12.2 Figures

Figure 1: An interpolated level set with simple topology.

Figure 2: Extra topological complications not directly implied by the data.

Figure 3: The possible arrangements of Highs on a face, with boundary contours.

Figure 4: Components of $\bf{X}_{f}^{\geq \tau}$ defined by edge-connected sets of Highs. Note this implies that the point p satisfies $f(p) < \tau$.

Figure 5: Inconsistent results when we interpolate components of $\bf{X}_{-f}^{> - \tau}$ defined by edge-connected sets of Highs. The point p satisfies $-f(p) \leq -\tau$ and thus $f(p) \geq \tau$, contradicting the previous figure.

Figure 6: Knit High above, Knit Low below: Initially there are two separate objects.

Figure 7: The two objects touch at the disambiguation value.

Figure 8: The two objects have merged.

Figure 9: The entire 4-hit face is engulfed.

Figure 10: Illustration of nomenclature for 4-hit face disambiguation.

Figure 11: The disambiguation point and value (for 2 dimensions) via linear interpolation across the face.

Figure 12: A Vertex Local Maximum: at thresholds above 20 there is no object, below 20 there is one object

Figure 13: A Local Minimum Critical Point: at thresholds above 10 there is a hole, below 10 the hole vanishes.

Figure 14: A Saddle Critical Point on a vertex: at thresholds above 20 there are two local objects, below 20 the two local objects merge.

Figure 15: A Critical 4-hit Face Saddle Critical Point.

Figure 16: A local maximum critical set, where region S has the constant value of 100.

Figure 17: A local minimum critical set, again where region S has the constant value of 100.

Figure 18: A Saddle Critical Set. Again region S has the constant value of 100.

Figure 19: Criticalities in the dataset (see table 1 for the critical values).

Figure 20: Zones of the 5 maxima shown Initially.

Figure 21: Zone of the criticality g has been added.

Figure 22: Zone of criticality h has been added

Figure 23: Zones of saddles i and j. At this point the objects have merged into one object with two holes. The two remaining criticalities are the minima k and l. Zone of k will include all remaining points with value greater than 50 and zone of l will include all the remaining points.

Figure 24: Criticality graph associated with the dataset.

Figure 25: Spinning one triangle in a cube face.

Figure 26: Illustration of a simple corner intersection of a surface with a cube. Note that each of the three faces have two hits. Each pair of hits in a face is connected back to the Articulation point. The Articulation point (center of moment of the three grouped hits, corresponding to one surface and one criticality beyond) is used to knit three triangles.

Figure 27: Illustration of a surface intersecting the waist of a cube, passing through the center of the cube, instead of passing through a corner.

Figure 28: Two examples of two separate surfaces passing through the corners of a cube. The left cube has a surface cutting through the two front faces of the cube. The right cube has a surface passing through the front surface, and another, possibly separate surface passing through the back face of the cube.

Figure 29: Triangle edge from AP to h is shared by two triangles, defined by the hits adjacent to h in each of the cube faces that shares the hit. These two hits are g and g'.

Figure 30: The triangle edge connecting two adjacent hits is shared by the two triangles defined by an articulation point in each of the two cubes that share the face on which the hits occur.

Figure 31: Overhead view of the eight edges emanating from a central hit point, into the 4 shared cubes. Each X marks a hit.

Figure 32: Cyclical enumeration of edges from AP to hits on the cube faces.