![](bg5.png)
ified results. However, the diameter or the computed
enclosures may be so wide as to be practically use-
less. We have applied to our algorithm the principle
of iterative refinement. After computing a diameter
of the error interval is less than a desired accuracy,
then a verified enclosure of the solution is given by
the approximation and the enclosure of its error.
In order to observe the performance of our algo-
rithm, we used a set of polynomials with more than
twenty different polynomials. However, we show
only five interesting polynomials in this paper due to
space limitations (see Table 1).
Table 1: Polinomial surfaces.
Surface Coefficients of polynomial
Whitney x
2
· z + y
2
Plucker x
2
· z − x · y + y
2
· z
Bicube x
4
+ y
4
+ z
4
− 1000
Mitchell 4 · (x
4
+ (y
2
+ z
2
)
2
) + 17 · x
2
· (y
2
+ z
2
)−
20 · (x
2
+ y
2
+ z
2
) + 17
Steiner x
2
· y
2
+ x
2
· z
2
+ x · y · z + y
2
· z
2
Figure 2 shows an evolution of rendered Bicube
surface in complex space for σ = 0 in every images
and δ values from 0 (real surface in real space) to
π
9
.
This sequence of images shows how complex roots
cover the real object like packing paper. However,
if the object is a Mitchell surface, we can see that
complex roots begin covering over object. Although,
the number of complex roots around horizontal-axis
increase quicker than those around vertical-axis (see
Figure 3).
The three following sequences of images for
Steiner, Whitney and Plucker surfaces are very inter-
esting. The complex roots of these surfaces not cover
over object, but they are around imaginary axes of
surfaces. For example, the Steiner surface shows the
three axes which appear in complex space (see Fig-
ure 4) and Whitney surface has complex roots only
around one imaginary axis (see Figure 5), like Plucker
surface (see Figure 6). It is important to show that the
number of complex root increase quicker for Steiner
surface than Whitney surface.
Finally, we show a scene with a Whitney surface in-
side a translucent sphere (see Figure 7). In this case,
we have chosen to render Whitney surface like a com-
plex object and sphere like a real object. This let us
see the evolution of complex object inside a real ob-
ject for several δ values, with refraction and reflection
effects.
5 CONCLUSIONS
We have designed and implemented a complex root
finder algorithm to render polynomial surfaces in
complex space. For this problem, it is not possi-
ble to use the Sturm sequences of a polynomial as a
root finder algorithm, because we try to find complex
roots. So we solve this problem as a eigenvalue prob-
lem, where we have used the polynomial root find-
ing algorithm proposed by Hammer with some addi-
tional extras. On the one hand, we have solved how to
find close zeros of zeros with higher accuracy. On the
other hand, we have extended this algorithm to find
all complex roots in intersection point.
This algorithm also allow us to render algebraic
surfaces defined as complex polynomial, where some
of coefficients of polynomial are complex numbers.
An additional possibility it is to redefine every rays
with complex origin points and complex direction
vectors.
Finally, we propose a new procedure to render im-
age with traditional ray tracing technique in complex
space. This technique allows to build a sequences of
images where we can analyse the evolution of com-
plex root of several polynomial surfaces in a three-
dimensional space. These images can use reflection,
refraction and translucent effects like a realistic im-
age.
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