# Linear Interpolation

## Prerequisites

### Linear Interpolation

#### Introduction

• Linear interpolation is a very simple type of interpolation.
• To find the value of a point which is not in the given set of points:
1. Find the two points that surround the point you are estimating: one will be smaller, one larger.
2. Draw a straight line between the two known points.
3. Find the value of y for the point that lies on the line at the given x.

#### Example

• If we are given a set of data points: X 1 2 3 4 5 6 7 y 28 21 96 96 120 87 50
Find the value for y when x is 2.4
1. The points the surround 2.4 are (2, 21) and (3, 96)
2. We calculate the gradient of this line:
3. And then find the value of y when x is 2.4:
4. When x is 2.4, y is 51

#### Code (using SciPy)

``````from scipy import interpolate

dataPoints = [
[1, 28],
[2, 21],
[3, 96],
[4, 96],
[5, 120],
[6, 87],
[7, 50]
]

interpolator = interpolate.interp1d(
[dataPoint[0] for dataPoint in dataPoints],
[dataPoint[1] for dataPoint in dataPoints],
'linear')

print(interpolator(2.4)) # prints 50.99999999999999
print(interpolator(2.6)) # prints 66.0``````

#### Code (using NumPy)

``````import numpy as np

class LinearInterpolator:
# an array of x values, in order
_sortedDomain = []

# a dictionary of x, y values
_dataPointsDict = {}

def __init__(self, dataPoints):
for dataPoint in dataPoints:
self._dataPointsDict[dataPoint[0]] = dataPoint[1]

self._sortedDomain = [dataPoint[0] for dataPoint in dataPoints]
self._sortedDomain = np.sort(self._sortedDomain)

def interpolate(self, x):
# find the index of the smallest element larger than x
closest = np.searchsorted(self._sortedDomain, x)

# find the two x values (before and after x
x1 = self._sortedDomain[closest - 1]
x2 = self._sortedDomain[closest]

# and their corresponding y values
y1 = self._dataPointsDict[x1]
y2 = self._dataPointsDict[x2]

# find the deltaX and deltaY
deltaX = x2 - x1
deltaY = y2 - y1

# and finally interpolate
xPart = x - x1

return y1 + yPart

dataPoints = [
[1, 28],
[2, 21],
[3, 96],
[4, 96],
[5, 120],
[6, 87],
[7, 50]
]

interpolator = LinearInterpolator(dataPoints)

print(interpolator.interpolate(2.4)) # 50.99999999999999
print(interpolator.interpolate(2.6)) # 66.0``````