An oscilloscope does not have a separate refresh rate. You're probably thinking of raster images that are drawn from a frame buffer as a series of horizontal lines. An oscilloscope is vector based, so the rate that the point moves is the same as the refresh rate - or at least it affects the refresh rate directly. The more points and lines that you have (the more drawing), the lower the refresh rate.
One trick is to disable the cathode ray (blank the z-axis) before stopping the motion of the line tracing. On systems that do the reverse - stop the motion before blanking the trace - you see a bright dot at the vertex of each line because the ray lingers longer at that point and creates more brightness.
Raster images are always blank anywhere that you don't actively draw. In addition, even when you do draw on part of the tube, the phosphor only holds the image for a brief time until it decays and is dark again. So, having blank sections of the screen are easy: just don't draw there.