Vision system for analyzing solid-of-revolution radius profile

The invention provides a method for visually monitoring the radius of an item which is rotating about a fixed axis and which has a trackable contour known to lie in a plane normal to the rotation, such as a part turned on a lathe. The present invention can also estimate the cross-sectional diameter of a growing crystal, and the height of the cross-section above the melt surface. In addition, the height and radius of the meniscus at the crystal/melt interface can be tracked by the system of the invention. The present makes it possible to further automate crystal growing processes in a manner that increases manufacturing efficiency, consistency, and overall quality. In general, the invention provides a machine vision method for estimating both a longitudinal position and a radius of a circular cross-sectional feature of a solid of revolution. The method includes the steps of acquiring an image of a circumferential feature of the solid of revolution using a camera having a focal length and disposed at a position characterized by a plurality of position parameters, and then finding a set of points along the image of the circumferential feature. The next step is determining an equation of an ellipse that substantially fits the set of points to provide a plurality of ellipse parameters, and then computing the longitudinal position and the radius of the circular cross-section of the solid of revolution using the focal length, the plurality of position parameters, and the plurality of ellipse parameters.

FIELD OF THE INVENTION 
This invention relates generally to automated manufacturing, and 
particularly to machine vision for industrial processes. 
BACKGROUND OF THE INVENTION 
Many manufactured parts are formed by rotating blank stock about a fixed 
linear axis while adding or subtracting material. The resulting parts are 
referred to as "solids of revolution." Examples include a clay pot 
produced on a rotating wheel, a baseball bat shaped on a lathe, or a 
silicon crystal grown from a rotating seed dipped into a silicon melt. 
Once a solid of revolution has been formed, it may be used directly as in 
the case of a baseball bat. Alternatively, it may be used as a template 
from which copies are made through various means such a plastic or metal 
casting. Such copies, having been formed without the aid of a rotating 
axis, will nonetheless also be well described as solids of revolution by 
virtue of their shape. 
A manufactured solid of revolution can be described by its profile which 
must generally conform to precise dimensional specifications. To ensure 
that the part conforms, and to avoid wasting material, a measurement of 
the radial dimension at various points along the axis of the part can be 
performed as the part is coming into being. This measurement can be 
performed in a variety of ways. For example, it is common to place a 
caliper in contact with the part to measure the diameter of the part 
immediately subsequent to a machining operation as the work progresses. 
Each of these measurement methods represents contact-based measurement. 
However, there are some manufacturing situations where contact-based 
measurement methods present significant disadvantages. For example, 
machining operations typically generate an abundance of heat, and 
consequently, the part becomes quite hot at and near the site on the part 
where the machining operation is occurring. To this heat is added any 
frictionally generated heat from moving contact of the lever arm or 
caliper with the rotating part. Since heat can cause thermal expansion, 
the resulting dimensional instability of the measurement device can lead 
to measurement inaccuracy. In other manufacturing situations, the part may 
be made from a radioactive, chemically active, pressure sensitive, 
thermally sensitive, or ultra-pure material, and consequently it is 
desirable to minimize or eliminate any contact with the part. In these and 
other situations, it can be advantageous to use a non-contacting method 
for measuring the diameter of a part at various positions along the length 
of the part, and/or at various times during the manufacturing process. 
A common non-contact approach for gauging parts described as solids of 
revolution involves viewing the object with a camera whose line-of-sight 
is normal to the object's axis of revolution. The resulting image then 
directly depicts the part profile. However, in order to obtain an accurate 
profile image, expensive optics must be used (e.g. telecentric lens) to 
achieve an image that well approximates an orthographic projection. 
Furthermore, this approach places strong constraints on camera position 
and viewing angle which are not always possible to achieve. 
An important application where it is necessary to gauge a solid of 
revolution is in the production of mono-crystalline material used in the 
manufacture of semiconductor devices. Such crystals are grown in a sealed 
vacuum chamber to prevent exposure of the molten silicon to reactive 
gasses such as oxygen which would ruin the crystal. Given the severe 
conditions within the chamber, it is highly desirable to employ a 
non-contacting method to measure the dimensions of the crystalline 
material as it is being manufactured. However, the non-contacting approach 
described in the previous paragraph, involving orthographic profile 
projection is not feasible because it is not generally possible to arrange 
for the camera to view normal to the axis of revolution of the growing 
crystal. Rather, the camera must view the crystal from a oblique angle 
determined by the position of a small quartz viewing window built into the 
vacuum chamber within which the crystal is grown. Under these 
circumstances, visual gauging methods must be compatible with whatever 
view of the crystal happens to be available. 
