Akima interpolation

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The Aikima interpolation is a special procedure of mathematics for interpolation of numerical values.

In 1970 Hiroshi Akima developed a new method that defines third-degree polynomials between the support points, but dispenses with the continuity of the second derivative in the support points. This condition creates the overshoots when there are strong changes in gradient. The Akima-interpolated curve runs through each support point and describes a very natural curve from support point to support point. If the continuity of the second derivative cannot be dispensed with, the Akima interpolation is already overwhelmed by its definition. If the second derivative of any kind of interpolation is to be included in the calculations, these must, however, be very error-tolerant. In such applications, the mean value from the second derivatives of the polynomials to the left and right of the support points also leads to sufficiently accurate results.

Akima went back to the basic idea of ​​interpolation in his approach: What happens when a curve is drawn manually? Obviously, only a local sub-interval of points is taken into account when drawing. It was this aspect that he put into mathematical form.

Define a polynomial of the form in the i-th interval

P _i (X _i ) = A _i + B _i * (XX _i ) + C _i * (XX _i ) + D _i * (XX _i ) ³

with the condition that this function runs through the i-th and i + 1-th support points, i.e. that the following applies:

P _i (X _i ) = Y _i

P _i (X _{i + 1} ) = Y _i +1

and that the gradient of the function in the support points has a slope that has yet to be determined,

P´ _i (X _i ) = t _i

P´ _i (X _{i + 1} ) = t _{i + 1}

this is how the polynomial coefficients can be calculated:

A _i = Y _i

B _i = t _i

C _i = (((Y _{i + 1} -Y _i ) / (X _{i + 1} -X _i )) - 2t _i -t _{i + 1} ) / (X _{i + 1} -X _i )

D _i = (((t _i + t _{i + 1} ) - 2 (Y _{i + 1} -Y _i )) / (X _{i + 1} -X _i )) / (X _{i + 1} -X _i )

This type of coefficient determination is known as Hermite interpolation .

Hiroshi Akima determines the slope t _i at the i-th point with the help of two adjacent points on the right and left.

The core of Akima interpolation is the slope formula. From the straight line slopes m _i, it supplies the slope t _{i of} the interpolation at point X _i .

m _i = ${\displaystyle {\frac {Y_{i+1}-Y_{i}}{x_{i+1}-x_{i}}}}$

t _i = ${\displaystyle {\frac {|m_{i+1}-m|m_{i-1}+|m_{i-1}-m_{i-2}|m_{i}}{|m_{i+1}-m_{i}|+|m_{i-1}-m_{i-2}|}}}$

To understand how this expression works, consider the following four cases:

1. t _i = m _i-1 if m _i-2 = m _i-1 and m _i = m _{i + 1}
2. t _i = m _i if m _i-2 = m _i-1 and m _i = m _{i + 1}
3. t _i = m _i-1 if m _i-1 = m _i
4. t _i = m _i if m _i-2 = m _i-1 and m _i = m _{i + 1} ${\textstyle {\frac {1}{2}}}$

Since n slopes have to be calculated, two extrapolated points to the left and right of the value range are necessary. These are each determined by a polynomial of the second degree, which is determined using the last three support points. The following guidelines apply:

X ₃ - X ₁ = X ₂ - X ₁ = X ₁ - X ₂

respectively

X _n - X _n-2 = X _{n + 1} - X _n-1 = X _{n + 2} - X _n

The four coefficients for all n-1 polynomials are thus determined.

Individual evidence

1. ↑ Heinz Heise GmbH & Co. KG (ed.): C't magazine for computer technology . Issue No. 6. Hannover 1989.
2. ↑ Cubic Splines and Akima Interpolation. Retrieved July 2, 2020 .