The objective of this paper is to quantify the room shape in terms of visibility of a visual target within the room, thus finally to obtain the optimal room shape. A wide variety of visual targets are envisaged, such as a blackboard within a classroom, a stage in a theater, or a painting in a gallery. Their common point is that they are so important that their visibility impacts the room shape. In order to obtain planning guidelines of various rooms with various visual targets, the visibility theory is developed in both a unified and a deductive way. In concrete terms, (i) A reasonable function for expressing “point-visibility” is proposed from an arbitrary viewpoint. The proposed formula not only is understandable intuitively and operational, but represents generalization of approximation equation for the solid angle of the visual target. (ii) Next, a reasonable method of aggregating point-visibility is proposed. Therefore “area-visibility” as a value for evaluating the space as a whole is derived, doubly integrating the p-th power of point-visibility. Area-visibility can be used to evaluate both classrooms that need equality among students by focusing bad view areas, and galleries where the visitor can determine the viewing position by focusing good view areas. (iii) Finally, the optimal room shape is derived in which area-visibility is maximized. Here, we obtain the optimal aspect ratio of a rectangular plan room where the visual target is on one wall.
Through the process described above, this paper is successful in clearly describing area-visibility as the formula containing three parameters and obtaining the relationship between these values and optimal room shapes. These three consist of the two parameters derived from the point-visibility function（negative impact α when viewing the target at an angle, and negative impact β when viewing the target far away）, and one parameter introduced when aggregating point-visibility (degree of inequality p in the visibility distribution).
This means that “α and β representing human eyesight” and “p depending on rooms’ usage” can be directly linked to an “evaluation value of room shape known as area-visibility”, and thus the “optimal shape of the room”. In other words, once the preconditions are determined, we can obtain information immediately that is useful for planning and design. In consequence, this is considered to be valid knowledge that allows the visibility evaluation that previously was carried out based on experience to be performed objectively.
