Computer Graphics
CSE 5280 Viewing Fundamentals


The Stages of the Viewing Transformation

There are three conceptual stages in the 3D viewing process: clipping, projection and the transformation to 2D screen coordinates. This tutorial will cover the projection process.

Projection is the transformation of an n-dimensional coordinate system to an m-dimensional system (where m is less than n). In computer graphics applications (such as CAD and virtual reality) the most common type of projection is the transformation of 3D dimensional data to 2D data. This involves defining straight lines or `rays' called projectors which through points on the object being projected and through the projection plane to form the 2D image.

The most commonly used projections are the parallel projection and the perspective projection.

Parallel Projections

In this, the projection lines are all parallel to each other. This means that the image formed on the projection plane has the same relative dimensions as that of the original object (i.e. there is no concept of distance from the viewer).Center of projection is at infinity and the direction of projection (DOP) for all points is the same. This feature is important in applications such as CAD where it is important to preserve the object's scale and shape. Parallel projections fall under 2 major categories:

  1. Oblique - Non-orthographics parallel projections. Projection-plane normal and the direction of projection differ.
  2. Orthographic - The direction of projection is perpendicular to the view plane. When the direction of projection is parallel to any of the principal axis, this produces the front, top, and side views illustrated below. With any single projection we lose considerable information. Thus if we wish to display detailed information about the object multiple projections is a must (see below):

Perspective Projections

This type of projection includes the concept of distance from the viewer by scaling the projected image of objects by an amount relative to their distance from the view reference point. This transformation is determined by prescribing a center of projection and a view plane. The view plane is determined by the view reference point (VRP) and a view plane normal (VPN). The object point P is located in world coordinates at (x, y, z). The problem is to determine the image point coordinates P(x', y', z'). Map points onto view plane along projectors emanating from the center of projection (COP).
Pros - gives a realistic view and feeling for three dimensional form of object
Cons - does not preserve shape of object or scale (except where object intersects projection plane)
Different from a parallel projection because,

Number of Vanishing Points

Number of vanishing points = number of principal coordinate axes intersected by projection plane. One vanishing point, one principal coordinate axis is intersected by projection plane, two vanishing points and two principal coordinate axis intersected, and so on .....See figures below.

Vanishing Points and the View Point--1

View Volumes

3D Viewing Process

Projection Plane

Parallel Orthographic Projection

Perspective Projection View Volume

Semi-infinite view volume for perspective projection

Clipping the View Volume for Orthographic Parallel Projection

Limiting view volume useful for eliminating extraneous objects

Limiting View Volume for Perspective Projections