Engineering - Interpolator Formula

To interpolate the y

x

x

y

x_{1} | y_{1} |

x_{2} | y_{2} |

x_{3} | y_{3} |

Solve for y_{2}

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linear interpolation single interpolator | |

bilinear interpolation double interpolator |

Linear interpolation is a mathematical technique used to estimate an unknown value between two known data points on a straight line, assuming a constant rate of change between the points and that the function connecting them is linear.

Curve fitting, on the other hand, is a broader process of constructing a curve, or mathematical function, that best fits a series of data points. For example, linear interpolation can be considered a simple form of curve fitting where the curve is a straight line.

Linear interpolation and curve fitting are essential because they provide efficient ways to estimate values within a data set when exact data is unavailable, analyze data trends, and create graphical representations of data. These techniques are critical for approximation, data analysis, and visualization.

The linear interpolation equation is given by:

y = y1 + (x - x1) * ((y2 - y1) / (x2 - x1))

where (x1, y1) and (x2, y2) are the known data points, x is the unknown point's x-value, and y is the unknown point's y-value.

To solve for y using linear interpolation, follow these steps:

- Identify the two known data points, (x1, y1) and (x2, y2), surrounding the x-value for which you want to estimate the y-value.
- Substitute the known values into the linear interpolation equation.
- Solve for y.

- Extrapolating beyond the known data points can lead to inaccurate estimates.
- Using linear interpolation for non-linear data sets results in poor approximations.
- Not verifying the assumptions of a constant rate of change and linearity.
- Relying only on interpolation when more accurate methods or data are available.
- Misinterpreting the results, such as assuming the interpolated value equals the exact value.
- Sensitivity to outliers and extreme values in the data set.

- Computer graphics and image processing
- Finance (e.g., interest rate calculations)
- Engineering (e.g., temperature and pressure estimations)
- Geographical Information Systems (GIS) and mapping
- Medical imaging
- Video game development
- Weather forecasting
- Audio signal processing
- Computer-aided design (CAD)
- Data science and analytics

- Polynomial interpolation
- Spline interpolation (e.g., cubic and B-spline)
- Hermite interpolation
- Rational interpolation
- Nearest-neighbor interpolation

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