doi:10.1016/S0097-8493(00)00110-2 
Copyright © 2001 Elsevier Science Ltd. All rights reserved.
Technical Section
Visualisation of implicit surfaces
Ronald J. Balsysa and Kevin G. Suffern
, 
, b
a Faculty of Informatics, School of Computing and Information Systems, Central Queensland University, Rockhampton M.C., Qld. 4702, Australia
b Faculty of Information Technology, University Technology, Sydney (UTS), P.O. Box 123, Broadway NSW 2007, Australia
Available online 9 February 2001.
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Abstract
We discuss the polygonisation and rendering of implicit surfaces through the adaptive subdivision of octree nodes. Three new algorithms which form part of this process, are discussed. The first algorithm removes the cracks that form on the surfaces between adjacent nodes that are subdivided to different depths in the octree. This algorithm fixes the cracks for arbitrary differences in depth between the adjacent nodes. The second algorithm allows user specified selected polygon outlines to be drawn on the surfaces, instead of drawing all the outlines, or no outlines. The third algorithm fixes the gaps that can appear in the polygon outlines when only selected outlines are drawn. We also investigated the use of an exact calculation of the curvature of the implicit surfaces to drive the adaptive subdivision. We conclude that this is not worthwhile doing, and give reasons.
Fig. 1. Plotting node of octree showing signs of function at the corners of the node. In this case each face has 0 or 2 intersections and a satisfactory polygonisation results.
Fig. 2. (a) Crack formation in an adaptively subdivided surface, (b) crack removed by altering the polygon in the less subdivided node.
Fig. 5. Super-ellipsoid (2) with a=12, b=8, c=3, d=6, 
1=0.3, 2=2, (a) with cracks removed, (b) with cracks removed and polygons outlined.
Fig. 3. Face of a plotting node where there is a difference in depth of 2 between it and its neighbour nodes of the octree. Line segments in all nodes are shown.
Fig. 4. Hyperbolic paraboloid (1) with a=2, b=4, and a maximum subdivision depth of 6, (a) without crack removal, (b) with cracks removed.
Fig. 6. Hyperbolic paraboloid surface with maximum subdivision depth of 10, (a) all polygon outlines drawn, (b) and (c) selected outlines drawn.
Fig. 7. Equipotential surface of the form (3) for 6-point charges showing only polygon outlines perpendicular to the z direction.
Fig. 8. Ellipsoid (1) viewed from along the z-axis in a parallel projection with polygon outlines drawn in black, node outlines drawn in orange. Numbers along the x (horizontal)-axis are integer node values. See text.
Fig. 9. Half of a relativistically moving sphere with continuous outlines drawn perpendicular to one axis.
Fig. 10. Implicit blending surface (4) drawn with maximum depth 8 and continuous outlines.
Fig. 11. Equipotential surface of the form (3). (a) Shows gaps appearing in some outlines. (b) Gaps filled as discussed in text.
Fig. 12. Formation of gaps in polygon outlines. See text.
Fig. 13. (a) Ellipsoid (5) with a=8, b=12, c=3, and a maximum subdivision depth of 10. (b) Cyclide surface (6) with parameters a=15, r=3, f=5, and a maximum subdivision depth of 7.
Fig. 14. Super-ellipsoids (2) with a=12, b=8, c=5, d=6, 1=3. (a) 2=1, (b) 2=3.
Fig. 15. Spiky surface (7) with n=4, and a maximum subdivision depth of 6.
Fig. 16. Steiner's Roman surface (8) a maximum subdivision depth of 8.
Fig. 17. Heart surface (9) with a maximum subdivision depth of 8.
Fig. 18. Cross-section of a sphere and two nodes (a) and (b) that contain the surface.
Fig. 19. Sphere rendered using (a) planarity and divergence of surface normals, and (b) exact curvature, as the subdivision criteria.
Fig. 20. Section of the cubic surface (10) (a) polygonised using planarity with angle <80° (see Bloomenthal [3]) and divergence with angle <10°, and (b) using the maximum principle curvature with radius of osculating sphere >32 cube size.
Fig. 21. Cross-section of a plotting node that illustrates potential problems with the use of point sampling of curvature to drive the adaptive subdivision.
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