Let M be an m-dimensional smooth compact manifold embedded in Rd, where m is a constant known to us. Suppose that a dense set of points are sampled from M according to a Poisson process with an unknown parameter. Let p be any sample point, let ϱ be the local feature size at p, and let ϱε be the distance from p to the (n+ 1) th nearest sample point for some n between (m+12)+1 and (d+12). Using the n sample points nearest to p, we can estimate the tangent space at p and it holds with probability 1 - O(n- 1 / 3) that the angular error is O(ε2). The running time is bounded by the time to compute the thin SVD of an n×(d+12) matrix and the full SVD of an n× d matrix, which is usually O(d2n2) in practice. We implemented the algorithm and experimentally verified its effectiveness on both noiseless and noisy data. © 2016, Springer Science+Business Media New York.