Appendix B: Table of Figures

B Table of Figures

Figure 3.1 $Figure 3.1 Figure 3.1$ $PDF HTML$ : Closed triangulated polygonal chain49 $PDF HTML$

Figure 3.2 $Figure 3.2 Figure 3.2$ $PDF HTML$ : Quadrisection of $int Δ$ 49 $PDF HTML$

Figure 3.3 $Figure 3.3 Figure 3.3$ $PDF HTML$ : A simplified region containing two vertical lines and two continuous, rectifiable curves.52 $PDF HTML$

Figure 3.4 $Figure 3.4 Figure 3.4$ $PDF HTML$ : Geometry of the finite subcover of $𝑉_{𝑛} \subset 𝑊_{𝑛}$ for some $𝑛 \in ℕ$ .78 $PDF HTML$

Figure 3.5 $Figure 3.5 Figure 3.5$ $PDF HTML$ : $𝜁$ , $𝜉$ , and $𝑧$ when $| 𝜗 - 𝜏 | > 𝛿$ , with distances marked. The use of $+$ and $-$ denote a value more or less (respectively) than the preceding value.103 $PDF HTML$

Figure 4.1 $Figure 4.1 Figure 4.1$ $PDF HTML$ : The annulus $𝑉$ , with $𝛾_{1}$ , $𝛾_{2}$ , and $𝛾$ .112 $PDF HTML$

Figure 4.2 $Figure 4.2 Figure 4.2$ $PDF HTML$ : The orientation of a neighborhood that does not enclose $\infty$ after projection.165 $PDF HTML$

Figure 4.3 $Figure 4.3 Figure 4.3$ $PDF HTML$ : A semicircular contour with orientation marked.166 $PDF HTML$

Figure 4.4 $Figure 4.4 Figure 4.4$ $PDF HTML$ : An indented semicircular contour with orientation marked.167 $PDF HTML$

Figure 4.5 $Figure 4.5 Figure 4.5$ $PDF HTML$ : A wedge contour with orientation marked.169 $PDF HTML$

Figure 4.6 $Figure 4.6 Figure 4.6$ $PDF HTML$ : An indented wedge contour with orientation marked.171 $PDF HTML$

Figure 4.7 $Figure 4.7 Figure 4.7$ $PDF HTML$ : A rectangular contour with orientation marked.173 $PDF HTML$

Figure 5.1 $Figure 5.1 Figure 5.1$ $PDF HTML$ : The biholomorphism between $𝔻$ and $𝑄$ via $𝐹$ .201 $PDF HTML$

Figure 5.2 $Figure 5.2 Figure 5.2$ $PDF HTML$ : The two regions $𝑈_{1}$ and $𝑈_{2}$ sharing a boundary curve $𝛾$ , the disk $𝐷 (𝑧, 𝑅_{𝑧})$ for $𝑧 \in 𝛾$ , and the curve¹ $Γ$ . Solid large arrows and arrowheads denote orientation of the ${\tilde{Γ}}_{1}$ and ${\tilde{Γ}}_{2}$ regions.204 $PDF HTML$

Figure 6.1 $Figure 6.1 Figure 6.1$ $PDF HTML$ : The elements of $𝒢︀$ , relative to $𝐾$ and its neighborhood, $𝑈$ .208 $PDF HTML$

Figure 6.2 $Figure 6.2 Figure 6.2$ $PDF HTML$ : The striped region bounds $𝐾$ , while the thin lines bound the dotted region $𝑈$ .225 $PDF HTML$

Figure 6.3 $Figure 6.3 Figure 6.3$ $PDF HTML$ : The striped region represents $𝐻$ while the unshaded regions represent the set which $𝜁_{𝑘}$ can be in, each of which are denoted by small stars. The dotted disks represent the finite subcover of $𝐻$ : notice that every striped region is also dotted.226 $PDF HTML$

