To understanding the rendering process, you must master the procedure that specifies a camera and then constructs a transformation that projects a three-dimensional scene onto a two-dimensional screen. This procedure has two several components: First, the specification of a camera model; second, the conversion of the scene's coordinates from Cartesian space to the space of the camera; and finally the specification of a viewing transformation that projects that scene into image space
For a pdf version of these notes look here.
The Camera Model
We specify our initial camera model by identifying the following parameters.
The specification of , and forms a viewing volume in the shape of a pyramid with the camera position at the apex of the pyramid and the vector forming the axis of the pyramid. This pyramid is commonly referred to as the viewing pyramid. The specification of the near and far planes forms a truncated viewing pyramid which gives the region of space which contains the primary portion of the scene to be viewed (We note that objects may extend outside the trunchated pyramid. In many situations polygons will lie between the near plane and the camera, or, in distance, beyond the far plane.). The viewing transform, transforms this truncated pyramid onto the image space volume .
The Camera Transform
Given the definition of a camera , the camera transformation is a combination of a transform that first converts the coordinates of the Cartesian frame to the local coordinates of the camera's frame,
and second, applies the viewing transform. These two transformations are usually multiplied together to form a single matrix that is applied to all points of the scene.
Defining a Frame at the Camera Position
The main idea here is to define a frame at the camera position. Given such a frame , we generate a transformation that converts the Cartesian Frame coordinates to the camera's frame.
To define a frame at the camera position is easy - and there are actually an number of ways of doing this. One of the vectors is obvious - that is, we want
In order to define the other vectors that make up the frame, we must make an assumption. We assume that the vertical direction of the camera must be in the plane defined by and the vector . This frequently happens when you are taking a picture, if you think about it - and it actually fairly easy to arrange. See the following figure for an illustration of this process. In the figure, the dotted line is the direction of view, and should be placed on the negative axis by the transformation.
To define and we utilize the following steps
We note that this works well, except when you wish to have the camera look in the direction or . In these cases, either or and , and we cannot calculate a frame in this manner. However, we can utilize another vector as the ``up direction'' to utilize with to obtain .
Calculating the Matrix
To calculate the actual matrix that implements the transformation, we can write each of the vectors , and as a linear combination of , , and (Since the vectors defining are linearly independent). In addition, we can write the vector as a linear combination of , and . Thus we can calculate the values , where
The matrix that convets the coordinates of objects in the frame into coordinates for the frame is given by
We note, that by our construction, the frame is an orthonormal frame (all vectors are unit vectors and are mutually perpendicular) and in this case the equations above simplify tremendously. In particular, all the denominators , and we can simplify the numerators utilizing the identities
The camera transform is a Cartesian-frame-to-frame transform. This is combined with the viewing transform to give a transformation that converts a scene into image space.
the Graphics Notes Home Page
Return to the Geometric Modeling Notes Home Page
Return to the UC Davis Visualization and Graphics Group Home Page
This document maintained by Ken Joy
Mail us your comments
All contents copyright (c) 1996, 1997, 1998,
Computer Science Department
University of California, Davis
All rights reserved.