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Drawing curves with the Dynamic System Geogebra: the Newton’s method

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The whole paper must be written with Times New Roman 12, with the exception of the title
Drawing curves with the Dynamic System Geogebra: the Newton’s method
Francisco Regis Vieira Alves, Instituto Federal de Educação, Ciência e Tecnologia do Estado do Ceará – IFCE. Brazil, fregis@ifce.edu.br
ABSTRACT: In this paper, we discuss some applications related to a systematic standard method in Algebraic Geometry. In fact, we bring a comparative discussion between an old preoccupation related to describe a shape of a algebraic curve. Indeed, we find some authors that indicate and provide several static figures en virtue to guide the visual understanding. Actually, we can explore the potentialities of the Dynamic System Geogebra – DSG and compare, from a qualitative view, the constructions provided by this software. KEYWORDS: Visualization, DSG, Newton´s Methdods, Algebraic Curves. .
1 Introduction
In a general context, the graph behavior of certain classes of the mathematical objects is extremely relevant en virtue to acquire an intuitive understanding about its nature. In fact, when we dispose graphical representations of a object, we have the possibility to develop our discursive function related to a specific conceptual entity. In fact, when we visualize an object in 2D, we can realize a designation of a particular conceptual object. We can elaborate some statement and propositional sentences about the object. Moreover, we can act in a problem situation en virtue to develop some sequences of logical statement in a coherent way. Finally, as a human been, we can reflect about theses statement and its pragmatic values.
The whole paper must be written with Times New Roman 12, with the exception of the title
For example, what would we can declare about the following expression like
2233523323
0
XXYYYXYXXYXYXY
+ + + + + + + + =
? (*) Moreover, can a particular procedure produce a graphical representation related to this complex planar equation? In a particular way, we have indicated in the figure 1, some steps related to the Newton’s diagram. Its description can be finding in Vainsencher (2009, p. 40-41). In a general manner, this procedure provides some criterion to describe, near at the srcin, a trace of algebraic curve in the plane. However, when we lead of a complex case, like in the figure 1, become impossible to obtain entire information about the shape of the algebraic curve (*). In fact, below we determine some segments (in the left side) from the term
ijij
aXY
effectively present in the algebraic expression like (,)0
fXY
=
.
Figure 1. Visualization of a algebraic plane curve with DSG
Well, in the next section, with the DSG´s help, we will bring some examples of the useful use en virtue the perception and the visualization. Moreover, this indispensable cognitive ability is a powerful tool en virtue the teaching and learning at the academic level (ALVES, 2014)
The whole paper must be written with Times New Roman 12, with the exception of the title
2 Visualizing with DSG´s help
In the standard literature, we find several methods related the description of the plane curves. In fact, Frost (1918, p. 167) explains that “at the same time, I have generally expect the student only to examine the figure, in other to see how the elements were combined. I shall now give a few rules for the systematic treatment of the equation of a curve, only recommending them as convenient plans to adopt, and leaving to the ingenuity of the student such modifications as particular forms of equations may suggest.”. Undoubtedly, we see the relevance attributed to the figures as an element that can to guide the understanding. In fact, in the figure 2, we show extraordinary figures provided in his old mathematical book.
Figure 2. Some curves described by Frost (1918, p. 167-168) and his systematic method
The whole paper must be written with Times New Roman 12, with the exception of the title
Contrary the static appearance that we see in the figure 2, we can visualize, with the DSG´s help, a dynamic representation of a algebraic curve. Clearly, its trace is less complicated. However, when we analyze its graph behavior (on the left side), we can conjecture a serious computational problem. On the other hand. lets take the curve
C
described
6235
0
xxyy
− − =
. Given
tIR
∈
, we will find the intersections points of
C
with the line
ytx
=
. For this, we compute
6235623563555
0()()00
xxyyxxtxtxxtxtx
− − = ∴ − − = ↔ − − =
. So, if we consider
53535
0()0
xxxttxtt
≠ ∴ − − = ↔ = +
. Finally, we still obtain
46
ytxtt
= = +
. So, we obtain that
3546
()(,)
ttttt
α
= + +
. Now, we can obtain the velocity vector
35202435242
46(46)'()(35,46)3535
t
dydytttt dt ttttt dxdxttt dt
α
≠
+ += + + ∴ = = =+ +
. So, in some works (ALVES, 2014), we can describe the trace’s behavior from the velocity vector components or, the vector
''()(''(),''())
txtyt
α
=
.
Figure 3. Visualizing some limitations of the DSG related to the trace of curves
The whole paper must be written with Times New Roman 12, with the exception of the title
In fact, from the expression
3524
4635
dytt dxtt
+=+
if 000
dyt dx
= ↔ =
and from this, we expect a cuspid point at the srcin. Moreover, we can analyse the behavior of the components functions
2435
(35,46)
tttt
+ +
. On the other hand, lets consider the instructions in Vainsencher (2009, p. 40-41). En virtue that, we choose the terms
6235
,,
XXYY
and determine the following segments:,,
ABBCAC
. We determine the following parameter variation
01
t
≤ ≤
and, for which point
(6,0),(2,3),(0,5)
we construct the parametric form related to these segments. For example, in the case
(2,3),(0,5)(1)(2,3)(0,5)
tt
∴ − +
. From this expression, we indicate (on the right side), the pink trace corresponding to the
AB
variations. In the figure 4, we visualize and understanding how the geometric and graphical elements were combined in the dynamic construction.
Figure 4. Description of the Newton’s method with DSG´s help

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