed depth lands farther from the principal point, i.e. larger on the sensor — that is magnification. Focal length and field of view are inversely linked through the sensor size: doubling f roughly halves the angular field of view for the same sensor. For computer vision it is the focal length in pixels that matters, because algorithms reason about pixels, not millimeters; the millimeter figure on the lens barrel only becomes meaningful once you also know the sensor dimensions.