Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Scan line display of algebraic surfaces
SIGGRAPH '89 Proceedings of the 16th annual conference on Computer graphics and interactive techniques
Guaranteed ray intersections with implicit surfaces
SIGGRAPH '89 Proceedings of the 16th annual conference on Computer graphics and interactive techniques
Robust ray intersection with interval arithmetic
Proceedings on Graphics interface '90
Ray tracing trimmed rational surface patches
SIGGRAPH '90 Proceedings of the 17th annual conference on Computer graphics and interactive techniques
Functional composition algorithms via blossoming
ACM Transactions on Graphics (TOG)
An implicit surface polygonizer
Graphics gems IV
Using particles to sample and control implicit surfaces
SIGGRAPH '94 Proceedings of the 21st annual conference on Computer graphics and interactive techniques
ACM Transactions on Graphics (TOG)
Guaranteeing the topology of an implicit surface polygonization for interactive modeling
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
Shape Interrogation for Computer Aided Design and Manufacturing
Shape Interrogation for Computer Aided Design and Manufacturing
Ray tracing algebraic surfaces
SIGGRAPH '83 Proceedings of the 10th annual conference on Computer graphics and interactive techniques
Ray tracing parametric patches
SIGGRAPH '82 Proceedings of the 9th annual conference on Computer graphics and interactive techniques
Real-time GPU rendering of piecewise algebraic surfaces
ACM SIGGRAPH 2006 Papers
How to Solve a Cubic Equation, Part 5: Back to Numerics
IEEE Computer Graphics and Applications
Interactive Ray Tracing of Arbitrary Implicits with SIMD Interval Arithmetic
RT '07 Proceedings of the 2007 IEEE Symposium on Interactive Ray Tracing
Isotopic triangulation of a real algebraic surface
Journal of Symbolic Computation
Computation of singularities and intersections of offsets of planar curves
Computer Aided Geometric Design
G1 continuity conditions of adjacent NURBS surfaces
Computer Aided Geometric Design
Real-Time Ray Tracing of Implicit Surfaces on the GPU
IEEE Transactions on Visualization and Computer Graphics
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Piecewise algebraic B-spline surfaces (ABS surfaces) are capable of modeling globally smooth shapes of arbitrary topology. These can be potentially applied in geometric modeling, scientific visualization, computer animation and mathematical illustration. However, real-time ray casting the surface is still an obstacle for interactive applications, due to the large amount of numerical root findings of nonlinear polynomial systems that are required. In this paper, we present a GPU-based real-time ray casting method for ABS surfaces. To explore the powerful parallel computing capacity of contemporary GPUs, we adopt iterative numerical root-finding algorithms, e.g., the Newton-Raphson and regula falsi algorithms, rather than recursive ones. To facilitate convergence of the Newton-Raphson or regula falsi algorithm, their initial guesses are determined through rasterization of the isotopic isosurface, and the isosurface is generated based on regular criteria for surface domain subdivision. Meanwhile, polar surfaces are adopted to identify single roots or to isolate different roots, i.e., ray and surface intersections. As an important geometric feature, the silhouette curve is elaborately computed to floating-point accuracy, which can be applied in further anti-aliasing processes. The experimental results show that the proposed method can render thousands of piecewise algebraic surface patches of degrees 6-9 in real time.