# Introduction

This document briefly discusses different types of graphics pertinent to mathematics, the sciences and other technical fields for purposes of developing the WAI-ARIA taxonomy or other means of improving accessibility. The needs arising from educational applications are noted, motivating an approach to accessibility that transcends the limitations of text alternatives, which, since the mid 1990s, have remained the principal focus of attention in this domain.

# Categorizing Graphics and Identifying Emerging Access Requirements

Significant types of graphical representations are identified in the notes that follow. Systematic analysis of common concepts and the construction of a formal taxonomy are important tasks that lie outside the scope of this document.

## Data Visualization

Charts in their various forms depict data, which may be complex and may, in the context of a Web application, demand interaction from the user. The concepts pertaining to this category of graphics are currently under discussion by the task force and have been documented in separate proposals; hence they need not be examined further in the present document, other than to note the importance of this category to a wide range of fields in the natural and social sciences Some of the categories of graphics noted below overlap with the work currently being undertaken to support data visualization in the WAI-ARIA taxonomy.

## Analytic Geometry

Points, lines, curves, surfaces and other constructs presented in coordinate systems are commonplace in mathematics and associated disciplines. Formally, these constructs are sets of points on the real line, in the complex plane, in 3-space, etc. Depictions of mathematical functions and relations as graphs (in the sense of analytic geometry) are widespread, especially in elementary mathematics, including calculus. The components of such a diagram typically include the coordinate axes and the set of points corresponding to the given function or relation. More than one function or relation may be depicted in a single coordinate system, e.g., several linear functions may appear in the same plane, with a single pair of axes. Labelled points may be designated, e.g., minima, maxima, intercepts, discontinuities, etc.

Although typical graphics employing analytic geometry can be described in text, the development of interactive educational applications requiring students to select points in the plane or plot graphs of functions creates new demands regarding the accessibility of diagrams of this type. Interactive navigation as well as the construction of points, lines and curves in educational Web applications pose a challenge in particular for users who are blind. Haptic devices and tactile displays may be expected to become increasingly important in addressing this need. . Auditory presentation also has the potential to play a valuable role. Given the diversity of hardware and access techniques that may be available to the user now and, especially, in the foreseeable future, no single non-visual presentation (e.g., a tactile graphic) suffices to meet all needs. Furthermore, the requirements of conventional screen readers should not be disregarded.

The central concepts associated with analytic geometry are as follows:

• Coordinate axes and associated labels
• Points, lines, curves and surfaces on the line, in the plane or in space, respectively.
• Designated points with associated labels, e.g., the origin, intercepts, maxima and minima.
• Vectors.
• Tangent lines, perpendicular lines, shaded areas, rectangles.
• Grid lines
• Transformations: translation, rotation, reflection and dilation.
• Navigation along the axis or axes (two-way for a line, four-way for a plane, etc.).
• Navigation along a line or curve, for example to read values interactively or to select a point in an application.

This category of graphics associated with analytic geometry overlaps with data visualization: data can be plotted in a coordinate plane. Therefore, it is suggested that any taxonomic hierarchy developed to describe analytic geometry should include such data plots as special cases. Further, it should be recognized that lines and curves are part of the basic, implicit semantics of SVG and, as such, need not be represented explicitly in a taxonomy if they can be inferred directly from the SVG markup.

## Geometric Figures and Constructions

These graphics, typically representing two or three-dimensional geometrical figures, are very diverse. As in the preceding category, they can usually be well described in text alternatives or embossed in tactile form. However, the emergence of interactive educational applications, for example geometry-based games and tutorials, demands more than these conventional means of providing non-visual access can supply. In particular, applications that require the user to identify points in a geometrical figure or to construct such a figure present similar challenges to those described earlier in connection with analytic geometry.

Central concepts include:

• Points, rays, lines and line segments.
• Polygons, circles, curves, solid (i.e., three-dimensional) figures.