Figure 6.4 $Figure 6.4 Figure 6.4$ $PDF HTML$ : The construction of $𝐸_{𝑘}^{𝑗}$ . The entire polyline from $𝑝_{𝑘}$ to $𝜁_{𝑘}^{(𝑗)}$ is $𝛾_{𝑘}^{(𝑗)}$ 227 $PDF HTML$

Figure 6.5 $Figure 6.5 Figure 6.5$ $PDF HTML$ : A conceptual construction of ${𝐻_{𝑘}^{(𝑗)}}$ , which unions to $𝐻$ .227 $PDF HTML$

Figure 7.1 $Figure 7.1 Figure 7.1$ $PDF HTML$ : The construction of the arc subtending a positive angle on which $𝑔 (𝑧) < 𝑀$ . The striped region is $𝐾$ .244 $PDF HTML$

Figure 7.2 $Figure 7.2 Figure 7.2$ $PDF HTML$ : By choosing sufficiently small $𝜂 ∣ 𝛿$ , we obtain a grid such that the boundary curves are disjoint.269 $PDF HTML$

Figure 8.1 $Figure 8.1 Figure 8.1$ $PDF HTML$ : $𝑄$ has a greater heuristic distance to $𝑇_{𝑃} Σ$ for a more curved surface.285 $PDF HTML$

Figure 9.1 $Figure 9.1 Figure 9.1$ $PDF HTML$ : Illustration of the deformation of a homeomorphism between a rectangle with opposite sides identified and a torus.319 $PDF HTML$

Figure 9.2 $Figure 9.2 Figure 9.2$ $PDF HTML$ : The parallelogram $𝑃_{1}$ with 8 periods on its boundary with lattice $Λ$ .323 $PDF HTML$

Figure 9.3 $Figure 9.3 Figure 9.3$ $PDF HTML$ : The region $𝑈$ with a comb-shaped boundary.334 $PDF HTML$

Figure 9.4 $Figure 9.4 Figure 9.4$ $PDF HTML$ : Configuration of $𝑉$ , and possible $𝜇_{1}, 𝜇_{2}$ .336 $PDF HTML$

Figure 9.5 $Figure 9.5 Figure 9.5$ $PDF HTML$ : Possible positioning of $𝜑 (𝑉)$ relative to $Ω_{2}$ .336 $PDF HTML$

Figure 9.6 $Figure 9.6 Figure 9.6$ $PDF HTML$ : Possible configuration of $𝛾_{𝑟}$ .336 $PDF HTML$

Figure 9.7 $Figure 9.7 Figure 9.7$ $PDF HTML$ : Concept configuration of $𝜑 (𝛾_{𝑟})$ 336 $PDF HTML$

Figure 9.8 $Figure 9.8 Figure 9.8$ $PDF HTML$ : Two possibilities for the endpoints of $𝜑 (𝛾_{𝑟_{𝑗}})$ 339 $PDF HTML$

Figure 9.9 $Figure 9.9 Figure 9.9$ $PDF HTML$ : Two line segments $𝛼$ and $𝛽$ mapping to a Jordan curve bounding $𝑊$ 343 $PDF HTML$

Figure 9.10 $Figure 9.10 Figure 9.10$ $PDF HTML$ : The construction of $𝑁_{2}$ , $𝑉_{𝑞}$ , $𝜂_{2}$ , and $𝜓$ , and similarly, that of $𝑁_{1}$ , $𝑉_{𝑝}$ , $𝜂_{1}$ , and $𝜙$ .345 $PDF HTML$

Figure 10.1 $Figure 10.1 Figure 10.1$ $PDF HTML$ : A geometric visualization of the Parallelogram Law.363 $PDF HTML$

Figure 10.2 $Figure 10.2 Figure 10.2$ $PDF HTML$ : A set which is convex (left) and one which is not (right).364 $PDF HTML$

Figure 12.1 $Figure 12.1 Figure 12.1$ $PDF HTML$ : A Hankel contour.422 $PDF HTML$

Figure 12.2 $Figure 12.2 Figure 12.2$ $PDF HTML$ : A keyhole contour.422 $PDF HTML$