Let ε > 0 and 1 ≤ k ≤ n and let ${W_l}_{l=1}^p$ be affine subspaces of ℝⁿ, each of dimension at most k. Let $m=O\left({\epsilon }^{-2}(k+logp)\right)$ if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map $H:ℝⁿ\to {ℝ}^m$ such that for all 1 ≤ l ≤ p and $x,y\in W_l$ we have ||x-y||₂ ≤ ||H(x)-H(y)||₂ ≤ (1+ε)||x-y||₂, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε,k,p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces Y.