Figure 1: full ellipsoid outlines
Figure 2: hidden outlines computed using
simplistic version of Ferrari's algorithm
Figure 3: hidden outlines computed using
the combined algorithm described in this article
Don Herbison-Evans
don@it.uts.edu.au
Technical Report TR94-487
Basser Department of Computer Science
University of Sydney
Australia
(updated 27 December 2001)
A number of problems in computer graphics reduce to finding approximate real roots of quartic and cubic equations in one unknown. Various solution techniques are discussed briefly. The algorithms for analytic solution are discussed at length.
Methods are presented for controlling round-off error and overflow in the analytic solution of such equations. The solution of a quartic requires the solution of a subsidiary cubic equation. The cubics derived in the algorithms by Descartes, by Neumark and by Ferrari are examined for their stability.
An operation count of the resulting algorithms is presented, together with a C program performing a wide ranging comparison of their rounding-error behaviours.
This article addresses the problems of solving quartic and cubic equations in computer graphics. These equations are interesting as they can be solved by analytic algorithms, and in principle need no iterative techniques.
Quartic equations need to be solved when ray tracing 4th degree surfaces e.g., a torus. Quartics also need to be solved in a number of problems involving quadric surfaces. Quadric surfaces (i.e. ellipsoids, paraboloids, hyperboloids, cones) are useful in computer graphics for generating objects with curved surfaces (Badler, 1979). Fewer primitives are required than with planar surfaces to approximate a curved surface to a given accuracy (Herbison-Evans, 1982).
Bicubic surfaces may also be used for the composition of curved objects. They have the advantage of being able to incorporate recurves: lines of inflection. There is a problem, however, when drawing the outlines of bicubics in the calculation of hidden arcs. The visibility of an outline can change where its projection intersects that of another outline. The intersection can be found as the simultaneous solution of the two projected outlines. For bicubic surfaces, these outlines are cubics, and the simultaneous solution of two of these is a sextic which can only be solved by iterative techniques. For quadric surfaces, the projected outlines are quadratic. The simultaneous solution of two of these leads to a quartic equation.
The need to solve cubic equations in computer graphics arises in the solution of the quartic equations mentioned above. Also, a number of problems which involve the use of cubic splines require the solution of cubic equations.
One simplifying feature of the computer graphics problem is that often, only the real roots (if there are any) are required. The full solution of the quartic in the complex domain (Nonweiler, 1967) is then an unnecessary use of computing resources.
Another simplification in the graphics problem is that displays have only a limited resolution, so that only a limited number of accurate digits in the solution of a cubic or quartic may be required. A resolution of 1 in 1,000,000 should in principle be achievable using single precision floating point (32 bit) arithmetic, which would be more than adequate for most current displays.
The roots of quartic and cubic equations can be obtained by iterative techniques. These techniques can be useful in animation where scenes change little from one frame to the next. Then the roots for the equations in one frame are good starting points for the solution of the equations in the next frame. There are two problems with this approach.
One is storage. For a scene composed of "n" quadric surfaces, storage may be required for 4n(n - 1) roots between frames. A compromise is to store pointers to those pairs of quadrics which give no roots. This trivial idea can be used to halve the number of computations within a given frame, for if quadric `a' has no intersection with quadric `b', then `b' will not intersect `a'.
The other problem is more serious: it is the problem of deciding when the number of roots changes. There appears to be no simple way to find the number of roots of a cubic or quartic. The most well-known algorithm for finding the number of real roots, the Sturm sequence, involves approximately as much computation as solving the equations directly by radicals (Ralston, 1965). Without information about the number of roots, iteration where a root has disappeared can waste a lot of computer time, and searching for new roots that may have appeared becomes difficult.
Even when a root has been found, deflation of the polynomial to the next lower degree is prone to severe round-off exaggeration (Conte and de Boor, 1980).
Thus there may be an advantage in examining the techniques available for obtaining the real roots of quartics and cubics analytically.
