7.3 Orthogonal and Polyline Drawings
2 min read•august 12, 2024
Graph drawing gets even more precise with orthogonal and polyline techniques. These methods use specific rules to create clean, organized layouts. They're super useful for things like circuit diagrams and subway maps.
Orthogonal drawings use right angles and a grid system, while polyline drawings allow for more flexible bends. Both aim to make graphs easier to read and understand, balancing aesthetics with practical layout needs.
Orthogonal and Polyline Drawings
Orthogonal Drawing Principles
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- represents graphs with edges as sequences of horizontal and vertical line segments
- Edges in orthogonal drawings meet at right angles, creating a grid-like structure
- Vertices positioned at integer coordinates on a two-dimensional grid
- Orthogonal drawings maximize readability by reducing visual clutter
- Applications include circuit diagrams, subway maps, and architectural floor plans
Polyline Drawing Techniques
- allows edges to bend at arbitrary angles, offering more flexibility than orthogonal drawings
- Edges represented as sequences of connected line segments with varying orientations
- Polyline drawings balance aesthetic considerations with layout constraints
- Bend points serve as intermediate vertices along edges, allowing for complex routing
- Algorithms for polyline drawings often focus on minimizing crossings and preserving symmetry
Optimization and Grid-Based Approaches
- Bend minimization aims to reduce the total number of bends in a drawing, enhancing visual clarity
- Algorithms for bend minimization include network flow-based approaches and integer linear programming
- Grid drawing constrains positions and bend points to a discrete grid
- Grid-based layouts facilitate uniform spacing and alignment of graph elements
- Complexity of grid drawing algorithms depends on grid size and graph structure
Visibility Representations
Visibility Representation Fundamentals
- Visibility representation encodes graph structure using horizontal and vertical line segments
- Vertices represented as horizontal line segments, edges as vertical line segments
- Two vertices connected if their corresponding horizontal segments can "see" each other through a vertical line
- Visibility representations preserve and offer a compact encoding of graph topology
- Algorithms for constructing visibility representations often use planar graph embedding techniques
Kandinsky Model and Extensions
- Kandinsky model extends orthogonal drawings to allow for vertices of varying sizes
- Vertices represented as rectangles instead of points, accommodating labels or additional information
- Edges connect to vertices at designated ports, maintaining orthogonality
- Kandinsky drawings balance aesthetics with practical considerations for vertex representation
- Algorithms for Kandinsky drawings often involve flow-based approaches and iterative refinement
Topology-Shape-Metrics Approach
- Topology-shape-metrics (TSM) approach separates graph drawing into three phases: planarization, orthogonalization, and compaction
- Planarization phase determines the overall topology of the drawing, resolving crossings
- Orthogonalization phase assigns directions to edges and determines bend points
- Compaction phase assigns final coordinates to vertices and bend points, minimizing
- TSM approach allows for incremental improvements and optimization at each phase
- Applications of TSM include VLSI and software engineering diagrams
Key Terms to Review (16)
Area: Area is the measure of the extent of a two-dimensional surface within a given boundary, typically expressed in square units. In the context of orthogonal and polyline drawings, area becomes crucial as it relates to the space occupied by shapes and the efficiency of layouts. Understanding area is essential for analyzing the properties of geometric figures represented in these drawings, which can influence both aesthetic and functional aspects of design.
Aspect Ratio: Aspect ratio is the ratio of the width to the height of a geometric shape or drawing, typically expressed as two numbers separated by a colon. This concept is important because it influences how objects and figures are represented visually, affecting both clarity and aesthetics in various types of drawings, including orthogonal and polyline representations.
Bend angle: The bend angle is the measure of the angle at which a polyline segment changes direction, typically used in the context of orthogonal and polyline drawings. It plays a crucial role in defining the visual structure of graphs and diagrams, influencing the overall clarity and readability of representations. Understanding bend angles is essential for creating efficient layouts in graph drawing, where minimizing overlaps and maximizing space can enhance the communication of information.
Bend minimization problem: The bend minimization problem is a challenge in graph drawing that aims to minimize the number of bends in the edges of a graph when represented as a polyline. This problem is particularly relevant in creating orthogonal and polyline drawings, where the visual clarity and aesthetic appeal of graphs are enhanced by reducing the number of sharp turns or bends in the connecting lines. By addressing this issue, designers can create more readable and efficient representations of complex networks.
