  Shape manipulation is a central research topic in computer graphics. Topology
editing, such as breaking apart connections, joining disconnected ends, and
filling/opening a topological hole, is generally more challenging than geometry
editing. In this paper, we observe that the saddle points of the signed
distance function (SDF) provide useful hints for altering surface topology
deliberately. Based on this key observation, we parameterize the SDF into a
cubic trivariate tensor-product B-spline function $F$ whose saddle points
$\{\boldsymbol{s}_i\}$ can be quickly exhausted based on a subdivision-based
root-finding technique coupled with Newton's method. Users can select one of
the candidate points, say $\boldsymbol{s}_i$, to edit the topology in real
time. In implementation, we add a compactly supported B-spline function rooted
at $\boldsymbol{s}_i$, which we call a \textit{deformer} in this paper, to $F$,
with its local coordinate system aligning with the three eigenvectors of the
Hessian. Combined with ray marching technique, our interactive system operates
at 30 FPS. Additionally, our system empowers users to create desired bulges or
concavities on the surface. An extensive user study indicates that our system
is user-friendly and intuitive to operate. We demonstrate the effectiveness and
usefulness of our system in a range of applications, including fixing surface
reconstruction errors, artistic work design, 3D medical imaging and simulation,
and antiquity restoration. Please refer to the attached video for a
demonstration.