The manufacture of crystalline material is an important first step in the 
production of many products (laser, electronic chips, etc.). One widely 
used method for growing single large single crystals is known as "high 
temperature Czochralski crystal growth." This method is used, for example, 
to manufacture large crystals of Gallium Arsenide and Silicon. These 
crystals are then sliced to provide large wafers for use in the production 
of electronic chips. 
FIG. 1 provides a simplified diagram of a Czochralski crystal growing 
system. The Czochralski crystal growing method involves lowering a 
single-crystalline "see crystal" 10 until it touches the surface of a high 
temperature melt of source material 12. The seed crystal 10 is 
continuously rotated and slowly pulled upward as material from the melt 12 
adds to the length and diameter of the growing crystal. The seed 10, neck 
14, crown 16, and body 18 are all parts of the growing crystal. The 
growing crystal is pulled 32 and rotated 34 via a chuck 20 attached to a 
cable 22 that is connected to a drive motor 24. The high temperature melt 
of source material 12 is contained in a crucible 26 that is heated by a 
heater 28. Gradually, over many hours, a single large crystal with an 
elongated body 18 is pulled from the melt 12. 
To produce high quality crystals, it is necessary to continuously monitor 
the cross-section of the crystal body 18 where it contacts the melt 12. 
During early stages of crystal growth, the cross-section of the neck 14 
and crown 16 must be similarly monitored. In practice, the crystal/melt 
contact is often visible as a bright meniscus 30 that extends just above 
the plane of the melt surface to contact the underside of the growing 
crystal. The meniscus 30 arises from surface tension and it appears bright 
because its curved surface strongly reflects light emitted by the hot wall 
of the crucible 26 used to contain the molten source material. Even when a 
meniscus is not present, for example during the crown stage of crystal 
growth, the boundary between crystal and melt is still generally visible 
as an abrupt change in luminance at the boundary. 
The meniscus 30 can be used to estimate the crystal cross-sectional 
diameter. Known systems for estimating crystal diameter include systems 
that use a line-scan camera to sense luminance along a one-dimensional 
cross-section of the scene. The viewing angle of the camera is set to 
include the crystal and the melt to either side of the crystal. These 
systems all work by measuring the length of a chord that crosses the 
elliptical projection of the circular boundary between crystal and melt. 
By calibrating the relationship between chord length and crystal diameter, 
such systems have long been used to obtain a time-varying estimate of 
crystal diameter. 
A major limitation of line-scan systems is due to their inability to 
distinguish between various factors, in addition to crystal diameter, that 
can cause the measured chord length to vary. These factors include changes 
in melt level, and changes in crystal position. 
Melt level changes occur due to incorporation of molten source material 
into the solid crystal. The crucible may be gradually raised in an effort 
to maintain a constant melt level. However, melt level variation ranks as 
the largest source of error in the diameter estimates produced by 
line-scan systems. 
The crystal position varies because the crystal is suspended by a long 
cable or rod that continuously rotates, causing the crystal to also 
rotate. Due to the rotational energy being imparted to the crystal, any 
minor disturbance tends to cause the crystal to oscillate, or orbit, about 
the central vertical axis of the puller in a pendulum-like fashion. The 
amplitude of such oscillations can be larger than the entire diameter of 
the crystal during the early stages of crystal growth, causing the line 
scan system to see no only large variations in chord length but also the 
complete disappearance of the crystal from its one-dimensional view. 
Therefore, line-scan systems perform badly during the early (neck and 
crown) stages of crystal growth. 
The above limitations of line-scan systems for crystal diameter estimation 
can be overcome by two-dimensional imaging systems that detect and use the 
entire boundary defining the crystal/melt interface. Indeed, this 
interface is a rich source of information which is not fully exploited in 
existing systems, even when these systems employ a two-dimensional image 
sensor. For example, on existing system sold by Hamamatsu, requires a 
one-dimensional cross-section of the image in a manner that mimics the 
operation of a line-scan camera as describe above. 
Another existing system, previously developed by Cognex Corporation, 
Natick, Ma., the assignee of the present invention, acquires a 
two-dimensional image from which it estimates the width (in pixels) of the 
elliptical projection of the crystal meniscus. That Cognex system 
effectively deals with crystal orbit, but is unable to tell if a change in 
projection width results from a change in melt level or a true change in 
crystal diameter. 
