Computer graphics processing and selective visual display system – Computer graphics processing – Three-dimension
Reexamination Certificate
2000-10-17
2002-12-03
Zimmerman, Mark (Department: 2671)
Computer graphics processing and selective visual display system
Computer graphics processing
Three-dimension
C345S422000, C345S423000, C345S006000
Reexamination Certificate
active
06489961
ABSTRACT:
This application contains a computer program listing submitted on CD-ROM created Aug. 30, 2002.
This invention relates to computer graphics, and in particular, to the rendering of a straight line on a display in which the voxels are arranged on a cylindrical coordinate grid.
BACKGROUND
Many computer-implemented displays consist of two-dimensional arrays of individual picture elements, or pixels. To form an image, a rasterizer selectively colors the pixels. Because the individual pixels are so small, the display appears to a human viewer to be a continuous rendering of an image. This illusion is particularly effective for complex images of continuous tones such as photographs.
For simple geometric shapes, however, the pixelated nature of the display can become apparent to the human viewer. For example, if the rasterizer is instructed to draw a straight line, there is no guarantee that the points on that desired line will coincide with the pixels that are available for rendering the line. As a result, the desired line is often rendered by a rasterized line of pixels that are close to but not necessarily coincident with the desired line. This results in rasterized lines that have a jagged or echeloned appearance.
In the course of rendering an image, a large number of straight lines and line segments are often drawn. As a result, given a desired line, the rasterizer must frequently select those pixels that will minimize the jagged appearance of the resulting rasterized line. A straightforward mathematical approach is to use the equation of the desired line and the coordinates of the available pixels to minimize a least square error across all points on the line. While such an approach has the advantage of globally optimizing the selection of pixels on the rasterized line, the large number of floating point operations required causes this approach to be prohibitively time-consuming.
To meet constraints on speed, rasterizers typically implement rasterization methods that avoid time-consuming floating point operations. Among the methods that meet the foregoing constraints is that taught in Bresenham, J. E., “Algorithm for Computer Control of a Digital Plotter,” IBM System Journal, Vol. 4, pp. 25-30, 1965, the contents of which are herein incorporated by reference. Using only integer operations, the Bresenham algorithm reduces the choice of what pixel to select to an examination of the sign of a discriminant.
Even faster rasterization methods exist that select multiple points on the rasterized line based on the outcome of a single decision. There also exist a variety of rasterization methods aimed at rendering conic sections and quadric sections on a pixelated display.
However, the foregoing rasterization methods all rely on the assumption that the array of pixels is arranged in a uniform rectangular grid that can readily be modeled by a Cartesian coordinate system. This is a reasonable assumption given the prevalence of two-dimensional displays such as computer monitors and printers at the time these algorithms were developed.
Since then, however, volumetric, or three-dimensional displays have been developed. Such displays permit the generation, absorption, or scattering of visible radiation from a set of localized and specified regions within a volume. Examples of such systems are taught in Hirsch U.S. Pat. No. 2,967,905, Ketchpel U.S. Pat. No. 3,140,415, Tsao U.S. Pat. No. 5,754,147 and on pages 66-67 of
Aviation Week,
Oct. 31, 1960.
In a typical volumetric display
1
, shown in
FIG. 1
, a motor
2
spins an imaging plate
3
rapidly about an axis
4
. A light source
5
under the control of a rasterizer
6
illuminates selected spots
7
on the imaging plate
3
at successive instants. If the imaging plate
3
spins rapidly enough, and if the successive instants are separated by short enough time intervals, a continuous curve will appear to hang in mid-air.
FIG. 2
illustrates the projection
8
onto a plane perpendicular to the axis
4
of an exemplary line rendered by the rasterizer
6
. In
FIG. 2
, the imaging plate
3
is shown in six successive instants as it rotates around the axis
4
, now perpendicular to the page. At each of the six instants, the light source
5
, under the control of the rasterizer
6
, illuminates a spot
7
on the imaging plate
3
. As shown in
FIG. 2
, by aiming the light source
5
at the correct spot and firing the light source
5
at the right time, it is possible to trace out the line. It is the function of the rasterizer
6
to correctly aim and fire the light source
5
so as to trace out the line.
To aim the light source
5
, the rasterizer
6
needs a way to identify points in space. In other words, the rasterizer
6
needs a coordinate system. One possible coordinate system is a Cartesian coordinate system.
Using a Cartesian coordinate system, the rasterizer
6
would draw the line
8
in
FIG. 2
by specifying, for example, that at time t
1
, the light-source
5
should aim 30 units north, at time t
2
, the light-source
5
should aim 29 units north, 1 unit east and 1 unit in elevation, at time t
3
, the light-source
5
should aim 28 units north, 2 units east and 2 units in elevation, and so on. To ensure that the imaging plate
3
is at the appropriate location, the rasterizer
6
also has to specify the values of the times t
1
, t
2
and t
3
based on how fast the imaging plate
3
is spinning. Although the spinning of the imaging plate
3
can be resolved into a north-south component and an east-west component, this is a computationally taxing exercise that can easily be avoided by using a polar coordinate system.
As an alternative, the rasterizer
6
could draw the line in
FIG. 2
in cylindrical coordinates by specifying, for example, that at time t
1
, the light source
5
should aim 30 units away from the axis
4
at a 90 degree angle, at time t
2
, the light source
5
should aim 29.02 units from the axis
4
at an angle of 88.03 degrees and point up by 1 unit of elevation, and at time t
3
, the light source
5
should aim at 28.07 units from the axis
4
at an angle of 85.91 degrees and point up by 2 units of elevation. This would, of course, draw the same line that was drawn using Cartesian coordinates. The advantage of using cylindrical coordinates becomes apparent when it comes time to specify when to fire the light source
5
. Because the imaging plate
3
is spinning, it is far more natural to represent its motion in terms of degrees per second than it is to resolve its motion into a north-south component and an east-west component. For this reason, calculations involving motion of the imaging plate
3
in a volumetric display are best performed in terms of a cylindrical coordinate system.
The process of rendering a line in a cylindrical coordinate system is fundamentally different from that of rendering a line in a Cartesian coordinate system. In a Cartesian coordinate system, a line has a constant slope. A unit change in one direction always results in the same change in the remaining two directions, regardless of where that change occurs. This is not the case in a cylindrical coordinate system.
In a cylindrical coordinate system, the slope of a line can vary dramatically with position along the line. In contrast to a Cartesian coordinate system in which the slope of a line is constant at all points on the line, the slope of a line in a cylindrical coordinate system can vary significantly along the line. This is because the coordinate grid in a cylindrical coordinate system is not a spatially uniform grid, as is the coordinate grid in a Cartesian coordinate system.
For those portions of the line that are closest to the axis of the cylindrical coordinate system, the change in distance of the line from the origin changes very slowly with angle. For those portions of the line that are far from the axis of the cylindrical coordinate system, even a small change in angle results in a huge change in radial distance from the axis.
SUMMARY
A rasterizer according to the invention recognizes that the dominant direction of a desir
Baxter, III William V.
Giovinco Michael G.
Actuality Systems, Inc.
Fish & Richardson P.C.
Nguyen Kimbinh T.
Zimmerman Mark
LandOfFree
Rasterization of lines in a cylindrical voxel grid does not yet have a rating. At this time, there are no reviews or comments for this patent.
If you have personal experience with Rasterization of lines in a cylindrical voxel grid, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Rasterization of lines in a cylindrical voxel grid will most certainly appreciate the feedback.
Profile ID: LFUS-PAI-O-2954540