Quartics are the highest degree polynomials which can be solved analytically in general by the method of radicals i.e.: operating on the coefficients with a sequence of operators from the set: sum, difference, product, quotient, and the extraction of an integral order root. An algorithm for doing this was first published by Cardano in the 16th century (Cardano, 1545). A number of other algorithms have subsequently been published. The question arises: which algorithm is for best to use on a computer to finding the real roots in terms of speed and stability for computer graphics.
Very little attention appears to have been given to a comparison of the algorithms. They have differing properties with regard to overflow and the exaggeration of round-off errors. Where a picture results from the computation, any errors may be rather obvious. Figures 1, 2, and 3 show a computer bug composed of ellipsoids with full outlines, incorrect hidden outlines, and correct hidden outlines, respectively. In computer animation, the flashing of incorrectly calculated hidden arcs is most disturbing.
Figure 2: hidden outlines computed using
simplistic version of Ferrari's algorithm
Figure 3: hidden outlines computed using
the combined algorithm described in this article
Many algorithms use the idea of first solving a particular cubic equation, the coefficients of which are derived from those of the quartic. A root of the cubic is then used to factorize the quartic into quadratics, which may then be solved. The algorithms may then be classified according to the way the coefficients of the quartic are combined to form the coefficients of the subsidiary cubic equation. For a general quartic equation of the form:
x4 + ax3 + bx2 + cx + d = 0
the subsidiary cubic can be one of the forms:
Ferrari-Lagrange (Turnbull, 1947)
y3 + by2 + (ac - 4d)y + (a2d + c2 - 4bd) = 0
Descartes-Euler-Cardano (Strong, 1859)
y3 + (2b - 3a2/4)y2 + (3a4/16 - a2b + ac + b2 - 4d)y + (abc - a6/64 + a4b/8 - a3c/4 - a2b2/4 - c2) = 0
Neumark (Neumark, 1965)
y3 - 2by2 + (ac + b2 - 4d)y + (a2d - abc + c2) = 0
The casual user of the literature may be confused by variations in the presentation of quartic and cubic equations. Sometimes, the highest degree term has a non-unit coefficient. Sometimes the coefficients are labelled from the lowest degree term to the highest. Sometimes numerical factors of 3, 4 and 6 are included. There are also a number of trivial changes to the cubic caused by the following:
if
y3 + py2 + qy + r = 0
then
z3 - pz2 + qz - r = 0 for z = - y
and
z3 + 2pz2 + 4qz + 8r = 0 for z = 2y
Tables 1,2 and 3 shows the stable combinations of signs of the quartic coefficients for the computation of the coefficients of these subsidiary cubics.
Of the three subsidiary cubics, that from Ferrari's algorithm has two stable combinations of signs of a, b, c and d for the derivation of all of the coefficients of the cubic, p, q, and r. For this reason, attempts were made initially to use Ferrari's method for finding quadric outline intersections (Herbison-Evans, 1983).
The coefficients of the subsequent quadratics depend on two intermediate quantities, e and f, where
e2 = a2/4 - b - y
f2 = y2/4 - d
ef = (ay/4 + c/2)
The signs of each of the quartic coefficients a, b, c, d and y, the cubic root, may be positive or negative, giving 32 possible combinations of signs. Of these, only 12 can be clearly solved in a stable fashion for e and f by the choice of 2 out of the 3 equations involving them. In the remaining 20 cases, the most stable choices are unclear. This is shown in tables 4 and 5. Of the 12 stable cases, sadly only 2 are from the stable cases for the calculation of p, q and r. In the other 10 cases, the value of y may be unreliable.
Using Ferrari's method, the quadratic equations are :
x2 + Gx + H = 0
x2 + gx + h = 0
where
G = a/2 + e
g = a/2 - e
H = -y/2 + f
h = -y/2 -f
If a and e are the same sign, and b and y are the same sign, then g may be more accurately computed using:
g = (b + y)/G
If a and e are opposite signs, G can be more accurately computed from g in a similar fashion.
If y and f are the same sign, then H may be more accurately computed using:
H = d /h
If y and f are opposite in sign, then h can be computed similarly from H more accurately.
The solution of the quadratic equations requires the evaluation of the discriminants:
g2-4h and G2-4H
Unless h and H are negative, one or both of these evaluations will be unstable. Unfortunately, positive h and H values are incompatible with the 2 stable cases for the evaluation of p, q, r, e and f, so there is no combination of coefficients for which Ferrari's algorithm can be made entirely stable.
