I knew the formula but fumbled the explanation of what the parameter actually means geometrically.
Start by defining the interpolation parameter t, then present the linear interpolation formula P(t) = (1-t)P0 + tP1, and explain its geometric meaning as a point moving along the line segment. Finally, discuss extrapolation when t is outside [0,1], including the risks and potential use cases.
Pro tip: Mention that in ML, interpolation is used in techniques like mixup and latent space interpolation, and that extrapolation can be useful for data augmentation but must be done cautiously to avoid unrealistic samples.
Introduce t as a scalar that determines the position of the interpolated point relative to the two endpoints. Typically, t=0 gives the first point, t=1 gives the second point, and t in (0,1) gives points in between.
Write the linear interpolation formula: P(t) = (1-t)*P0 + t*P1, where P0 and P1 are the two 3D points. This can also be written as P(t) = P0 + t*(P1 - P0).
Describe how the formula traces a straight line between P0 and P1. As t varies from 0 to 1, P(t) moves uniformly from P0 to P1, and the distance from P0 is proportional to t times the total distance.
Explain that when t < 0 or t > 1, the point lies outside the segment, on the line extending beyond P0 or P1. This is extrapolation, which can produce points far from the original data and may not be meaningful in all contexts.
Mention relevant ML use cases such as mixup data augmentation, interpolating between embeddings, or generating intermediate samples in generative models, and note the trade-offs of extrapolation.
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