Map-making as Graph Drawing Alan Saalfeld Mathematical Cartographer

Some (possibly) useful math Recent, (at least as yet unexploited), discoveries in geometric graph theory. Want to convey a sense of the elegant structural properties of geometric graphs. Really surprising and really cool results. Proof outlines and key properties Lots of pictures and key results. Application opportunities: what next…??

Building Topology from Geometry Given spaghetti (a bunch of line segments) determine the fundamental structure of the embedded plane graph: –vertices (line segment end points and intersections) –edges (line segments and portions of the original line segments that do not cross) –polygons (connected components of the graph set complement in the plane)

Building Geometry from Topology? Finding a realization of a graph G(V,E) Given an abstract planar graph, G(V,E) find a graph drawing of it using straight-line segments for edges –Two edge segments may intersect in (at most) one common endpoint. V = {v 1, v 2, v 3, v 4 } E = { {v 1, v 2 }, {v 3, v 4 }, {v 1, v 4 }, {v 1, v 3 }, {v 3, v 2 }, {v 2, v 4 } } v3v3 v1v1 v4v4 v2v2 A realization of G(V,E)

Triangular Graphs A triangular graph is a maximal plane graph that has only three vertices on its convex hull. A triangular graph is a triangulation of its vertex set. A triangular graph with n vertices has (3n-6) edges and (2n-5) elementary triangles.

Extending to Triangular Graphs Start with any plane graph with s vertices: Add edges until the graph’s convex hull is completely triangulated: Add three new extreme vertices so that their triangle contains the graph’s convex hull, then add more edges to finish triangulating the (s+3) vertices:

Schnyder’s Discoveries (for triangular graphs) Normal Angle Labelings Directed Edge Labelings 3-Tree Edge Decompositions 3-Path Region Decompositions Partial and Total Order relations Vertex property and Edge Property Discrete Barycentric Coordinates

Normal Angle Labelings Result

Normal Angle Labelings Rules

Key Result #1 Every triangular graph has a normal angle labeling.

Directed Edge Labelings Angle Labels Edge Labels Impossible Angle Labels Possible Angle Labels

Key Result #2 Every normal angle labeling of a triangular graph induces a normal directed-edge labeling.

3-Tree Edge Decompositions

Key Result #3 Every triangular graph can be decomposed into three edge- disjoint spanning trees of the three different types of directed edges.

Key Result #4 From every interior vertex v of a triangular graph, there are three non- crossing single color directed paths from v to the three extreme vertices of the triangular graph.

3-Path Region Decompositions The three paths to extreme vertices The 3-region partition created by the paths

Key Result #5 For each interior vertex v, the three non-crossing single-color directed paths from v decompose the triangular graph into three regions.

A Region Probability Measure  A function  : {R i (v) | v  V, i = 1, 2, 3}  [0,1] Monotonicity: If R i (v)  R i (u), then  (R i (v)) <  (R i (u)). Global additivity: For all v  V, 3 Σ  (R i (v)) = 1. i=1 R 1 (v) R 2 (v) R 3 (v)

A Triangle Probability Measure   : {Δ i | i = 1, 2, …, 2n – 5 }  (0,1] Positivity: For every triangle Δ, 0 <  (Δ). Global additivity: 2n-5 Σ  (Δ j ) = 1. j=1

Every triangle probability measure induces a unique valid region probability measure. The constant triangle probability measure,  (Δ) = 1/(2n-5) for all 2n-5 triangles Δ, induces the “triangle- count-fraction” region probability measure. Lots of measures:

 (R i (v)) = # triangles in R i (v)/(2n–5)  (R i (v)) = # vertices for R i (v)/total* vertices  (R i (v)) = # edges for R i (v)/total* edges  (R i (v)) = area of R i (v)/total area  (R i (v)) = population of R i (v)/total population *Choose counting method to keep the total constant. More region measures:

Key Result #6 Every triangle probability measure or region probability measure  permits the assignment of three barycentric-like coordinates to every interior vertex of the triangular graph.

Barycentric Coordinates Every point in a triangle is a unique convex combination of the vertex points {p 1,p 2,p 3 }: p=  1 p 1 +  2 p 2 +  3 p 3 where  i ≥0, and 1=  1 +  2 +  3. barycentric coordinates (  1,  2,  3 ) are called the barycentric coordinates of p with respect to {p 1,p 2,p 3 }. Converting to barycentric coordinates or back to Cartesian coordinates is easy. x1x1 y1y1 1 x2x2 y2y2 1 x3x3 y3y3 1 xy1 x2x2 y2y2 1 x3x3 y3y3 1 = 11 x1x1 y1y1 1 x2x2 y2y2 1 x3x3 y3y3 1 x1x1 y1y1 1 xy1 x3x3 y3y3 1 = 22  3 = 1-  1 -  2

Barycentric Coordinates are just Relative Triangle Areas xixi yiyi 1 xjxj yjyj 1 xkxk ykyk 1 xy1 xjxj yjyj 1 xkxk ykyk 1 A1A1 A 1 +A 2 +A 3 = = 11

X Y Isolines in RED for Y constant Isolines in BLUE for X constant x = 2 y = -1 Isolines of Cartesian coordinates

Isolines of barycentric coordinates p1p1 p2p2 p3p3  1 = ⅝  3 = ½  2 = ⅜ Isolines in RED for  2 constant Isolines in GREEN for  3 constant Isolines in BLUE for  1 constant

Vertex property For any vertex v, and any vertex u, there exists some i for which  (R i (v)) <  (R i (u)) and some other j for which  (R j (u)) <  (R j (v)). If u  R j (v), u  v, then  (R i (u)) <  (R i (v)). u

Edge Property For any edge {v,w} and any vertex u not belonging to that edge, there exists some i for which both  (R i (v)) <  (R i (u)) and  (R i (w)) <  (R i (u)).

Key Result #7 If the three barycentric-like coordinates are used as actual barycentric coordinates for the interior vertices, then the edges may be added as straight-line segments for a topologically- correct realization of the graph.

Partial Order relations Each different colored tree is associated with one partial order  (R 2 (w)), and  (R 3 (v)) >  (R 3 (w)).

Partial Order relations For partial order < i on the interior vertices the blue edges are precisely the covering relations of < i. We say that w covers v with respect to < i if v < i w, and there is no u between v and w with respect to < i.

Key Result #8 If the three barycentric-like coordinates are used as actual barycentric coordinates for the interior vertices, then the edge identities may be completely determined from three special partial orders derived from those barycentric coordinates.

More research directions Explore additional measures. Can you find a measure for cartograms? The starting triangulation can even consist of topological triangles (with curved edges). The realization will still have straight edges.

Additional research opportunities Explore additional measures. Apply to underlying regular grids: Look at special connectivity properties of the 3 colored trees: e.g., every pair of points is connected by one single-colored path.