# トポロジ¶

 目的： ベクタデータにおけるトポロジの理解 キーワード： ベクタ、トポロジ、トポロジルール、トポロジエラー、検索半径、スナップ距離、シンプルフィーチャ

## 概要¶

Topology はGISにおけるベクタの地物（ポイント、ポリライン、ポリゴン）の接続または隣接のような空間的関係を表現したものです。トポロジカルまたはトポロジベースのデータはデジタル化エラーを検出して修正するのに便利です(e.g. 道路レイヤ内の２本のラインが完全に交差していない)。トポロジは、ネットワーク分析のような、ある種の空間分析を実行するのに必要です。

Imagine you travel to London. On a sightseeing tour you plan to visit St. Paul’s Cathedral first and in the afternoon Covent Garden Market for some souvenirs. Looking at the Underground map of London (see figure_topology_london) you have to find connecting trains to get from Covent Garden to St. Paul’s. This requires topological information (data) about where it is possible to change trains. Looking at a map of the underground, the topological relationships are illustrated by circles that show connectivity.

Figure London Underground Network 1:

ロンドン地下鉄ネットワークのトポロジ。

## トポロジエラー¶

There are different types of topological errors and they can be grouped according to whether the vector feature types are polygons or polylines. Topological errors with polygon features can include unclosed polygons, gaps between polygon borders or overlapping polygon borders. A common topological error with polyline features is that they do not meet perfectly at a point (node). This type of error is called an undershoot if a gap exists between the lines, and an overshoot if a line ends beyond the line it should connect to (see figure_topology_errors).

Figure Topology Errors 1:

Undershoots (1) occur when digitised vector lines that should connect to each other don’t quite touch. Overshoots (2) happen if a line ends beyond the line it should connect to. Slivers (3) occur when the vertices of two polygons do not match up on their borders.

The result of overshoot and undershoot errors are so-called ‘dangling nodes’ at the end of the lines. Dangling nodes are acceptable in special cases, for example if they are attached to dead-end streets.

Topological errors break the relationship between features. These errors need to be fixed in order to be able to analyse vector data with procedures like network analysis (e.g. finding the best route across a road network) or measurement (e.g. finding out the length of a river). In addition to topology being useful for network analysis and measurement, there are other reasons why it is important and useful to create or have vector data with correct topology. Just imagine you digitise a municipal boundaries map for your province and the polygons overlap or show slivers. If such errors were present, you would be able to use the measurement tools, but the results you get will be incorrect. You will not know the correct area for any municipality and you will not be able to define exactly, where the borders between the municipalities are.

これは、位相的に正しいデータを作成しているため、独自の解析のために重要なだけでなく、あなたがデータを渡す人々のためだけではありません。彼らはあなたのデータと分析結果が正しいことを期待します！

## トポロジルール¶

Except for some special GIS data formats, topology is usually not enforced by default. Many common GIS, like QGIS, define topology as relationship rules and let the user choose the rules, if any, to be implemented in a vector layer.

• 自治体マップのエリアの境界は重なってはいけません。

• 自治体マップのエリアの境界にギャップ（細長い隙間）があってはいけません。

• 境界を示すポリゴンは閉じていないといけません。境界線のアンダーシュートやオーバーシュートは許可されません。

• ベクタの線レイヤにおいて等高線のラインはインターセクト（互いに交差）してはいけません。

## トポロジのツール¶

Many GIS applications provide tools for topological editing. For example in QGIS you can enable topological editing to improve editing and maintaining common boundaries in polygon layers. A GIS such as QGIS ‘detects’ a shared boundary in a polygon map so you only have to move the edge vertex of one polygon boundary and QGIS will ensure the updating of the other polygon boundaries as shown in figure_topological_tools (1).

Another topological option allows you to prevent** polygon overlaps** during digitising (see figure_topological_tools (2)). If you already have one polygon, it is possible with this option to digitise a second adjacent polygon so that both polygons overlap and QGIS then clips the second polygon to the common boundary.

Figure Topological Tools 1:

(1) Topological editing to detect shared boundaries, when moving vertices. When moving a vertex, all features that share that vertex are updated. (2) To avoid polygon overlaps, when a new polygon is digitised (shown in red) it is clipped to avoid overlapping neighbouring areas.

## スナップ距離¶

Snapping distance is the distance a GIS uses to search for the closest vertex and / or segment you are trying to connect when you digitise. A segment is a straight line formed between two vertices in a polygon or polyline geometry. If you aren’t within the snapping distance, a GIS such as QGIS will leave the vertex where you release the mouse button, instead of snapping it to an existing vertex and / or segment (see figure_snapping_distance).

Figure Snapping Distance 1:

## 検索半径¶

Snapping distance and search radius are both set in map units so you may need to experiment to get the distance value set right. If you specify a value that is too big, the GIS may snap to a wrong vertex, especially if you are dealing with a large number of vertices close together. If you specify the search radius too small the GIS application won’t find any feature or vertex to move or edit.

## 一般的な問題 / 注意すべき点¶

Mainly designed for simplicity and for fast rendering but not for data analysis that require topology (such as finding routes across a network). Many GIS applications are able to show topological and simple feature data together and some can also create, edit and analyse both.

## わかりましたか?¶

ここでは以下のことを学びました:

• トポロジ は隣接するベクタの地物の空間的関係を示すものです。

• GISでのトポロジは **tトポロジツール**によって提供される。

• トポロジは デジタイズのエラーを検出し修正する ために利用されます。

• For some tools, such as network analysis, topological data is essential.
• スナップ距離検索半径 はトポロジ的に正しくベクタデータをデジタイズするのに役立ちます。

• シンプルフィーチャ デーはは真のトポロジデータ・フォーマットではないが、GISアプリケーションで一般的に使われます。

## やってみよう¶

ここでは人に教える際のアイデアいくつか述べていきます:

• トポシートマップであなたの地域のバス停に注目しましょう。すると２点のバス停間の最短ルートを見つけることができます。

• Think of how you would create vector features in a GIS to represent a topological road network of your town. What topological rules are important and what tools can your learners use in QGIS to make sure that the new road layer is topologically correct?

## より詳しく知りたい場合は¶

• Chang, Kang-Tsung (2006). Introduction to Geographic Information Systems. 3rd Edition. McGraw Hill. ISBN: 0070658986
• DeMers, Michael N. (2005). Fundamentals of Geographic Information Systems. 3rd Edition. Wiley. ISBN: 9814126195

ウェブサイト:

QGISユーザーガイドでは, QGISで提供されるトポロジ編集の詳細な情報が含まれています.