SUMMARY OF THE INVENTION 
The invention provides a machine vision method for estimating both a 
longitudinal position and a radius of a circular cross-section of a solid 
of revolution. The method includes the steps of acquiring an image of a 
circumferential feature of the solid of revolution using a camera having a 
focal length and disposed at a position characterized by a plurality of 
position parameters, and then finding a set of points along the image of 
the circumferential feature. The next step is determining an equation of 
an ellipse that substantially fits the set of points to provide a 
plurality of ellipse parameters, and then computing the longitudinal 
position and the radius of the circular cross-section of the solid of 
revolution using the focal length, the plurality of position parameters, 
and the plurality of ellipse parameters. In a preferred embodiment, the 
step of finding a set of points along the image of the circumferential 
feature includes the steps of creating a hypothetical ellipse at a 
location in the image, creating a plurality of edge detection windows, 
each edge detection window being located so as to include a portion of the 
hypothetical ellipse. Then, within each edge detection window, an edge is 
located so as to provide an edge position representing a point along the 
image of the circumferential feature. In another preferred embodiment, the 
step of determining an equation of an ellipse that substantially fits the 
set of points uses the step of minimizing a fitting error using a 
numerical optimization technique. In a further preferred embodiment, the 
numerical optimization technique is a technique that minimizes the square 
of the fitting error. In another preferred embodiment, the step of 
computing the longitudinal position and the radius of the circular 
cross-section of the solid of revolution uses only one of the major axis 
length and the minor axis length. In further preferred embodiments, it is 
advantageous to repeat the previous steps to provide a plurality of recent 
longitudinal position values and radius values, and then to average the 
plurality of recent longitudinal position values and radius values to 
provide an average recent longitudinal position value and the average 
recent radius value, periodically reporting the average recent 
longitudinal position value and the average recent radius value of the 
solid of revolution. In applications where there are no readily detectable 
circumferential features, it can be useful to first project structured 
light onto the surface of the solid of revolution so as to create the 
circumferential feature. 
In another general aspect of the invention, a machine vision method for 
estimating both a melt level and a radius of a circular cross-section of a 
growing semiconductor crystal is provided. The method includes the steps 
of acquiring an image of a circumferential feature of the growing 
semiconductor crystal using a camera having a focal length and disposed at 
a position characterized by a plurality of position parameters; finding a 
set of points along the image of the circumferential feature; determining 
an equation of an ellipse that substantially fits the set of points to 
provide a plurality of ellipse parameters; and computing the longitudinal 
position and the radius of the circular cross-section of the semiconductor 
crystal using the focal length, the plurality of position parameters, and 
the plurality of ellipse parameters. 
The invention is useful for visually monitoring the diameter of an item 
which is rotating about a fixed axis and which has a trackable contour 
known to lie in a plane normal to the rotation axis (for example, machined 
parts turned on a lathe). If a trackable contour were not directly 
available on the turning part, one or more contours could be created with 
structured light. 
The present invention can simultaneously estimate the cross-sectional 
diameter of a crystal, and the height of the melt surface, both in 
physical length units (e.g. cm). The meniscus at the crystal/melt 
interface is tracked by the system of the invention. Although the meniscus 
is essentially circular in the plane of the melt, the images of the 
meniscus to be processed are perspective projections of the meniscus. The 
present invention rigorously exploits this geometric perspective 
distortion. 
The present inventions solves the problems of the prior art described 
above, thereby making it possible to further automate crystal growing 
processes in a manner that increases manufacturing efficiency, 
consistency, and overall quality. In particular, the invention provides 
improved estimates of crystal diameter and melt level that lead to more 
effective control of the crystal growing process.

DETAILED DESCRIPTION OF THE FIGURES 
FIG. 2 illustrates a "solid of revolution" 40 with axis of revolution 42. 
As illustrated in FIG. 3, every plane perpendicular to the axis of 
rotation 52 of a slid of revolution 50 cuts the object's surface in a 
parallel circle, e.g. 54, 56, 58. The surface of an object of revolution 
is therefore covered by an infinite set of parallel circles, where each 
circle is associated with a distinct position along the axis of rotation. 
