How can I draw a curve which satisfies several definite integrals?

Question

Given data like the following:

I would like to draw a curve with the property that integrating over the intervals of the bars (eg 04:00 - 07:00) will give the same answer as that given by the bar chart (in this case, 62).

I was planning to do this using numpy and scipy. I had a play with the interpolation library, but decided it wasn't what I was after.

Does anyone know of a tool or algorithm for making such a curve?

Answer 1

Let's figure out the linear system for computing the coefficients of a piecewise quadratic curve. Let there be n bars, where bar i (counting from 0 ) extends from x[i] to x[i+1] , with height y[i] . Piece i is the quadratic function

lambda z: a[i]*z**2 + b[i]*z + c[i]

where a[i], b[i], c[i] are coefficients. We get n equations fixing the area.

a[i]*(x[i+1]**3 - x[i]**3)/3 + b[i]*(x[i+1]**2 - x[i]**2)/2 + c[i]*(x[i+1] - x[i])
    == y[i]*(x[i+1] - x[i]) for i in range(n)

We get n-1 equations matching up the values at boundaries.

a[i]*x[i+1]**2 + b[i]*x[i+1] + c[i]
     == a[i+1]*x[i+1]**2 + b[i+1]*x[i+1] + c[i+1] for i in range(n-1)

We get n-1 equations matching up the derivatives at boundaries.

2*a[i]*x[i+1] + b[i] == 2*a[i+1]*x[i+1] + b[i+1] for i in range(n-1)

There are two more degrees of freedom. We could, eg, require that the beginning and ending derivatives be zero.

2*a[0]*x[0] + b[0] == 0
2*a[n-1]*x[n] + b[n-1] == 0

Use NumPy to solve these equations (this probably involves turning this system into a matrix).