It might appear that the problem can be alleviated by observing that reversing the signs of a and c simply reverses the signs of the roots, but may alter the stability of the intermediate quantities. However, all the algorithms appear to have identical stabilities under this transformation.
This algorithm also has one combination of quartic coefficients for which the evaluation of the subsidiary cubic coefficients is stable. However, the calculation of these coefficients involves significantly more operations than Ferrari's or Neumark's algorithms. Also, the high power of 'a' in the coefficients makes this algorithm prone to loss of precision and also overflow.
In this algorithm, if the greatest root of the cubic, y, is negative, the quartic has no real roots. Otherwise, the coefficients of the quadratics involve the quantities m, n1 and n2:
x2 + mx + n1 = 0 and
x2 - mx + n2 = 0
where
m = y(1/2)
n1 = (y + A + B/m)
n2 = (y + A - B/m)
and
A = b - 3a2d/8
B = c + a3/8 - ab/2
There appears to be no way of making the evaluation of A, B, n1 and n2 stable. Some quantities are bound to be subtracted, leading to possible loss of precision.
Attempts were also made to stabilise the algorithm of Neumark. In this, the coefficients of the quadratic equations are parameters g, G, h and H where:
G = (a + (a2 - 4y)(1/2))/2
g = (a - (a2 - 4y)(1/2))/2
H = (b-y)/2 + (a(b-y) - 2c)/(2(a2 - 4y)(1/2))
h = (b-y)/2 - (a(b-y) - 2c)/(2(a2 - 4y)(1/2))
Any cancellation due to the additions and subtractions can be eliminated by writing:
G = g1 + g2
g = g1 - g2
H = h1 + h2
h = h1 - h2
where
g1 = a/2
g2 = ((a2 - 4y)(1/2))/2
h1 = (b - y)/2
h2 = (a(b-y)/2 - c)/(a2 - 4y)(1/2)
and using the identities
G = y
H = d
h = -y/2 -f
Thus if g1 and g2 are the same sign, G will be accurate but g will lose significant digits by cancellation. Then the value of g can be better obtained using:
g = y/G
If g1 and g2 are of opposite signs, then g will be accurate, and G better obtained using:
G = y/g
Similarly, h and H can be obtained without cancellation from h1, h2 and d.
The computation of g2 and h2 can be made more stable under some circumstances using the alternative formulation:
h2 = (((b - y)2 - 4d)(1/2))/2
Furthermore
g2 = (a - c)/((b - y)2 - 4d)(1/2)
Thus g2 and h2 can both be computed either using
m = (b - y)2 - 4d
or using
n = a2 - 4y
If y is negative, n should be used. If y is positive and b and d are negative, m should be used. Thus 7 of the 32 sign combinations give stable results with this algorithm. These are shown in tables 6 and 7. For other cases, a rough guide to which expression to use can be found by assessing the errors of each of these expressions by summing the moduli of the addends:
e(m) = b2 + 2|by| + y2 + 4|d|
e(n) = a2 + 4|y|
Thus, if
|m|.e(n) > |n|.e(m)
then m should be used, otherwise n is more accurate.
following Tartaglia, let the cubic equation be
y3 + py2 + qy + r = 0
The solution has been published elsewhere on the web but here is be expressed (Littlewood, 1950) using:
u = q - p2/3
v = r - pq/3 + 2p3/27
and the discriminant:
j = 4(u/3)3 + v2
If this is positive then there is one root, y, to the cubic, which may be found using:
y = ((w - v)/2)(1/3) - (u/3)(2/(w - v))(1/3) - p/3
where
w = j(1/2)
As w is chosen here to be positive, this formulation is suitable if v is negative. However, the calculation in this form can lose accuracy if v is positive. This problem can be overcome by the rationalization:
(w - v)/2 = (w2 - v2)/(2(w + v)) = 2(u/3)3/(w + v)
giving the alternative formulation of the root:
y = (u/3)(2/(w + v))(1/3) - ((w + v)/2)(1/3) - p/3
A computational problem with this algorithm is overflow while calculating w, for
O(j) = O(p6) + O(q3) + O(r2)
If the cubic is the subsidiary cubic of a quartic, the different algorithms have differing overflow behaviours:
Ferrari: O(j) = O(a4d2) + O(a3c3) + O(b6) + O(c4) + O(d3)
Descartes: O(j) = O(a12) + O(b6) + O(c4) + O(d3)
Neumark: O(j) = O(a4) + O(b6) + O(c4) + O(d3)
Before evaluating the terms of w, it is useful to test p, q and r against the appropriate root of the maximum number represented on the machine ('M'). The values of u and v should similarly be tested. In the event that some value is too large, various approximations can be employed: e.g.
