Modeling time and topology for animation and visualization with examples on parametric geometry

Isotopic equivalence by Bézier curve subdivision for application to high performance computing

“…Once the dynamic visualization begins, the containment of the PL graphics within a pipe surface of constant radius enables run time warning messages, as the geometry approaches the boundary of this tubular neighborhood, increasing the prospect for a significant topological change. This is far superior to prevailing animation techniques that concentrate on self-intersection analysis on a per frame basis, which are known to be error prone (Jordan et al, 2008;Lasseter, 1987).…”

“…The classical pipe surface was invoked as the boundary of a tubular neighborhood as an algorithmic constraint for a PL ambient isotopic approximation of a static spline curve (Maekawa et al, 1998). That static view has been extended so that many perturbations of the PL approximant within the pipe surface continue to maintain ambient isotopic equivalence (Jordan et al, 2008) and is applied here. Once the dynamic visualization begins, the containment of the PL graphics within a pipe surface of constant radius enables run time warning messages, as the geometry approaches the boundary of this tubular neighborhood, increasing the prospect for a significant topological change.…”

Computational Topology Counterexamples with 3D Visualization of Bézier Curves

“…Much of the motivation for considering these counterexamples came from applications in scientific visualization [9,10,11]. A primary focus was to establish tubular neighborhoods of knotted curves so that piecewise linear (PL) approximations of those curves within those neighborhoods remained ambient isotopic to the original curves.…”

Unknots with highly knotted control polygons

“…Ambient isotopy is a fundamental concept in knot theory [4]. Practical applications of ambient isotopy appear in geometric modeling, visualization and animation [5].…”

“…Formally, this is the ambient isotopy of linear extrapolation of [P 2 , P 3 ] until the x co-ordinate of P 2 is sufficiently negative and the y co-ordiante of P 3 is sufficiently negative 5 . The remaining three crossings are alternating, as can be verified by linear interpolation, so that K is ambient isotopic to a piecewise linear trefoil [7].…”