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Why is it better to have 1280 x 960 for a (non-flat) CRT rather than 1280 x 1024?
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The normal monitor's width/height ratio is 4:3. 1280x960 is also 4:3 while 1280:1024 is not; the result of 1280x1024 will be that a circle on the screen is not a circle but an oval (the distortion as described by Matir above).
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What role does the refresh rate play?
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The vertical refreshrate is 'how often the picture is written to the screen'; or better, how often the screen is refreshed. Lower refresh rates result in the perception of flickering (screen on/off). If I look at a screen with 60Hz refresh, it really hurts my eyes.
To perceive no flicker, it's advised by some wise men to use at least 72Hz. This, however, is subjective. Some people still see a flicker at 75Hz (or even 80Hz) and it also depends (my experience) on the color scheme that is used. I can 'see' flickering when using a white background with black foreground at 70Hz, but I don't 'see' it when using a grey background with a black foreground @ 70Hz.
It's further said that refresh rates higher than 85Hz don't give much improvement in the perception.
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Would a lower resolution increase the refresh rate? (Assuming a higher refresh rate is better.)
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It will increase the max refreshrate that you can use.
There are three parameters that determine which resolution at which refresh rates can be handled by a monitor. These are the bandwidth (or dotclock or pixelclock), the horizontal refresh (in X-configuration called HorizSync) and the vertical refresh (in X-configuration called VertRefresh).
Suppose that your monitor can handle 1024x768 @ max 80Hz. This means it can handle 1024x768x80 pixels per second is 62914560 pixels per second; this is roughly the bandwidth (63MHz). Whatever you do should fit in there.
Reducing the resolution to 800x600 will give a theoretical max vertical refreshrate of the 62914560 divided by 800x600 is 131Hz. Increasing the resolution to 1280x960 will in a similar way result in a max vertical refreshrate of 51 Hz.
We can calculate the horizontal refresh as well.
With 1024x768@80Hz we have to write 768x80 lines per second is 61440 lines per second which is your horizontal refresh rate (approx 61kHz). At 1280x960@51Hz, this will be 48960 lines per second (approx 49kHz).
Your monitor should support all calculated numbers to be able to display the resolution that you want at the refresh rate that you want. Not taken into account are some compensations, so the actual requirements are usually a bit higher. E.g. a HorizSync of 61kHz will actually be 65kHz).
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Is it okay to customize a resolution even though my motherboard did not mention of such supported resolution?
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You have a chance that you will see nothing as the videocard / video circuit does not understand what it's given.
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What is the maximum standard resolution a 15.4" laptop can use?
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Check the specs of a few laptops. Further, a laptop uses TFT screens and those only display one resolution proporly (the number of pixels in the TFT). Using a higher resolution usually will give a bad quality picture, using a lower one will usually give a smaller picture (black borders around it).
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Is a 15.4" laptop's screen the same length as a 17" desktop's?
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If both have a 4:3 ratio, it's more or less the same.