FIG. 3 also illustrates the parallel projection 60 of the solid of 
revolution 50 in a direction normal to the axis of revolution 52. The 
projections of the bright circles 54, 56, 58 are seen respectively as 
lines 62, 64, 66. The length of each of these lines 62, 64, 66 in the 
plane of projection equals the diameter of the solid of revolution at the 
corresponding position along the axis of revolution 52. Given the lengths 
of projection lines at all positions along the axis of revolution 52, we 
obtain the radius profile plot 70 in FIG. 4, where the radius profile r(z) 
70 plots the radius r 74 of the solid of revolution 40 as a function of 
position z 72 along the axis of revolution 42. 
Referring to FIG. 5A, the profile monitoring system 80 of the invention 
receives a video input sequence I.sub.XY 82 from a camera (not shown), and 
receives a set of fixed view-geometry parameters 84, including viewing 
angle .theta., focal length f of the lens of the camera, and the distance 
d from the focal point of the lens to the point where the camera axis z' 
intercepts the object's rotation axis z. The output of the system r(z) 86 
is a time-varying estimate of the radius r of the object as a function of 
position z along the object's axis or rotation. 
FIG. 5B is a specialization of the profile monitoring system 80 of FIG. 5A, 
referred to as the crystal monitoring system 88, which monitors the 
diameter of a growing crystal. The inputs 90 and fixed view-geometry 
parameters 92 are identical to those in the general profile monitoring 
system 80. However, the output consists of two measurements, the radius of 
the crystal R.sub.C 94 where it contacts the melt, and the height of the 
melt surface H.sub.M 96 relative to an arbitrary fixed origin. These 
outputs 94, 96, like the input sequence I.sub.XY 90, vary with time. 
The viewing geometry of the camera with respect to the solid of revolution 
40 is illustrated in FIG. 6. Three coordinate systems are shown including 
a world coordinate system (x,y,z) having three mutually orthogonal axes, a 
viewer-centered coordinate system (x',y',z') having three mutually 
orthogonal axes, and an image-plane coordinate system (x",y",z") having 
three mutually orthogonal axes. The z-axis 98 in the world coordinate 
system is assumed to be the axis of revolution (for a crystal, this is 
also the axis along which the crystal is pulled as it grows). Several 
constant parameters are also depicted in FIG. 6, including the viewing 
angle .theta., the focal length f of the lens 100, and the distance d. 
When d is added to f, the result is the distance from the lens 100 to the 
intersection between the line-of-sight and the z-axis. 
Referring to FIG. 10, we now describe the method to estimate the position 
and the radius r of a cross-section of a solid of revolution based on its 
elliptical projection in an image plane. This method relates the 
mathematical parameters of the ellipse to those of the circular 
cross-section of the solid of revolution, and shows how to estimate the 
latter from the former. The specific methods used to analyze an input 
image to estimate the parameters of an elliptical projection will be 
described below. 
Suppose we are given a circular cross-section 102 of the solid of 
revolution 40 with radius r 104 as shown FIG. 8A. The circular 
cross-section 102 is assumed to be at position z along the axis of 
revolution of the solid of revolution 40. Then, based on the camera 
geometry described with reference to FIG. 6 above, the image of the circle 
102 is obtained by centrally projecting the circle 102 into the image 
plane (x",y") of FIG. 8B to obtain an ellipse 106. 
In terms of the circle radius r 104 and position z and the fixed imaging 
parameters described with reference to FIG. 6, the major and minor 
semi-axes 108 and 110 (a and b, respectively), and the vertical location 
y0 of the ellipse in the image plane (x",y"), are given by the following 
set of equations: 
##EQU1## 
This system of equations can be inverted to obtain z and r in terms of a, 
b, y0, and the other fixed imaging parameters f, d, and .theta.. Multiple 
solutions exist, but the following solution is applicable to the present 
problem. 
##EQU2## 
The above solution (Eq. 2) is used in the radius profile estimation system 
of the invention to determine the radius r of each visible circular 
cross-section of the solid of revolution 40 visible in an image, and its 
associated position z along the axis of revolution. In the crystal 
monitoring system shown in FIG. 1, the radius Rc is interpreted as the 
radius r and the height H.sub.m of the crystal where it contacts the melt 
surface is interpreted as the position z along the axis of revolution. The 
parameters f, d, and .theta. are known constants based on the camera 
geometry as described with reference to FIG. 6. However, the parameters a, 
b, and y0 describe the elliptical projection of the circular cross-section 
of the solid of revolution in the image. These parameters must be 
estimated from images of the part under inspection, such as the bright 
meniscus 30. The method used by the invention to analyze images to 
estimate a, b, and y0 is described next. 
