This paper introduces a new algorithm for manifold learning and nonlinear dimensionality reduction. The algorithm learns the local geometry of a manifold by constructing tangent spaces at each data point and aligning those tangent spaces to obtain global coordinates of the data points with respect to the underlying manifold. The algorithm is illustrated using curves and surfaces in 2D/3D Euclidean spaces and higher dimensional spaces. The paper also addresses several theoretical and algorithmic issues for further research and improvements. Key words: nonlinear dimensionality reduction, principal manifold, tangent space, subspace alignment, singular value decomposition. The paper discusses the problem of manifold learning and dimensionality reduction, emphasizing the challenge of discovering the structure of a manifold from a sample of data points. It highlights the limitations of traditional linear dimensionality reduction techniques and the need for nonlinear methods. The paper presents two lines of research in manifold learning: one based on estimating pairwise geodesic distances and another based on analyzing overlapping local structures. The local linear embedding (LLE) method is discussed as an example of the latter approach. The paper's approach is inspired by and extends the work of Ref. [2,7], which opens new directions in nonlinear manifold learning. The paper argues that nonlinear dimensionality reduction should be combined with the process of reconstructing the nonlinear manifold, as the two processes interact in a mutually reinforcing way. The paper proposes a new algorithm called local tangent space alignment (LTSA) that constructs a principal manifold passing through the middle of the data points and finds a global coordinate system that characterizes the data points in a low-dimensional space. The paper also discusses the error analysis of LTSA, the computation of global coordinates, and presents numerical experiments to validate the algorithm. The paper concludes by addressing several theoretical and algorithmic issues for further research and improvements.This paper introduces a new algorithm for manifold learning and nonlinear dimensionality reduction. The algorithm learns the local geometry of a manifold by constructing tangent spaces at each data point and aligning those tangent spaces to obtain global coordinates of the data points with respect to the underlying manifold. The algorithm is illustrated using curves and surfaces in 2D/3D Euclidean spaces and higher dimensional spaces. The paper also addresses several theoretical and algorithmic issues for further research and improvements. Key words: nonlinear dimensionality reduction, principal manifold, tangent space, subspace alignment, singular value decomposition. The paper discusses the problem of manifold learning and dimensionality reduction, emphasizing the challenge of discovering the structure of a manifold from a sample of data points. It highlights the limitations of traditional linear dimensionality reduction techniques and the need for nonlinear methods. The paper presents two lines of research in manifold learning: one based on estimating pairwise geodesic distances and another based on analyzing overlapping local structures. The local linear embedding (LLE) method is discussed as an example of the latter approach. The paper's approach is inspired by and extends the work of Ref. [2,7], which opens new directions in nonlinear manifold learning. The paper argues that nonlinear dimensionality reduction should be combined with the process of reconstructing the nonlinear manifold, as the two processes interact in a mutually reinforcing way. The paper proposes a new algorithm called local tangent space alignment (LTSA) that constructs a principal manifold passing through the middle of the data points and finds a global coordinate system that characterizes the data points in a low-dimensional space. The paper also discusses the error analysis of LTSA, the computation of global coordinates, and presents numerical experiments to validate the algorithm. The paper concludes by addressing several theoretical and algorithmic issues for further research and improvements.