High voltage bipolar sawtooth wave generation

I'm currently trying to make a simple CRT oscilloscope and I need to generate a bipolar sawtooth wave in order to drive the horizontal deflection coil. I don't exactly know what the voltage has to be, but I've heard that it's above 100V p-p. Is there any simple way to do this? The wave needs to be as linear as possible. It would be nice If I could change the frequency with a potentiometer.

I also need to figure out how to amplify an input signal into something that can fully deflect the vertical coil as well. Are there any commonly available high voltage bipolar op-amps?

Look at a schematic of such scopes, or the schematic of an old TV.

But this is not an easy thing to do.
If you have the electronics for that tube, you might use parts of it.

Oscilloscopes don't tend to use coils since they are not fast enough, they use deflection plates - having said that what tube are you actually using?

If its a TV tube then you won't get much bandwidth on the vertical deflection I suspect, and you'll have the problem of compensating for the inductance and non-linear magnetic properties of the ferrite core if it has one. The electron beam deflection depends on the current in the coil and the voltage across the coil controls the rate of change of current at higher frequencies where inductance dominates.

The general approach for driving a high voltage output for deflection plates is to generate the signals at low voltage and amplify last thing. For deflection plates a high voltage cascode differential amplifier is one design I've seen. High voltage and high bandwidth creates special demands on amplifier design though, not a first project!

The tube is a 5" black and white tube and it does indeed use coils. For my first attempt this shouldn't be a problem though. I just finished tearing the circuit boards down to the bare minimum for the tube to function, and I'm left with a CD1379cp at the center of it all (CD1379CP pdf, CD1379CPDescription, CD1379CPDatasheet, CD1379CP view ::: ALLDATASHEET :::). So far I've tried putting a voltage into pin 1 but I always end up with a centered horizontal line; the deflection is greater if I put the same voltage directly across the vertical deflection coil. And I'm still at a loss as to how I should adjust the horizontal frequency, I'm no good at this analog stuff!

You can get a cheap ic like the icl8038(I think it was called) that will output multiple waveforms and Is easily adjustable, tho the upper frequency limit isn't too high, then you just need to amp it up
Or pwm it, amp it, then smooth it which would be more efficient I think

Ok, but amplification is still a problem.

Find an old diagram for a solid state scope... having said that...
you might have "ripped" too much.
The device has a deflection circuit that might gave been used the horizontal frequency is 15,734 Hz and the vertical rate is 60 hz the color burst frequency 3.579545 MHz is the key as it is the source for the local color injection... and it is the master clock for both the horiz remember the odd 734 instead of the old 15750... turns out that 15734 is a multiple of 3.59 color burst osc.
I digress too far really but some technical background is necessary.
The Horizontal frequency won't shift by much before high voltage regulation issues, brightness and more important Focus issues will be an issue...
Now comes the vertical and that one is much easier as it can be changed in frequency by factor of 2 both ways and in most tv sets the vertical output circuit is an oscillator and a power amp. Frequently in the same IC. It is very likely that there is some kind of Synch generator driving both the horiz and vert sweep sections of the TV and locked to the color burst frequency, So you get an inkling of what you are facing.
Last is the easiest and that is the video amplifier which is most likely an IC... back in the day it was 2 or 3 class A amplifiers with some video EQ... peaking coils to improve the high frequency response which leads to a clearer for lack of a better word... like copies of copies of old VCR tapes with out the EQ. I mention it because a video driver for a scope should be as flat as possible...
In Short, Yes it can be done, the above list is really a list of the shortcomings of attempting to convert an old tube or CRT type TV set to an O'scope. There are many Many shortcomings there.
Making a Good scope from a processor is very possible.... Great TFT LCD devices are coming available and there are some Very impressive chips available.
But the available data is limited by the processor controlling the device functions and the A/D conversions both speed, depth and quality (Noise and distortion due to Nyquist like effects)...
If you 'sweep' the sampling clock on an A/D converter around a multiple of an input being measured you will find "Aliased" products in the A/D data stream that are interactions between the sample and the sample frequency...
And Finally we get to the meat in this mess... An Arduino isn't fast or the A/D deep enough to keep track of itself let alone be a valid measuring tool for much beyond measuring voltage or time and neither is any more accurate than the Vref for the A/D converter and or the clock used by the Micro.
You could spend a lot less money and time by finding an "Oldie but Goodie" like I did, I looked for a month or so and found a Tektronics 2213, a Basic dual trace 60 Mhz scope....
Old but highly reliable, I paid $55.00 for it and $50.00 for shipping and thought I got a good deal. It's worked very well for 3 years now. In my view a good analog scope is much better for certain types of work and a digital scope is better for other things like deep event storage for those really hard to find problems. Many years ago a man I worked for wanted to rent a "storage" scope (it has two types of phosphors... and I said why... we don't need to see it, we know when the FF resets prematurely... Fix that and the issue is dead... the issue turned out to be a snubber network and a .1u
f cap across the FF power leads. After those changes were made all was well, Permanently.

Doc