History beats a single point — next-step prediction

Background

The lesson's motivating example: predicting a ball's next position from its current point alone is underdetermined — you can't tell which way it's moving. Give the model the trajectory and a simple constant-velocity estimate works: the next point is the last point plus the last observed step (velocity).

$$\hat{p}_{t+1} = p_t + (p_t - p_{t-1})$$

Problem statement

Implement next_position(trajectory) returning the constant-velocity prediction of the next point.

Input

• trajectory — array-like of past points (length $\ge 2$). Each point is a scalar or a coordinate vector (e.g. [x, y]).

Output

Returns an np.ndarray (or scalar array) the shape of one point: $p_t + (p_t - p_{t-1})$.

Examples

Example 1 — 2D straight line

Input:  trajectory = [[0, 0], [1, 1], [2, 2]]
Output: [3 3]

Explanation: last step was $[1,1]$, so the next point continues to $[3,3]$.

Example 2 — 1D

Input:  trajectory = [1, 2, 3]
Output: 4

Explanation: velocity $1$ → next is $4$.

Constraints

• Use only the last two points: last + (last - prev).
• Return the same shape as a single point.

Notes

• This is the simplest "temporal model" — it uses history (the previous step) rather than just the present. RNNs generalise this by learning what to remember.