Simultaneous Precise Solutions to the Visibility Problem of Sculptured Models

We present an efficient and robust algorithm for computing continuous visibility for two- or three-dimensional shapes whose boundaries are NURBS curves or surfaces by lifting the problem into a higher dimensional parameter space. This higher dimensional formulation enables solving for the visible regions over all view directions in the domain simultaneously, therefore providing a reliable and fast computation of the

visibility chart

, a structure which simultaneously encodes the visible part of the shape’s boundary from every view in the domain. In this framework, visible parts of planar curves are computed by solving two polynomial equations in three variables (

t

and

r

for curve parameters and

θ

for a view direction). Since one of the two equations is an inequality constraint, this formulation yields two-manifold surfaces as a zero-set in a 3-D parameter space. Considering a projection of the two-manifolds onto the

tθ

-plane, a curve’s location is invisible if its corresponding parameter belongs to the projected region. The problem of computing hidden curve removal is then reduced to that of computing the projected region of the zero-set in the

tθ

-domain. We recast the problem of computing boundary curves of the projected regions into that of solving three polynomial constraints in three variables, one of which is an inequality constraint. A topological structure of the visibility chart is analyzed in the same framework, which provides a reliable solution to the hidden curve removal problem. Our approach has also been extended to the surface case where we have two degrees of freedom for a view direction and two for the model parameter. The effectiveness of our approach is demonstrated with several experimental results.