TVL means TV lines and is used by the professional imaging industry to define the real visible resolution of a CRT display device rather than theoretical possibilities or pixel based formulas which do not apply to this technology.
A CRT works just like any other electron tube except that the plate where the electron beam goes to is actually a viewing screen. The heater of the tube releases electrons which forms a beam. That beam is shaped into a circle and that is what we call spot size which determines what the resolution will be. An old and very popular method used to decrease the spot size and also provide better focus are beam forming magnets that help to shape the beam into a more perfect cylinder and help keep the electron stream within that cylinder. As you move the beam further from the center of the screen it starts to take the shape of an ellipse, an artifact, which will also start to lower the resolution unless you do something to counteract this simple geometrical problem. Another problem is the distance also changes between the heater and the face of the tube which changes the focus of the beam. Curving the face of the tube will reduce this artifact. You can also change the focus voltage based on where the beam is on the tube and this is called dynamic focus. At a certain point there will be nothing you can do prevent any of this. The greater the distance the beam has to move from the center the more likely you are to lose focus and shape hence resolution.
CRT's for the longest time were circular rather than rectangular for this very reason. This is also the basis of TVL. If the vertical and horizontal scan is the same size than you have the same resolution in either plane but TV is 4:3. TVL resolution represents 3:3, or a 1 to 1 ratio, because this is the true resolution capability of the CRT within that defined area. When you go past the defined area the beam spot changes shape and focus too much to actually pass the same amount of lines in the middle of the screen. This is why computer monitors have a CRT size spec and then viewing area spec - they use more of the center or high resolution area of the tube and stay away from the edges. So TVL for TV displays and viewing area for computers is a function of
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In the 60's somebody had the bright idea to change the CRT from circular to rectangular to decrease material costs in manufacturing since the rest of the face of the tube wasn't being used anyway. These were 90 degree tubes referring to the sweep of the beam which directly affects the physical depth requirement of the product. To reduce depth with larger models manufacturers started using 120 degree tubes. Geometrically our frames still had curves though and the screen was also curved. People noticed that with graphics, letter boxed movies and news ticker tapes, hence the flat perfectly straight framed tubes we have now. By being flat and/or having a wide sweep such as current 120 degree tubes beam shape suffers even more as you get towards the edge. Beam focus can be improved via dynamic focus technology. These artifacts are easily seen using the Avia disc dot pattern on nearly all direct view CRT's and many projectors regardless of them being analog or HD. To eliminate or reduce this artifact requires dynamic beam forming technology typically available only on professional or high end CRT displays.
This brings us to the difference between projection and direct view. The beam on projection CRT's moves over a significantly smaller surface area than direct view. These artifacts are therefore minimized and when we add the 2 technologies previously described it can be nearly perfect. While a direct view can also perform as well it requires far more effort and cost to make it happen - $30k for the Sony 38" HD monitor. The least expensive CRT display with the most resolution will be projection in general.
So how does TVL apply to us now? Due to digital nearly all things are described as pixels rather than analog transitions, TVL, these days which is not the correct way to describe CRT resolution performance. TVL has no bearing or use for digital fixed pixel displays. While it is unfortunately rarely used or stated for consumer CRT products it does help us to understand why CRT's perform the way they do.