I am a beginner and I recently started reading about catadioptric systems. While I understand how they can be used to simulate longer focal lengths, I am still confused as to how the entire frame is captured, since the mirror obstructs the center. The question is probably trivial, but I haven't been able to find an answer online.
Every point on the transmissive surface (the part that lets light into the lens) of the front of any lens is illuminated by every point of light in the field of view. The lens then (more or less successfully) focuses all of those points of light into a virtual image on the sensor or film that recreates the scene in the field of view.
Light from an item that appears in the top of the scene as viewed through the lens doesn't just pass through the top of the lens. Light from that spot strikes all of the front surface of the lens at various angles. The same is true for every other point in the field of view - light from all of those points strike every point on the front of the lens. How well the lens refocuses all of the light that falls on its entire front surface onto an image sensor or piece of film determines the optical quality of the lens.
If the center part of the front of a lens is obstructed, as is the case with catadioptric lenses, light from items in the center of the field of view still make it into the lens because some of the light from those items strikes the edges of the lens and is then focused by the lens onto the center of the sensor or film.
There is light that is lost from those items in the center of the lens' field of view. It is the light that would have struck the lens at the most direct angle (i.e. parallel to the optical axis of the lens) and this loss is reflected in the properties of images created using catadioptric designs. We call light that is parallel to a lens' optical axis 'collimated' light. There is also light that is lost from items that are not in the center of the lens' field of view, but in this case the lost light is from a greater angle relative to the optical axis of the lens. The greater the angle a light ray has to the optical axis of the lens, the 'less collimated' that light is said to be.
The effect of collimated light that is from items near the center of the lens' field of view being blocked and less collimated light being blocked from items not near the center of the lens' field of view give catadioptric lenses their unique optical properties. One of the most noticeable of those unique properties is the 'donut' shape of bokeh produced when using such a lens. Losing the most collimated light from the center of the lens' field of view also affects the resolution of objects in the center of the lens' field of view. The greater the difference is between the distance to the source of the light and the distance at which the lens is focused, the more the loss of that collimated light affects the resolution and brightness of that part of the image.
There will not be a black spot in the image, and the (very narrow) area of the image which in focus is not affected. But the out of focus areas are affected by the central mirror.
Out of focus points of light, which would form "bokeh balls" when shooting with a regular lens, become rings. Some photographers, such as Shihya Kowatari from Japan, use this to artistic effect.
Many regular shooters find the out of focus areas as interpreted by catadioptric lens distracting (they can run a bit wild).
A cone of light rays gathered by the sides of the front lens element is reflected to the center mirror, which, reflects the rays further to the rear lens element. The curvature of the mirrors allow the projected image to change its size several times and also to exclude the reflection of the center mirror from the reflection itself.
Here is a good example of how the light travels inside, taken from this Wikipedia article.
Think of the vista being imaged as being completely covered by tiny transparent dots. Reflected light from each dot radiates out. Some rays originating from every point will hit the lens. Moreover, from that point some rays will hit the center of the lens, some will it the left edge, and some will hit the right center of the lens. In other words, some light rays from every point hit everywhere on the lens’s surface. Each will have its path of travel changed by the curve of the lens. All the light rays from a given point will follow a new path. A ray trace shows that these will converge and image as a single point. This is an image point that we can photograph or magnify with an eyepiece lens. Not all these rays will make it to the image plane. Some will be obstructed by dirt on the lens. Some will be obstructed by an obstacle like a secondary mirror. The image is formed regardless of the missing rays. The chief effect of the missing rays is that the image will be dimmer, because some rays were not available to contribute their light energy.
You can test this by placing a central blocking obstruction on your camera lens. You can paste a circular piece of paper on your camera lens and take a picture. You will find that the camera will still image. It is common practice to dot a lens with paint to make fine adjustments as to image brightness. Enlarger printers that use cluster lenses project several images of a negative simultaneously for high speed printing of school pictures. You know, the packages of small and large pictures made for sale at the school. Cluster lenses must be speed- equalized, otherwise, some image will be too dark and some to light. We would dot the ones that pass too much light with black paint. Sounds goofy -- but it works without significant degrading of the image.