- On each chart of the atlas, the restriction of F and C is a spline surface patch.
- The evaluation of F is independent of the choice of the charts.

The domain manifold is at the bottom. Two charts are shown, the spline surfaces are defined on them. The piecewise polynomial surface is on the top. The evaluation of the green region is independent of the choice of the charts. |

- Compute conformal structure of the domain manifold M.
- Select one holomorphic 1-form, remove zero point neighbor.
- Construct affine atlas by integrating the holomorphic 1-form.
- Define knots on one chart, consistently extend to all charts.
- Given one point on M, evaluate the spline surface on the point by choosing arbitrary chart which covers it by polar form.

(a)Holomorphic 1-form | (b) Domain manifold (Singular points marked in red) | (c) Domain manifold without singular points | (d) Manifold spline | (e) Spline overlaid with control net | (f) The singular points are filled |

**Experimental**** Results**

Knot (genus 1)

(a) Holomorphic 1-form | (b) Domain manifold | (c) Manifold spline | (d) Control net |

Sculpture (genus 3)

(a) Holomorphic 1-form | (b) Domain manifold (Singular points marked in red) | (c) Manifold spline | (d) Manifold spline (The red curves correspond to the edges in the domain manifold) | (e) Spline overlaid with the control net |

The data samples and binary codes are available for research purpose. Please email your thoughts, comments to gu AT cs DOT sunysb DOT edu.