Bending algorithm: A bending algorithm is a method used to adjust and optimize the layout of polyline or orthogonal drawings by modifying the angles and segments of lines to improve aesthetic appeal or clarity. This technique aims to create smoother transitions and reduce visual clutter, making diagrams more readable. By effectively bending the paths, the algorithm ensures that connections between points maintain clarity while preserving essential geometric relationships.
Circuit layout: A circuit layout is a representation of a network of electrical components and their connections, often depicted in two dimensions to show how they will be physically arranged on a board. This layout is crucial for ensuring that the components function correctly and efficiently when the circuit is built. It also relates to both orthogonal and polyline drawings as these styles are commonly used to visualize the connections and arrangements in circuit designs.
Crossing Number: The crossing number of a graph is the minimum number of edge crossings in any drawing of the graph in the plane. This concept is crucial for understanding how graphs can be represented visually and relates directly to planarity, which affects the way graphs can be interpreted and analyzed. The crossing number helps quantify the complexity of a graph's layout and is important in various applications, such as graph drawing algorithms and geometric representations of graphs.
Edge: An edge is a fundamental element of geometric objects, representing a line segment that connects two vertices in a shape or polytope. Edges play a crucial role in defining the structure and properties of various geometric forms, influencing aspects like dimensionality, connectivity, and overall geometry.
Fáry's Theorem: Fáry's Theorem states that every planar graph can be represented as a straight-line drawing in the plane, where no two edges cross each other. This theorem is crucial for understanding how to visualize graphs in a way that maintains their structural integrity while ensuring clarity. It connects with various aspects of graph theory and provides a basis for creating clean visual representations, particularly in orthogonal and polyline drawings as well as during planarity testing and embedding processes.
Graph visualization: Graph visualization refers to the representation of graph structures in a visual format, which makes it easier to understand and analyze relationships between nodes and edges. This technique plays a crucial role in the analysis of complex data sets, where visual representations can reveal patterns and insights that are not easily discernible in textual or numerical formats. In particular, orthogonal and polyline drawings are specific methods of graph visualization that help in organizing how graphs are laid out, emphasizing clarity and reducing visual clutter.
Grid layout: A grid layout is a systematic arrangement of elements in a structured format, typically using a matrix of rows and columns. This organization facilitates the clear representation of relationships and connections among various components, making it especially useful in visualizations such as orthogonal and polyline drawings, where clarity and precision are paramount.
Orthogonal Drawing: An orthogonal drawing is a way of representing a graph in a two-dimensional space where all edges are drawn as sequences of horizontal and vertical line segments. This type of drawing emphasizes clear visual representation and can be particularly useful for understanding complex structures, especially in the context of planarity and embedding where clarity and organization of spatial relationships are crucial.
Planarity: Planarity refers to the property of a geometric object, particularly graphs, being drawable on a flat surface without any edges crossing each other. This concept is essential for understanding how various shapes and figures can be arranged and visualized in two-dimensional space, impacting the analysis of geometric structures and their relationships.
Polyline drawing: A polyline drawing is a sequence of connected line segments that can be straight or curved, representing a shape or figure in a two-dimensional space. These drawings are commonly used in various fields such as computer graphics and geometric modeling, allowing for the creation of complex shapes through simple linear connections. Each segment in a polyline can be defined by its endpoints, making it easy to manipulate and analyze geometric properties.
Straight-line drawing algorithm: A straight-line drawing algorithm is a method used in computer graphics and discrete geometry to represent graphs in a way that each vertex corresponds to a point and each edge is represented as a straight line connecting its two vertices. This type of representation is particularly important for creating clear and visually appealing diagrams that help convey the structure of the graph. Such algorithms aim to minimize edge crossings, ensure good vertex placement, and optimize the overall aesthetics of the drawing.
Vertex: A vertex is a fundamental point in geometry where two or more edges meet, often forming a corner in a geometric shape. In the context of various geometric constructs, vertices serve as crucial reference points for defining shapes and their properties, influencing aspects like dimensionality, connectivity, and the overall structure of geometric objects.