Question on significance of different ways of measuring Price Elasticity of Demand I'm a pre college student, self studying economics, I have a question regarding the different forms of measurement of Price Elasticity of Demand.

The basic equation for PED with respect to a point $(x_1,y_1)$ on a demand curve (qty x axis, price y axis), given another point $(x_2,y_2)$ is:
$\frac{(x_1-x_2)/x_1}{(y_1-y_2)/y_1}$
This gives us the responsiveness of qty demanded relative to price at an initial point $(x_1,y_1)$.

However, my text mentions that depending on the choice of initial point, either $(x_1,y_1)$ or $(x_2,y_2)$, the PED is different. Hence, in order to overcome such a problem, we utilize arc elasticity of demand, which uses the midpoint as a initial value.

My question is this:
Isn't PED supposed to vary along a demand curve? If we rearrange it as (Change in qty/Change in price)(Price/Quantity), we can see that it does depend on which point on the curve it is. Hence, why is it regarded as a "problem" that depending on which initial value we use, we get different PED?

And by extension, I read that the definition of arc elasticity is the elasticity of one variable to another between two given points. I don't fully understand what is the significance of using the midpoint as an initial value, other than it netting the same PED for two distinct points. What does it actually mean when we take the difference of two points and divide it by the midpoint?

And on a final note, is point price elasticity of demand recommended when we have the functional relation of price to quantity?

I've been looking on the internet, and I haven't really been able to find/just don't understand anything about this. Sorry if this is elementary, but I haven't got anyone I know to ask.
Ideally the points $(x_1,y_1)$ and $(x_2,y_2)$ are very close. The assumption is that that they are so close that the elasticity of the demand function on the curve between them is nearly constant. This is what elasticity is trying to measure, local changes, because on the far side of the range of the demand function consumer behavior may be unobserved and totally different. E.g. to my knowledge no one has tried to sell a bottle of Coke for \$40,000 so we don't know what value the demand function would take. If you would choose a very distant point $(x_3,y_3)$ instead of the local $(x_2,y_2)$ you would get a very different value of elasticity.

So what does the book mean by initial value? Ideally the elasticity measured in points $(x_1,y_1)$ and $(x_2,y_2)$ would be constant because they are very close. Of course in practice this will not always be the case, meaning
$$
\frac{(x_1-x_2)/x_1}{(y_1-y_2)/y_1} \neq \frac{(x_1-x_2)/x_2}{(y_1-y_2)/y_2}.
$$
Instead of taking one of these values as the elasticy of the curve we can improve our estamite by taking the arc elasticity which usually yields a value between these two.