Parabolic blending

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Parabolic blending is a process of creating a kink-free curve through a series of points. Depending on the problem, it can be used as an explicit representation y = y (x) or as a parametric representation x = x (t), y = y (t). Two quadratic polynomials are formed in each interval between two points No. i and i + 1 :

${\displaystyle p(x)={\frac {(x-x_{i})(x-x_{i+1})}{(x_{i-1}-x_{i})(x_{i-1}-x_{i+1})}}y_{i-1}+{\frac {(x-x_{i-1})(x-x_{i+1})}{(x_{i}-x_{i-1})(x_{i}-x_{i+1})}}y_{i}+{\frac {(x-x_{i-1})(x-x_{i})}{(x_{i+1}-x_{i-1})(x_{i+1}-x_{i})}}y_{i+1}}$

${\displaystyle q(x)={\frac {(x-x_{i-1})(x-x_{i+1})}{(x_{i}-x_{i-1})(x_{i}-x_{i+1})}}y_{i}+{\frac {(x-x_{i-1})(x-x_{i})}{(x_{i+1}-x_{i-1})(x_{i+1}-x_{i})}}y_{i+1}+{\frac {(x-x_{i})(x-x_{i+1})}{(x_{i+2}-x_{i})(x_{i+2}-x_{i+1})}}y_{i+2}}$

In order to represent the function y (x), which is only given by individual points, between point no.i and point no.i + 1, both parabolas are linearly mixed in this interval:

${\displaystyle y(x)={\frac {x_{i+1}-x}{x_{i+1}-x_{i}}}p(x)+{\frac {x-x_{i}}{x_{i+1}-x_{i}}}q(x)}$

The very first and the very last point do not become part of the curve, but control the tangent .

The process originally comes from the ship's design, but its quality is not sufficient to generate or smooth ship lines, because the curvature is discontinuous. In the ship design, it represented a primitive, simple and fast method for visualizing cracks in the frame.

See also

• Spline interpolation
• Polynomial interpolation
• Bezier curve