Perspective View of a Square by Two Distinct Viewers

        Consider a square being viewed from the side by two viewers, view₁ and view₂, from different perspectives as in the given figure. Each viewer would see the vertices of the square along a line. The perspective of viewer, view₁, of the vertices of the square is the points along a line in the order from top-down D, A, B, C; whereas, the perspective of viewer, view₂, of the vertices of the square is the points along a line in the order from top-down (or left-right) A, B, D, C.
        Drag view₂ to see how the viewer's perspective changes as view₂ moves around the square. Note how the order of the vertices changes as viewer 2's perspective changes.

       The points along the line from which the viewer sees the vertices of the square will be defined in a later section as a pencil of points. Each diagram of a viewers perspective (observation point, view lines, and pencil of points) will be called a perspectivity with the point representing the viewer called the center of the perspectivity.
        In the section on perspectivities and projectivities, we will study relationships between the views (pencil of points) of the two viewers. How can the perspective of one viewer be projected onto the perspective of the other? This leads to the more general concept of projectivity, which will be defined as a product of perspectivities.

Timothy Peil, 4 February 2013, Created with GeoGebra