**Proofs**

The integral of the curve written above is very nearly .6 times the sum of the endpoints over the interval from -1..1 which is about 60% what the integral would be for a straight line connecting the two points by the average value formula...

Note the integral of the average value over the interval is multiplied by 2, or F+I because the interval is from -1 to 1 which is two units wide..

And the other concavity:**Notes**

It's interesting how it compares to other types of interpolation, for polynomial interpolation for example you need 3 points for a second order curve, whereas this only uses the two endpoints and has a closed form. I think that might make it computationally simpler if one has a fast way to calculate the ln and exponential functions involved...

**Future investigation**

I'd like to know whether in terms of arclength whether this curve is somehow optimal in terms of smoothness...

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