Abstract: To achieve smooth curve and surface modeling through subdivision of control vertices and the unit control tangent vectors at the control vertices, an algorithm is proposed that extends the traditional linear subdivision scheme to a nonlinear subdivision scheme. In each subdivision step, the algorithm first linearly subdivides the vertices, then linearly subdivides the unit control tangent vectors followed by a normalization step to obtain unit tangent vectors. Subsequently, a normal plane is constructed at each linearly subdivided vertex using the obtained unit tangent vector as plane normal, and the original vertices are projected onto the normal plane by arc interpolation. Finally, the new subdivision vertices are obtained by weighted averages of the projection points on the normal planes. The nonlinear subdivision schemes based on point tangents can preserve circles and circular cylinders, and they have the same convergence and smoothness orders as linear subdivision schemes. The experimental examples show that the point-tangent subdivision schemes are beneficial for the construction of complex smooth spatial curves and the modeling of pipe-like surfaces.