Referring to FIG. 9, the estimation of image ellipse parameters begins by 
generating a hypothetical ellipse estimate 117 in the acquired image of 
the meniscus 118 with a specific set of parameters (a, b, y0). In general, 
the image can be of any circumferential feature of a solid of revolution. 
If no circumferential feature is readily evident on a solid of revolution, 
structured light can be projected onto the solid of revolution to simulate 
a circumferential feature. 
An ellipse gauging tool, described below, is then applied to the image at 
the hypothesized position y0 and with the hypothesized major and minor 
axes a and b, respectively. The ellipse gauging tool consists of a set of 
edge tools 116 for finding an edge within a designated search window, such 
as the Edge Tool sold by Cognex Corporation. The edge tools 116 are 
arranged at intervals around the circumference of the hypothesized ellipse 
117. The search direction of each edge tool 116 is normal to the local 
tangent to the hypothesized ellipse 117. 
The arrangement of edge tools 116 is illustrated in FIG. 9 in relation to 
the elliptical projection of the circumferential feature, such as the 
meniscus of a growing crystal. As shown in the FIG. 9, the meniscus 118 
generally appears as a bright elliptical band in the image. According to 
standard practice, each edge tool 116 computes the gradient of the image 
intensity profile along its search direction, and then finds the peak in 
the resulting gradient profile that satisfies all user-configured 
parameters such as edge direction. If this peak exceeds a configurable 
threshold, then the edge tool is judged to have detected an edge and the 
position of the edge in the 2D image is recorded. Based on the position 
results from all edge tools 116, the ellipse gauging tool obtains a set of 
points on the circumference of the visible ellipse 118 in the image. 
Because each edge tool 116 seeks the strongest edge along its full length, 
a set of points on the image ellipse 118 is obtained even when the 
hypothesized ellipse 117 has a slightly different position and size than 
the ellipse 118 actually present in the image. 
In step (130) of FIG. 10, if three or more of the edge tools 116 detect 
edges in the image, then the ellipse gauging tool uses these points to 
estimate a best-fitting ellipse in the mean-squared-error sense. In 
particular, the ellipse fitting procedure finds the parameters (a, b, y0) 
that minimize the mean squared error between the elliptical fit and the 
set of edge points. Once the parameters of the best elliptical fit to the 
set of detected edge points has been derived, this completes the ellipse 
gauging operation for the current image. The newly estimated ellipse 
parameters are then used to set the positions and orientations of the 
tracking edge tools for the next acquired image as described above. 
Therefore, if the elliptical projection of a circular cross-section of a 
solid of revolution moves, grows, or changes aspect (defined as the ratio 
a/b) over time, then the positions and orientations of edge tools 116 will 
track these changes, enabling the ellipse gauging tool to successfully 
estimate the parameters (a, b, y0) of the elliptical projection from each 
successively acquired image. 
If fewer than three of the edge tools 116 detect edges in the image, then 
the ellipse gauging tool is unable to fit an ellipse 117. It is therefore 
necessary to hypothesize a new set of ellipse parameters and then apply 
the ellipse gauging tool using the new ellipse position and size. This 
hypothesis-test procedure continues until the ellipse gauging tool is 
successful. 
Various methods for generating hypothetical ellipses can be employed. The 
crystal monitoring system of FIG. 5B can advantageously use blob analysis 
to detect bright regions in the image and to determine the center of 
gravity, width, and height of each such region. Ellipse hypotheses that 
match these parameters are first tested with the ellipse gauging tool. If 
none of these hypotheses are verified, then hypotheses are generated 
randomly over a bounded three-dimensional parameter space (a,b,y0) that 
reflects the range of positions and sizes of elliptical projections that 
circular cross-section of solids of revolution are expected to produce 
based on the part-type under inspection. Once the elliptical projection of 
a circular cross-section of a solid of revolution is found in the image, 
the ellipse gauging tool proceeds to track changes in the ellipse 
parameters over time as described in the previous paragraph. 
