An Introduction to Planar Geometric Projections

Planar geometric projections are a distinct class of projections that maintain straight lines when mapping an image onto a viewing surface. This class differs from non-planar projections in the type of viewing surface and projectors used for displaying a three-dimensional object. In a planar geometric projection, straight lines form the projectors and map the object's image to a planar viewing surface called the viewplane. In contrast, a non-planar projection uses curved projectors to map an image onto a curved surface.

Converting to the View Reference Coordinate System

Step 1. Define the u-, v-, and n-axes of the view reference coordinate system
The development of a planar geometric projection requires that each vertex of an object be converted form the world coordinate system to the view reference coordinate system. In the world coordinate system, vertices are defined by their "real world" x-, y-, and z-coordinate values (see Figure 1). When converting from the world coordinate system to the view reference coordinate system, each point of the "real world" object is mapped to a two-dimensional viewplane.

Figure 1 Specifying the view reference coordinate system. The intersection of the u-, v-, and n-axes defines the VRP. The VPN is represented by the n-axis.

To determine the position of the viewplane in the world coordinate system, the first step is to locate the origin, called the view reference point (VRP), of the view reference coordinate system. After positioning the origin, the n-axis (corresponding to the z-axis of the world coordinate system) is defined by a vector called the viewplane normal (see Figure 1). The direction of the viewplane normal n (which is perpendicular to the viewplane) is defined as a displacement of x units along the x-axis, y units along the y-axis, and z units along the z-axis. As with many vector operations, the viewplane normal is then normalized. The normalized vector n is represented by the equations

so that the unit vector has a length of 1. The v-axis, which represents the vertical axis of the view reference coordinate system, is specified by the normalized vector v. To determine v, use an upward vector, up, as a meaningful suggestion for the upward direction. Then use up to find up'; the closest vector lying in the viewplane (Hill, 1990). The equation for finding the up' vector is

where (up · n) is the dot product of the upward vector and the viewplane normal. The vector v is equal to the normalized version of up'. The third axis of the view reference coordinate system, the u-axis, is perpendicular to the vectors forming the n- and v-axes. The direction of the u-axis is determined by the equation

where u , the vector defining the u-axis, is equal to the cross product of vectors n and v. This equation defines the direction of the axes for a left-handed coordinate system. In this case, the positive direction for the n-axis is receding from the viewer. To establish a right-handed coordinate system in which the positive direction of the n-axis is approaching the viewer, the components of the vector u are multiplied by a negative one. After these calculations, the view reference coordinate system is specified in terms of the world coordinate system.

Step 2. Convert the object's vertices to positions in the view reference coordinate system
With the exception of multiview projections, transformations are required to convert each endpoint of a 3D object from the world coordinate system to the view reference coordinate system. The formulas described by Hill, 1990

where p is the (x, y, z) coordinates of a point in the world coordinate system, r is the (x, y, z) coordinates of the VRP, and (u, v, n) define the point in the view reference coordinate system.

Subclasses of Planar Geometric Projections

Figure 2 The subclasses of planar geometric projections.