6.838, Computational Geometry; Fall 2001

Point Location

Sergi Elizalde & David Pritchard

Searching - Idea

Back to the original problem: now that we have a trapezoidal decomposition of S, how can we determine which trapezoid our query point q lies in?

We would like to make some kind of decision tree (DAG), composed of simple queries that we can use to narrow down the location of q (like a binary search)

Each node of this tree represents an individual query: it has out-degree 2, corresponding to a yes-no question

The leaves should point to individual trapezoids - once we hit one of these leaves we have determined which trapezoid q is in, and thus which face q is in.

For each leaf, we store a pointer to the face that trapezoid lies in

We will use two kinds of queries:

- point query: does q lie to the left or the right of a given point?

- segment query: does q lie above or below a given line segment?

Although we could accomplish what we need with just the second kind of query, the combination of these two queries works very will with our trapezoidal map

A possible decision tree D for the trapezoidal map T is shown below:

Result of perfmorming a search against the point q is shown below:

Query against point p₁:

It lies to the right, so query against point q₁:

It lies to the left, so query against segment s₁:

It lies below, so query against point p₂:

It lies to the right, so query against segment s₂:

It lies above, so we know q lies in trapezoid D.