# Compute Point-to-Plane Distance and Fit a Robust Plane Use NumPy-style vectorized operations to solve both parts of this 3D geometry exercise. ### Constraints & Assumptions - Points are supplied as a finite floating-point array with shape (n, 3). - A plane is represented by coefficients (a, b, c, d) for ax + by + cz + d = 0. - In Part 2, approximately 70% of the points come from one physical plane and the rest are other objects or noise. - The distance threshold is provided in the same units as the coordinates. - Degenerate inputs must be detected rather than silently producing NaN values. ### Clarifying Questions to Ask - Is the threshold an absolute geometric distance? - What numerical tolerance should define a degenerate plane or collinear sample? - Must the returned normal have a canonical sign and unit length? - Is randomized fitting acceptable, and must runs be reproducible? ### Part 1: Vectorized Distances Given points and plane coefficients, return the unsigned perpendicular distance from every point to the plane without a Python loop over points. #### Hints - Express the numerator for all points as one matrix operation. - Determine which property of the coefficient vector makes the result invariant to rescaling the equation. #### What This Part Should Cover - Correct dimensions and normalization - Vectorized computation - Degenerate-normal handling ### Part 2: Robust Plane and Outliers Estimate the dominant plane despite roughly 30% outliers, then return the indices whose distance from the final plane exceeds the threshold. Describe a deterministic or seeded implementation and how you refine the estimate. #### Hints - A small subset can propose a hypothesis. - Separate hypothesis selection from final parameter estimation. #### What This Part Should Cover - Robustness to a large outlier fraction - A reproducible model-selection rule - Refinement and outlier classification - Numerical edge cases ### What a Strong Answer Covers - A correct geometric derivation - Careful NumPy shapes and floating-point checks - A robust estimator with explicit scoring and tie-breaking - Complexity, reproducibility, and failure behavior ### Follow-up Questions - How many random hypotheses are needed for a desired success probability? - How would heteroscedastic sensor noise change the inlier rule? - How would you fit several planes rather than only the dominant one?