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There are two lines on the wikipedia pages for "cardinal points" and "focal length" that seem to contradict each other, and I would be extremely grateful if someone could explain to me why they do not. In the page for cardinal points, it says:

If the medium surrounding the optical system has a refractive index of 1 (e.g., air or vacuum), then the distance from the principal planes to their corresponding focal points is just the focal length of the system. In the more general case, the distance to the foci is the focal length multiplied by the index of refraction of the medium.

This makes sense to me. I also get that these principal planes can often be located outside of the lens with some clever optics, allowing for lenses that are physically shorter than their focal length. However, on the page for focal length, the page reads:

When a photographic lens is set to "infinity", its rear nodal point is separated from the sensor or film, at the focal plane, by the lens's focal length. Objects far away from the camera then produce sharp images on the sensor or film, which is also at the image plane.

I don't see how these can both be true, because if the focal point, the point as I understand it to be where all the light converges, was on the film plane, an image wouldn't be rendered, it would just be an indistinguishable point of light. Does the light not have to travel a distance past the focal point to the film plane in order to form an image?

I think it is possible that I am getting my front and rear nodal points confused, or that I have a larger fundamental misunderstanding about how focal length is measured. Thank you so much for your help!

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"because if the focal point, the point as I understand it to be where all the light converges, was on the film plane, an image wouldn't be rendered, it would just be an indistinguishable point of light."

This understanding is incorrect... at all points on an objective lens exists all of the light required to form an mage (a portion of the total). That is why you can have a 200mm f/4 lens (50mm objective element) and a 200mm f/2 lens (100mm objective element). It is more accurate to understand objective lens area (aperture, f#) as being "stacking images."

The point at where all light converges is then where all source points from all areas of the objective element converge as a single point at the image plane. i.e. a point source in the scene converges as a point on the sensor.

This is a related diagram I made regarding DoField/DoFocus, but it shows the concept. The narrow aperture light/paths also exist w/in the wide aperture image; I just didn't include them for clarity/simplicity. Only the blue source is in true focus; and the narrow aperture image is darker (grey) because there are fewer images (light paths) focused/stacked/combined at the image plane.

enter image description here

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The focal length of a lens is a measurement taken when the lens is imaging a far distance object like a star. If the lens structure is a single symmetrical (convex – convex) then this measurement is taken from the center of the lens to the focused image. A distant object is at an infinite distance when its light rays arrive at the camera lens as a bundle of parallel rays.

For all practical math purposes an object is at an infinite distance 1000 meter (1000 yards) distant. The ray trace starts from a single point on the subject and is then extended to show how it traverses the lens. The ray trace is then continued showing its path downstream from the lens. If properly focused, the trace inside the camera will depict a triangle with its apex just kissing off on the surface of the digital image sensor or film. The key point being, the ray trace is just a single point on the subject.

In actuality every point on the subject could be ray traced. Such a ray trace reveals that each point on the subject has a ray trace that resembles a cone of light. You see, the lens works by fracturing the light from a subject (vista) into a googolplex of cones of light. Each has an apex. Since all lenses have optical defects called aberrations, the apex of each ray trace as they kiss off on the sensor is never a point; it is actually a tiny circle of light juxtaposed with others and it has scalloped boundaries. Because it is seen as an imperfect circle jumbled with others, it is called a circle of confusion. The image derived from the lens is thus, countless cones of light each with an apex that kisses off on the sensor. As a rule, a ray trace to show the focal length is just a trace of a ray passing through the center (axis) of the lens. All the other rays are not shone.

Opticians are unable to eliminate these aberrations. The best that can be done is to mitigate each. This is done by designing the lens so that it consists of several glass elements. Some are dense glass, some are less dense, some are convex with positive power, and some are concave with negative power. Some are cemented together, some are air-spaced. The air-space has a figure (shape) formed by the surfaces of lens that sandwich it in. This lens shaped air-space also acts like a weak lens. There are seven major types of aberrations. To mitigate is takes seven or more glass lenses of different powers. Because the camera lens is a complex array of glass, the measuring points used to find object distance and image distance are two cardinal points or nodal. Their placement likely does not fall in the center of the lens barrel. The forward nodal is the measuring point for object distance. The rear nodal is the measuring point for image distance.