Many of these constructs should be recognizable from the SVG markup used to describe them geometrically. Therefore, they may not warrant inclusion in the ARIA taxonomy.

## Vertex/Edge Graphs

These are graphs in the formal, mathematical sense. They have a wide range of applications in diverse disciplines. Examples include trees (occurring widely in computer science), networks and flow diagrams. The access needs associated with graphs become apparent as increasingly complex cases are considered. Although the pertinent information can be conveyed through descriptions, interactive reading and navigation becomes all the more advantageous as the number of edges and vertices grows. This need suggests that graphs should be included in the WAI-ARIA graphics taxonomy. Again, the user's means of interaction and navigation vary according to the available hardware as well as the individual's needs. For example, navigation via a touch surface (with auditory, haptic or spoken feedback) offers direct, two-dimensional exploration that cannot be accomplished as easily or effectively with a keyboard.1

The information that should be made available for the purpose of making graphs accessible includes:

• The set of vertices, including any labels associated with vertices.
• The edges (represented, for example, as pairs of vertices).
• Whether the edges are directed or undirected.
• Labels associated with edges
• Interactive navigation to adjacent vertices (i.e., those incident on the vertex that currently has focus).
• Navigation between components of a disconnected graph.
• Navigation to previous/next sibling, parent and children in trees.

## Measuring Instruments

Graphics representing measuring instruments occur in textbooks and, more importantly for present purposes, scientific simulations. Such simulations have been developed for use in educational applications. The value depicted by the measuring instrument may change as a simulation progresses. The following information may be captured in order to make graphical depictions of measuring instruments accessible:

• The type of measuring instrument, e.g., ruler, thermometer, volt meter.
• The scale (minimum and maximum values as well as the step size in which values are reported).
• The unit of measure.
• The currently reported value, if any.
• Any labels associated with the diagram.

## Venn diagrams

Due to their potential complexity, venn diagrams can usefully benefit from interactive reading and navigation in order to facilitate comprehension, which may be difficult if text alternatives (long descriptions) are used. Venn diagrams represent relations between sets, including union and intersection. Information that should be captured in a vocabulary for formally describing venn diagrams includes:

• The elements of each set and their associated labels.
• Subset/superset relations.
• Unions and intersections of sets depicted in the venn diagram.
• The universal set, if any.
• Interactive navigation to adjacent sets, i.e., those appearing geometrically next to the set that currently has focus.
• Hierarchical navigation down into subsets and up into supersets, relative to the set that is currently at the point of regard.
• Navigation into the intersecting regions of intersecting sets.
• Interactive reading of elements; selecting and moving elements within the diagram.
• Interactive venn diagram construction.

# Performing Actions on Graphical Objects

In general, interactive applications such as simulations require the user to perform actions upon objects that are represented graphically. These objects are very diverse and may not be describable by the WAI-ARIA taxonomy under development for SVG. Nevertheless, labels and descriptions of such objects may be supplied, enabling them to be identified by the user.

Having identified an object to be acted upon, e.g., a graduated cylinder in a chemistry simulation, the user needs to be able to

1. Enumerate the actions that can be performed on the object, for example, adding a given quantity of a specified chemical; and
2. Perform any of the available actions, for example by causing a prescribed quantity of the simulated chemical to be added to the contents of the graduated cylinder.

As the preceding discussion and example indicate, the yet to be defined mechanism for associating actions with objects and enabling them to be invoked by the user, needs to be quite general. It should not depend on how the object is represented, for example on whether it is assigned a WAI-ARIA role drawn from the graphics taxonomy, or made accessible via a label or description. Actions involving two or more objects should be supported, e.g., to enable the user in the above-mentioned example to add a chemical from a specified container into the graduated cylinder. (The container itself would be an object capable of description or annotation via WAI-ARIA roles defined in the taxonomy.)

1. Additional research is needed into how vertex/edge graphs are used in interactive applications and the types of interaction typically required.