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We quote from a text on computer graphics (M. Beeler et al., 鈥淗AKMEM,鈥� MIT Artificial Intelligence Report AIM-239, 1972):

Here is an elegant way to draw almost circles on a point-plotting display.

CIRCLE ALGORITHM:

NEW X = OLD X - K*OLD Y;

NEW Y = OLD Y + K*NEW X.

This makes a very round ellipse centered at the origin with its size determined by the initial point. The circle algorithm was invented by mistake when I tried to save a register in a display hack!

(In the preceding formula, k is a small number.) Here, a dynamical system is defined in 鈥渃omputer lingo.鈥� In our terminology, the formulas are

$$\begin{gathered} \mathit{x}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathit{x}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{-}\mathit{k}\mathit{y}\mathbf{\left(}\mathit{t}\mathbf{\right)} \\ \mathit{y}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathit{y}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{+}\mathit{k}\mathit{x}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)} \end{gathered}$$

a. Find the matrix of this transformation. [Note the entry$\mathit{x}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}$in the second formula.]

b. Explain why the trajectories are ellipses, as claimed.

Step by step solution

01

Define stability:

Stability is one of the most basic requirements for numerical models, which extends to mostly linear problems. If every solution that was initially close to it is always close to it. It is said to be stable and asymptotically stable although initially every solution close to it meets $\mathit{t}\mathbf{鈫�}\mathbf{鈭�}$

02

Find the matrix:(a)

Given,

$\begin{array}{rcl}x(t+1)& =& x\left(t\right)-ky\left(t\right)\\ y(t+1)& =& y\left(t\right)+kx(t+1)\\ & =& y\left(t\right)+k\left(x\right(t)-ky(t\left)\right)\\ & =& kx\left(t\right)+\left(1-k^2\right)y\left(t\right)\end{array}$

The matrix is$A=\left[\begin{array}{cc}1& -k\\ k& 1-k^2\end{array}\right]$

03

Explain the trajectories are ellipses:(b)

We can see that,

$$\begin{gathered} \det A=1 \\ A=\left[\begin{array}{cc}1& -\sin 蠁\\ \sin 蠁& {\cos }^2蠁\end{array}\right] \end{gathered}$$

We can assume,

$蠁鈭�\left(0,\frac{\mathrm{蟺}}{2}\right)$

This is an eliptic rotation matrix, but the ellipse that it concerns has semiaxes that aren't parallel to the coordinate axes. Still, we can conclude that the given equations will form an ellipse.

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