The optician, likely uses lenses of different powers and this causes the nodal points to be shifted around. A true telephoto lens, as compared to a long lens of the same focal length, has its rear nodal moved forward. It can even fall in air ahead of the lens. This shortens the lens barrel making it less awkward than its long lens counterpart. Often, a wide-angle lens has a focal length too short to reach the image sensor / film. The optician shifts the rear nodal to lengthen the back-focus (distance last lens to film/sensor).

The focal length is measured from the rear nodal to the apex of the cone of the image forming rays. When we focus on an object closer than infinity, the cone of the image forming rays is elongated due to the fact that then has limited refractive powers. Refract is Latin to bend backwards or inwards.

The key point for you: Ray traces to show focal length are simplified drawings, likely only the axial rays are shown. The way a lens works is to fracture the subject into countless points. Each sends out light rays that traverse the camera lens. Each traces out a cone of light. There will be a googolplex of cones of light and thus a googolplex of circles of confusion. When focusing on objects closer than infinity we focus by moving the lens further away from the film / sensor. Nobody said this stuff is easy!

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I believe I now see the confusion (sorry, slow on the uptake).

Most lens and imaging diagrams do indeed give the impression that all the light comes to a point at the focus point, it's even in the name. However that's not what is actually happening. The smallest point of light is actually an image, not a point.

The focal length of a lens refers to the fixed image distance of focus for an infinite object distance. This focal point is actually a focal image.

This diagram from an old physics book shows this better than most:

Focus & Image

Note that last (f) At infinity. The focus point F' is where the image forms, it's not actually a point.

(c) & (d) explain macro photography.

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When talking about light converging on the image plane in photography, we're talking about light from a particular point in the camera's field of view converging at the plane that contains the film or digital imaging sensor. In photography, this plane is called the focal plane or image plane. In the scientific field of optical physics, the terms focal plane and focal point are defined quite differently. When one reads such terms it is important to understand which usage for them is being employed.

Light from a singular point within the camera's field of view falls on the entire surface of the front of the lens. If the lens is properly focused at the distance which that point source of light is from the camera, then the light from that singular point that falls on the entire surface of the front of the lens converges on the same point at the image plane. Light from other points in the camera's field of view that are the same distance also converges on points on the image plane, but the points at which the light from different point sources converge on the image plane are not the same point. Light that is at an angle so that it is in the upper left corner of the camera's field of view will converge in the lower right corner of the image plane. Light that is at an angle so that it is in the top center of the camera's field of view will converge at the bottom center of the camera's image plane. Light that is at the center right of the camera's field of view will converge at the center left of the camera's image plane, and so on. Only light that is located on the lens' optical axis will converge in the center of the camera's image plane.

When a lens is focused at infinity, then the point source of light in question is sufficiently far enough away so that the light from that point which is reaching the front of the lens is sufficiently collimated to be indistinguishable from a source of light that is infinitely far away. Consider a star. We think of it as a point source of light. But stars are huge! They're much larger in diameter than the front element of any lens I've ever seen! Thus the light rays from a star (except our own sun, which is about one-half degree of arc in diameter as observed from the Earth's surface) that reach a lens here on Earth are almost perfectly parallel. This is what we refer to as collimated light.

We're not talking about all light falling on the front of the lens from every conceivable angle converging on the same point at the imaging plane. We're talking about light from one specific point within the camera's field of view falling on one specific point of the camera's film or sensor.

It seems you might be getting tripped up by the two different types of ray diagrams that are common. They look similar, but depict two very different things. One traces multiple rays of collimated light from a single point source at infinity. The other traces single rays striking the front of the lens from each of multiple points within the lens' field of view. In the first case, the rays converge on the sensor/film/focal plane. In the second case the rays from opposite sides of the lens cross over halfway between the lens and the sensor/film/focal plane. These two types of diagrams are not showing the same thing.

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