We quote from a text on computer graphics (M. Beeler et al.,...

We quote from a text on computer graphics (M. Beeler et al., "HAKMEM," MIT Artificial Intelligence Report AIM- 239,1972 ): Here is an elegant way to draw almost circles on a point-plotting display. CIRCLE ALGORITHM: \[ \begin{array}{l} \text { NEW } x=0 \text { LD } x-K * 0 L D Y \\ \text { NEW } Y=0 L D Y+K * N E W X \end{array} \] 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 "computer lingo." In our terminology, the formulas are \[ \begin{array}{l} x(t+1)=x(t)-k y(t) \\ y(t+1)=y(t)+k x(t+1) \end{array} \] a. Find the matrix of this transformation. [Note the entry $x(t+1)$ in the second formula. b. Explain why the trajectories are ellipses, as claimed.

We quote from a text on computer graphics (M. Beeler et al., "HAKMEM," MIT Artificial Intelligence Report AIM- 239,1972 ):
Here is an elegant way to draw almost circles on a point-plotting display.
CIRCLE ALGORITHM:
\[
\begin{array}{l}
\text { NEW } x=0 \text { LD } x-K * 0 L D Y \\
\text { NEW } Y=0 L D Y+K * N E W X
\end{array}
\]
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 "computer lingo." In our terminology, the formulas are
\[
\begin{array}{l}
x(t+1)=x(t)-k y(t) \\
y(t+1)=y(t)+k x(t+1)
\end{array}
\]
a. Find the matrix of this transformation. [Note the entry $x(t+1)$ in the second formula.
b. Explain why the trajectories are ellipses, as claimed.

Key Concepts

Dynamical Systems

Dynamical systems involve studying the evolution of points in a given space according to specific rules, often iteratively. In this context, the system is defined by update rules that change the state over discrete time steps, which is a fundamental concept in modeling processes in mathematics and computer graphics.

Matrix Representation of Linear Transformations

A key tool in analyzing such systems is the matrix representation of linear transformations. This approach enables the transformation of a state vector via multiplication by a matrix, encapsulating the dynamics in a compact, algebraic form. It simplifies the analysis and composition of transformations.

Invariant Quadratic Forms

Invariant quadratic forms are expressions that remain unchanged under a given transformation. In the analysis of linear dynamical systems, showing that a certain quadratic form is preserved helps reveal that the trajectories lie on conic sections such as ellipses or circles, which explains the geometric behavior of the system.

Eigenvalue Analysis

Eigenvalue analysis involves studying the eigenvalues and eigenvectors of the transformation matrix to understand how the system behaves over time. This analysis can indicate whether the transformation preserves areas, rotates the state space, and maintains the shape of trajectories, all of which are crucial for understanding why paths evolve into ellipses.

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