  This paper presents a method to generate high quality triangular or
quadrilateral meshes that uses direction fields and a frontal point insertion
strategy. Two types of direction fields are considered: asterisk fields and
cross fields. With asterisk fields we generate high quality triangulations,
while with cross fields we generate right-angled triangulations that are
optimal for transformation to quadrilateral meshes. The input of our algorithm
is an initial triangular mesh and a direction field calculated on it. The goal
is to compute the vertices of the final mesh by an advancing front strategy
along the direction field. We present an algorithm that enables to efficiently
generate the points using solely information from the base mesh. A
multi-threaded implementation of our algorithm is presented, allowing us to
achieve significant speedup of the point generation. Regarding the
quadrangulation process, we develop a quality criterion for right-angled
triangles with respect to the local cross field and an optimization process
based on it. Thus we are able to further improve the quality of the output
quadrilaterals. The algorithm is demonstrated on the sphere and examples of
high quality triangular and quadrilateral meshes of coastal domains are
presented.