if |p| > 27M(1/3), then y ~= -p
if |v| > M(1/2), then y ~= v(1/3)
if |u| > 3M(1/3)/4, then y ~= 4(1/3)u/3
If the discriminant j is negative, then there are 3 real roots to the cubic. These real roots of the cubic may then be obtained via parameters s, t and k:
s = (-u/3)(1/2)
t = -v/(2s3)
k = arccos(t)/3
giving
y1 = 2s.cos(k) - p/3
y2 = s(-cos(k) + 3(1/2)sin(k)) - p/3
y3 = s(-cos(k) - 3(1/2)sin(k)) - p/3
Note that if the discriminant is negative, then u must also be negative, guaranteeing a real value for s. This value may be taken as positive without loss of generality. Also, k will lie in the range 0 to 60 degrees, so that cos(k) and sin(k) are both positive. Thus we have:
y1 >= y2 >= y3.
If the cubic is a subsidiary of a quartic, either y1 or y3 may be the most useful root. Unfortunately, b = 2p in Neumark's algorithm, so although y1 may be the largest root, it may not be positive. Then if b and d are both negative, it would be advantageous to use the most negative root: y3.
The functions sine and cosine of arccos(t)/3 may be tabulated to speed the calculation (Herbison-Evans, 1983). Sufficient accuracy (1 in 107) can be obtained with a table of 200 entries with linear interpolation, requiring 4 multiplications, 8 additions and 2 tests for each function. When t is near its extremes, the asymptotic forms may be useful:
for t -> 0 :
sin(arccos(t)/3) ~= (1 - t/3(1/2))/2 + O(t2)
cos(arccos(t)/3) ~= 3(1/2))/2 + t/6 + O(t2)
for t -> 1 :
sin(arccos(t)/3) ~= (2(1 - t))(1/2)/3 + O((1-t)2)
cos(arccos(t)/3) ~= 1 - (1-t)/9 + O((1-t)2)
If the discriminant, j, is expanded in terms of the coefficients of the cubic, it has 10 terms. Two pairs of terms cancel and another pair coalesce, leaving 5 independent terms. In principle, any pair of subsets of these may cancel catastrophically, leaving an incorrect value or even an incorrect sign for the discriminant. This problem can be alleviated by calculating the 5 terms separately, and then combining them in increasing order of magnitude (Wilkinson, 1963). When solving quartics, the discriminant should be expanded in terms of the quartic coefficients directly. This gives fifteen terms, which can be sorted by modulus, and combined in increasing order.
There have been many algorithms proposed for solving quartic and cubic equations, but most have been proposed with aims of generality or simplicity rather than error minimisation or overflow avoidance. The work described here gives a low rate of error using single precision floating point arithmetic for the computer animation of quadric surfaces.
The operation counts of the best combination of stabilized algorithms are summarized in table 8.
A further comment may be useful here concerning the language used to implement these algorithm. Compilers for the language C often perform operations on single precision variables ('float') in double precision, converting back to single precision for storage. Thus there might be little speed advantage in using 'float' variables compared with using 'double' for these algorithms. Fortran compilers may not do this. For example, using a VAX8600, the time taken to solve 10,000 different quartics was 6 seconds, for Fortran single precision (using f77), 15 seconds for C single precision (using cc), and 16 seconds for C using double precision.
A check on the accuracy of the roots can be done at the cost of more computation. Each root may be substituted back into the original equation and the residual calculated. This can then be substituted into the derivative to give an estimate of the error of the root or used for a Reguli-Falsi or better still a Newton-Raphson correction.
