## Tuesday, June 2, 2015

### +- 40% curve

I found a curve that smoothly piece-wise interpolates a set of  a waves values along the integers and proved something interesting about it...

Given final and initial y values for each interval between integers, the curve smoothly connects those points such that the integral of the curve over the interval is either very close to 40% greater than the integral of the function that is a straight line connecting the two points or very close to 40% less, depending on whether it is concave up or down and which side of the x axis the curve is on... The two formulas for concavity work as follows: The concavity will be up if the Final point is greater than or equal to the Initial point, in which case the other formula will be concave down, and vice versa... I've chosen the above example to have alternating concavities using alternating formulas, it doesn't work for always increasing or always decreasing functions...
**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...