Flow Simulation in TINs Drainage Queries in TINs: from local to global and back again S. Yu, M. van Kreveld, and J. Snoeylink

Motivation Provide important information on water resources, flood areas, erosion and other natural processes Natural resource management Road building, logging, or other activities have potential to affect defining stream Manual quantification of terrain drainage is time consuming

TINs Piecewise-planar approximation to a terrain where each facet of the approximation is a triangle

Assumptions At any point, water follows the steepest descent Watercourses can merge, forming trees Watercourses end only at local minimum At any point there is a unique direction of steepest descent (no flat faces)

General Definitions trickle path(p): path beginning at point p, following steepest descent and ending at local minimum watershed(p): set of points which drain to p watercourse network: all points that form watersheds

Observations steepest descents in a triangle lie on parallel lines (planar object) edge become channels when flow comes from both adjacent triangles trickle paths cannot cross drain objects: faces edges vertices faces faces edges

Definitions

peak – local maximum (vertex) pit – local minimum (vertex) basin – watershed of a pit Method Approximate steepest descent by passing flow triangle to triangle in direction of steepest descent

Observations point p collects flow from region bounded by steepest ascent in adjacent triangles trickle paths from local channels determine the watershed network Lemma: Watercourse network is a collection of disjoint trees rooted at pits whose leaves are local channels

Watercourse Definitions junction – point of the network that has at least two separate incoming parts segment – maximal part that doesn’t contain junctions catchment area – the part of the terrain that drains directly into the segment

Processing a TIN Queries/Operations What is the area of the watershed of point q? What is the flow rate through q if water falls at uniform rate r? Construct the watershed of point q What points have watershed area of size A?

Idea Partition a TIN surface into pieces, for which simple functions summarize drainage behavior

Partitioning a TIN Tailored Definitions drain(p) – the point where trickle path(p) meets the network catchment area(s) – all points that drain(p) is on s set of basins (one per pit) – form a (coarse) partition set of catchment areas (per segment) – form a finer partition

base segments – for every vertex v define a node drain(v) which cut segments into base segments (further refinement) right/left strips – set of points p such that drain(p) lies on s and trickle path(p) comes from the right/left Lemma: A strip is a region bounded by a base segment from a local channel, a segment of a local ridge, and two steepest descent paths.

Corollary: Basins, watersheds, and catchment areas are bounded by local ridges and steepest descents Proof: all are composed of left/right strips of segments Corollary: Watershed area, projected area, or steady- state flow rate can be summarized by piecewise quadratic functions on base segments

Proof: For a query point q, base segment s determines left and right strips that are partially inside the watershed of q Every other strip is either entirely outside or inside and contributes a constant amount contribution(s) – at^2 + bt + c t is a parameter indicating the position of q on s

Computation of Flow Network For each triangle t, Calculate the center point p, Call t.trickle path(p) Locate tree in network (forest) Add (sub)path as branch to tree

Query: Given a discrete set of points P each with a corresponding mm/hour rain value, and a point q, approximate the flow at q according to time. Init: T = find flow tree containing query point q For each triangle t containing p in P, set inflow, slope, flow speed, out capacity Flow Simulation

run(): For each time frame, Start with any leaf triangle t in T, t.flow = t.inflow t.outflow = min(t.flow, t.outcapacity) t.flow -= t.outflow Until t is a pit, t.flow += t.inflow + t.sources.outflow t.outflow = min(t.flow, t.capacity) t.flow -= t.outflow Continue up the tree merging sub-trees and summing input flow

Results

Demonstrating trickle path

Working with grid elevation map (PGM)

TIN created from grid points containing triangles

14820 trickle paths longer than 5

31248 Triangle TIN created from grid

19998 Triangle TIN created from grid

8710 Triangle TIN created from grid points

TIN created from grid points containing triangles

5561 trickle paths longer than 10

TIN created from grid points containing triangles

TIN created from grid points containing triangles

Extensions and Future Research Absorption of water by different ground types Account water volume (spilling, pruning) Horizontal triangles, stream bifurcation

Conclusions TINs can provide a good, efficient approximation of drainage networks Large maps require significant optimizations/approximations