Design and use of a learning object for finding complex polynomial roots

Julio Benítez, José L. Hueso, Eulalia Martínez, Jaime Riera

Some interesting curves

Epitrochoids

The curve described by a point $P$ attached to a circle of radius $b$ rolling around a fixed circle of radius $a$ is called epitrochoid.


Figure 1: Generation of an epitrochoid


The parametric equations of the epitrochoid are:


$$ \begin{array}{lll} x(t) & = & (a+b)\cos t - c \cos\left(\frac{a+b}{b}t\right) \end{array} $$


$$ \begin{array}{lll} y(t) & = & (a+b)\sin t - c \sin\left(\frac{a+b}{b}t\right), \ \ t \in R \end{array} $$

where $c$ is the distance from $P$ to the center of the rolling circle.

When $b=c,$ point $P$ is on the moving circle and the curve is named epicycloid. For example, the epitrochoid with $a=b=c=r$ is the cardioid, the trajectory of a point of circle of radius $r$ that rolls around another circle of the same radius. Some other known epicycloids are the nephroid, with parameters $a=2b=2c,$ and the ranuncloid, whose parameters are in proportion $5:1:1.$

If $\frac{a}{b}$ is rational, the epitrochoid of parameters $(a,b,c)$ can be obtained in the complex plane as the image of the unit circle by the polynomial $p(z)=Az^m+Bz^n,$ where $\frac{m}{n}$ is the irreducible expression of $\frac{a+b}{b},$ $A=-c,$ and $B=a+b.$

The cardioid

The parameters of the cardioid are $(a,b,c)=(1,1,1).$ Then,


$$ \frac{a+b}{b} = \frac{2}{1},$$

so that, we can take $m=2$, $n=1$. The polynomial coefficients are $A=-c=-1$, and $B=a+b=2.$ Then, the polynomial is $p(z)=-z^2+2z.$

Define this polynomial in the applet by entering its coefficients in the Algebra panel or by dragging the corresponding points in the Graphics panel to:

  • $a_0 = 0+0i$
  • $a_1 = 2+0i$
  • $a_2 = -1+0i$
  • $a_3 = 0+0i$
  • $a_n = 0+0i$

Setting $r=1$ one obtains the cardioid in the image window


Figure 2: The cardioid

The nephroid

The parameters of the nephroid are $(a,b,c)=(2,1,1).$ Then,


$$ \frac{a+b}{b} = \frac{3}{1},$$

so that, we can take $m=3$, $n=1$. The polynomial coefficients are $A=-c=-1$, and $B=a+b=3.$ Then, the polynomial is $p(z)=-z^3+3z.$


Figure 3: The nephroid

The ranuncloid

The parameters of the ranuncloid are $(a,b,c)=(5,1,1).$ Then,

$$ \frac{a+b}{b} = \frac{6}{1},$$

so that, we can take $m=6$, $n=1$. The polynomial coefficients are $A=-c=-1$, and $B=a+b=6.$ Then, the polynomial is $p(z)=-z^6+6z.$


Figure 4: The ranuncloid

Epitrochoid generated by a polynomial

Reciprocally, a polynomial of the form $p(z)=Az^m+Bz^n, \ \ m>n,$ generates the epitrochoid with parameters

$$ \begin{array}{lll} a & = & B r^n \left(1-\frac{n}{m}\right) \end{array} $$

$$ \begin{array}{lll} b & = & \frac{n}{m}B r^n \end{array} $$

$$ \begin{array}{lll} c & = & -A r^m \end{array} $$

as the image of the circle of radius r centered in the origin.

For example, $p(z)=-z^7+z^3$ maps the unit circle on the epitrochoid with parameters $(a,b,c)=(4/7,3/7,1).$


Figure 5: Epitrochoid generated by $p(z)=-z^7+z^3$

Spirograph

To simulate the drawing of the curve, we can animate the slider of the argument. Right click on that slider and check Animation On in the Basic tab.

You will see the point Pz moving along the curve.

For a more realistic simulation, hide all the objects in the image window, except the argument slider and the point Pz (hide its label also).

Set on the trace of that point and restart the animation.


Figure 6: Spirograph simulation

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