A program was written to perform a comparison of the stabilities of the 3 algorithms for the solution of quartic equations. Quartics were examined which had all combinations and permutations of coefficients from the set:
108, 104, 1, 10(-4), 10(-8), -108, -104, -1, -10-4, -10-8
Of the 10,000 equations, the 3 algorithms agreed on the number of real roots in 8,453 cases. Of these, 1,408 had zero real roots. Of the remaining 7,045 equations, Ferrari's algorithm had the least worst error in 1,659 cases, Neumark's 2,918, Descartes' 88, and in the other 2,380 cases, 2 or more algorithms had equal worst errors.
It may be observed that 2 of the 7 stable cases for Neumark's algorithm coincide with 2 of the 12 stable cases for Ferrari's algorithm, making only 17 stable cases in all out of the 32 possible sign combinations. Further work on this topic may be able to increase this number.
Thanks are due to the Departments of Computer Science at the Universities of Sydney (Australia) and Waterloo (Canada) where much of this work was done. Initial investigations on which the work was based were made by Charles Prineas. Thanks are also due to the late Alan Tritter for discussions, Zoe Kaszas for the initial preparation of this paper, Alan Paeth for formatting this paper for publication, and to Professor John Bennett for his continual encouragement.
Badler, N.I. & Smoliar, S.W., Digital Representations of Human Movement, ACM Computing Surveys, Vol. 11, No. 1, pp. 24-27 (1979)
Cardano, G., Ars Magna, Nurmberg (1545)
Conte, S.D. & de Boor, C., Elementary Numerical Analysis, McGraw-Hill, p. 117 (1980)
Herbison-Evans, D., Real Time Animation of Human Figure Drawings with Hidden Lines Omitted, IE.E. Computer Graphics and Applications, Vol. 2, No. 9, pp. 27-33 (1982)
Herbison-Evans, D., Caterpillars and the Inaccurate Solution of Cubic and Quartic Equations, Australian Computer Science Communications, Vol. 5, No. 1, pp. 80-9-(1983)
Littlewood, D.E., A University Algebra, Heineman, London, p. 173 (1950)
Nonweiler, T.R.F., Roots of Low Order Polynomial Equations, Collected Algorithms of the ACM (C2), Algorithm 326 (1967)
Neumark, S., Solution of Cubic and Quartic Equations, Pergamon Press, Oxford (1965)
Ralston, A., A First Course in Numerical Analysis, McGraw-Hill, p. 351 (1965)
Strong, T., Elementary and Higher Algebra, Pratt and Oakley, p. 469 (1859)
Turnbull, H.W., Theory of Equations, Oliver and Boyd, London, Fourth Edition, p. 130 (1947)
Wilkinson J.J., Rounding Errors in Algebraic Processes, Prentice-Hall, p. 17 (1963)
___________________________________________ | | a+ | a- | | |________________________________ | | b+ | b- | b+ | b- | ___________________________________________ | | d+ | p | p r | p q | p q r | | c+ |_____________________________________ | | d- | p q | p q | p | p | ___________________________________________ | | d+ | p q | p q r | p | p r | | c- |_____________________________________ | | d- | p | p | p q | p q | ___________________________________________ Table 1 For Ferrari' algorithm, the coefficients of the subsidiary cubic that can be stably computed from the coefficients of the quartic.
___________________________________________ | | a+ | a- | | |________________________________ | | b+ | b- | b+ | b- | ___________________________________________ | | d+ | | p r | | p | | c+ |_____________________________________ | | d- | | p q r | | p | ___________________________________________ | | d+ | | p | | p | | c- |_____________________________________ | | d- | | p | | p q | ___________________________________________ Table 2 For Descartes' algorithm, the coefficients of the subsidiary cubic that can be stably computed from the coefficients of the quartic.
___________________________________________ | | a+ | a- | | |________________________________ | | b+ | b- | b+ | b- | ___________________________________________ | | d+ | p | p r | p r | p | | c+ |_____________________________________ | | d- | p q | p q | p | p | ___________________________________________ | | d+ | p r | p | p | p r | | c- |_____________________________________ | | d- | p | p | p q | p q | ___________________________________________ Table 3 For Neumark's algorithm, the coefficients of the subsidiary cubic that can be stably computed from the coefficients of the quartic.