The overall operation of the invention is summarized in FIG. 10. First, an 
image is acquired of a portion of the circular cross-section of a solid of 
revolution (126). Then, in step (128), the image is analyzed to find a set 
of points on the boundary of the elliptical projection of the circular 
cross-section assumed to be visible on the surface of a part that has the 
form of a solid of revolution. In the next step (130), these points are 
then used to estimate the parameters a, b, and y0 of the ellipse that 
minimizes the fitting error in the minimum means squared error sense. 
Then, in step (132), these parameters are then used as inputs to Equation 
(2) above to derive the physical radius of the circular cross-section of 
the solid of revolution, as well as its position along the central axis of 
the solid of revolution. Repeating such measurements at many locations 
along the full length of the solid of revolution, one generally obtains a 
radius profile as in the case of the general radial profile inspection 
system, or a time-varying radius and melt level of a semiconductor 
crystal, as in the specialized case of a crystal growth inspection system. 
FIG. 7 shows the embodiment of the invention that is specialized to monitor 
a growing semiconductor crystal as it is pulled from a melt. First, an 
image of the meniscus at the base of the growing crystal is acquired 
(112). Then, a set of points is found along the image of the meniscus 
using a plurality of edge tools 116, as described above. Next, a numerical 
optimization procedure, such as a least squares fit of the equation of an 
ellipse to the set of points is performed (120), thereby outputting a set 
of ellipse parameters a,b, y0. Then, the instantaneous radius of the 
crystal at the meniscus, and the melt level are computed using Eq. 2 
above. Optionally, temporal smoothing can be performed over a sequence of 
such acquired images. 
In general, it can be useful to temporally smooth the estimates obtained 
from multiple image acquisitions to reduce the effects of measurement 
noise. For example, with regard to crystal growth, recall that the crystal 
is generally suspended by a cable. Therefore, it can wobble and/or 
oscillate slightly in a pendulum-like fashion. Also, the edge data that 
the elliptical fits are based on contain some noise. Therefore, the 
preferred implementation of the system of the invention averages the 
instantaneous crystal radius and melt level estimates over a configurable 
number of recently acquired images before reporting the smoothed melt 
level and crystal radius estimate. In the crystal inspection system, this 
averaging operation takes the form of a standard moving average filter. In 
particular, the current reported value Vi, for the ith acquired image is 
obtained as a weighted sum of the measurement Mi from the current image 
and the previous reported estimate Vi-1 as follows: 
EQU Vi=(1-wt)*vi-1+wt*Mi, 
where wt is a constant between 0 and 1 which is chosen to produce the 
desired amount of temporal smoothing. 
The radius profile monitoring system of the invention is widely applicable 
to situations that involve non-contact dimensional measurements of solids 
of revolution. Because the system of the invention takes into account the 
geometry of central projection, the invention allows great freedom in 
camera placement, viewing angle, and selection of lenses. 
The procedure used for fitting an ellipse to a set of points involves first 
solving a system of four linear equations whose coefficients are computed 
from the set of point coordinates. The solution of the linear system is 
then used in a set of nonlinear equations to derive the parameters of the 
ellipse including the center point (x0, y0) and the major and minor axis 
lengths a, and b. The ellipse fitting problem and its minimum mean square 
error solution will now be described in detail below. 
The equation of an ellipse is: 
##EQU3## 
or equivalently, 
EQU x 2+c1*x+c2*y 2+c3*y+c4=0. 
where 
EQU c1=-2*x0 
EQU c2=a 2/b 2 
EQU c3=2*(a 2/b 2)*y0 
EQU c4=x0 2+(a 2/b 2)*y0 2-a 2. 
Given n points (xi, yi), we define the squared error (q) as: 
##EQU4## 
The minimum of q is obtained by solving for constants c1, c2, c3, c4 that 
satisfy the normal equations: 
##EQU5## 
where d(q)/d(ci) denotes partial differentiation of q w.r.t. ci. 
The above set of simultaneous equations in four unknowns is written in 
matrix form as: 
##EQU6## 
After solving for c1, c2, c3, c4, we next use these results to solve for 
the ellipse parameters using the following relations: 
EQU x0=-c1 /2 
EQU y0=-c3 /(2*c2) 
EQU a=sqrt(-c4 +c1 2/4+c3 2/(4*c2)) 
EQU b=sqrt((-c4+c1 2/4+c3 2/(4*c2))/c2) 
Other modifications and implementations will occur to those skilled in the 
art without departing from the spirit and the scope of the invention as 
claimed. Accordingly, the above description is not intended to limit the 
invention except as indicated in the following claims. 
##EQU7##