Linear Interpolation

Linear Interpolation is a fairly simple concept. It is used to estimate values that fall between two known values. The idea is to draw a straight line between the two known points and use that line to find the value at a specific point in between.

Say you have a ${\mathbb{R}}^1$ dimensional space with two known points: $(x_0,y_0)$ and $(x_1,y_1)$. The formula for linear interpolation to find the value $y$ at a point $x$ between $x_0$ and $x_1$ is given by:

$$y=y_0+\frac{(y_1-y_0)}{(x_1-x_0)}\cdot (x-x_0)$$

Parameterized for time, you can express it as:

$$y(t)=(1-t)\cdot y_0+t\cdot y_1\text{for }t\in [0,1]$$

LERPs can be extended to higher dimensions as well. For example, in ${\mathbb{R}}^2$, if you have two points $(x_0,y_0)$ and $(x_1,y_1)$, you can interpolate both the x and y coordinates separately:

$$x(t)=(1-t)\cdot x_0+t\cdot x_1$$ $$y(t)=(1-t)\cdot y_0+t\cdot y_1$$

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LERPs are widely used for the field of Splines, paticularly in Bezier Curves by “stacking” multiple LERPs together to create smooth curves.