How to interpolate a curve between two points?

I have two points, for e.g. let's take an equity Index like SPX 500. I have the values of the Index 2021 Q1 and a value for 2022 Q2. Now, I want to interpolate the values for the rest of the quarters which is 4 values but not in a straight line. I want put a curve is place. Is there any methodology for this?
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Ambassador to the humans
You would need to be more specific about the kind of curve you want. With only two values how should we know how the curve should look?
Lets assume the start value of the equity Index is 100 and end value is 50. I need to fit 4 points in between. And it should be decreasing concave curve similar to the bond price vs yield curves.


Ambassador to the humans
Any other restrictions you can come up with? There are an infinite number of curves that meet your current criteria.
Restrictions would be:

1) All 4 points should lie between 100 & 50.
2) They should be in a descending order i.e. 100> p1 > p2 > p3 > p4 > 50.
3) If possible the relative difference between 100 & p1 should be maximum as compared any other 2 adjacent points such as p2 & p3 or p3 & p4 etc.


No cake for spunky
When you only have two points you can as dason said fit an infinite number of curves through them (you can try this to see it). Nothing you added as a restraint really changes this. I think if you draw the two points on a graph and start fitting lines through them this will be obvious.


Less is more. Stay pure. Stay poor.
Well unless you have more data to inform the process you actually aren't interpolating, just guessing. How do you truly know, say point 3 isn't 155? This really feels fishy. How do you know the relationship isn't approximately linear? If you had a series of values you could then fit a curve linear line if it was appropriate with say a slope coefficient that is multiplicative. But given two points with no other data you are just making things, up right?