Piecewise planar 3D objects are very common in digital furniture manufacturing. In this paper, we propose a novel method for automatic reconstruction of 3D objects with planar facets from unorganized point clouds. We formulate this problem into a point clustering problem where the key difficulty lies in consolidating co-planar points into a cluster. In order to achieve this purpose, the first step is to triangulate the input point cloud into a mesh (may have over connectivity) that is a super-set of the underlying manifold mesh surface. Then the hundreds of thousands of normal vectors of triangles, after being mapped onto a Gauss sphere, are capable of reporting reliable facing directions of the faces of the final 3D model. After grouping the points based on co-planarity, we fit each cluster with a planar facet and then assemble them into a piecewise planarization representation of the whole model. We further introduce an additional regularization term to meet the orthogonality requirement on a particular occasion, and then transform this problem into a purely convex optimization problem. Our method is efficient and requires just a few parameters. Extensive experimental results show that it is able to handle point clouds with various levels of noise and yield a desirable piecewise planar 3D model with a clean and compact representation.