_________________________________________________________________
| | a+ | a- |
| y+ |______________________________________________________
| | b+ | b- | b+ | b- |
_________________________________________________________________
| | | | | | |
| | d+ | ef | ef | | |
| c+ |___________________________________________________________
| | | 2 | 2 | 2 | 2 |
| | d- | ef f | ef f | f | f |
_________________________________________________________________
| | | | | | |
| | d+ | | | ef | ef |
| c- |___________________________________________________________
| | | 2 | 2 | 2 | 2 |
| | d- | f | f | ef f | ef f |
_________________________________________________________________
Table 4
For Ferrari's algorithm, the intermediate
quantities that can be stably computed from
the coefficients of the quartic and a
positive root of the cubic.
_________________________________________________________________
| | a+ | a- |
| y- |______________________________________________________
| | b+ | b- | b+ | b- |
|________________________________________________________________
| | | | 2 | | 2 |
| | d+ | | e | ef | ef e |
| c+ |___________________________________________________________
| | | 2 | 2 2 | 2 | 2 2 |
| | d- | f | e f | ef f | ef e f |
|________________________________________________________________
| | | | 2 | | 2 |
| | d+ | ef | ef e | | e |
| c- |___________________________________________________________
| | | 2 | 2 2 | 2 | 2 2 |
| | d- | ef f | ef e f | f | e f |
|________________________________________________________________
Table 5
For Ferrari's algorithm, the intermediate
quantities that can be stably computed from
the coefficients of the quartic and a
negative root of the cubic.
____________________________________________________ | | a+ | a- | | y+ |_________________________________________ | | b+ | b- | b+ | b- | ____________________________________________________ | | d+ | g1 | g1 h1 | g1 | g1 h1 | | c+ |______________________________________________ | | d- | g1 | g1 g2 h1 h2 | g1 | g1 h1 h2 | ____________________________________________________ | | d+ | g1 | g1 h1 | g1 | g1 h1 | | c- |______________________________________________ | | d- | g1 | g1 h1 h2 | g1 | g1 h1 h2 | ____________________________________________________ Table 6 For Neumark's algorithm, the intermediate quantities that can be stably computed from the coefficients of the quartic and a positive root of the cubic.
_______________________________________________________ | | a+ | a- | | y- |____________________________________________ | | b+ | b- | b+ | b- | _______________________________________________________ | | d+ | g1 g2 h1 | g1 g2 | g1 g2 h1 h2 | g1 g2 | | c+ |_________________________________________________ | | d- | g1 g2 h1 h2 | g1 g2 | g1 g2 h1 h2 | g1 g2 | _______________________________________________________ | | d+ | g1 g2 h1 h2 | g1 g2 | g1 g2 h1 | g1 g2 | | c- |_________________________________________________ | | d- | g1 g2 h1 h2 | g1 g2 | g1 g2 h1 h2 | g1 g2 | ________________________________________________________ Table 7 For Neumark's algorithm, the intermediate quantities that can be stably computed from the coefficients of the quartic and a negative root of the cubic.
___________________________________________________________________
| | additions | multiplications | functions | |
| | and | and | e.g. | tests |
| | subtractions | divisions | root,sine | |
|________________________________________________________________
| | best 11 | best 7 | | best 14 |
| cubic | | | all 2 | |
| | worst 12 | worst 11 | | worst 19 |
|__________________________________________________________________
| rest | best 12 | best 20 | best 1 | best 27 |
| of | | | | |
| quartic | worst 16 | worst 34 | worst 2 | worst 39 |
|__________________________________________________________________
|quadratic| 3 | 5 | 2 | 3 |
| ( x2 ) | | | | |
===================================================================
| | best 29 | best 37 | best 7 | best 37 |
| totals | | | | |
| | worst 34 | worst 55 | worst 8 | worst 64 |
===================================================================
Table 8
Operation counts for a best combination
of stabilized algorithms.