Eclipse the Ellipse

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Recall that for $0\le t<2\pi$ \begin{align*} x(t) &= \cos (t)\\ y(t) &= \sin (t) \end{align*}

gives a parametric plot of a unit circle. Describe the plot of \begin{align*} x(t) &= 3\cos (t)\\ y(t) &= \sin (t) \end{align*}

for $0\le t<2\pi$ .

Now describe the plot of \begin{align*} x(t) &= 2\cos (t)\\ y(t) &= 5\sin (t) \end{align*}

for $0\le t<2\pi$ .

We claim that an ellipse centered at the origin is defined by points $(x,y)$ satisfying \[ \left (\frac{x}{a}\right )^2 + \left (\frac{y}{b}\right )^2 = 1. \] Are the parametric curves we found above ellipses? Explain why or why not.

Here we have some plots showing two concentric circles and an ellipse that touches both. \[ \begin{tabular}{ccc} \par \begin{tikzpicture} \begin{axis}[ width=2.5in, clip=false, domain=(0:2*pi), axis lines=center, unit vector ratio*=1 1 1, xlabel=$x$, ylabel=$y$, every axis y label/.style={at=(current axis.above origin),anchor=south}, every axis x label/.style={at=(current axis.right of origin),anchor=west}, ] \pgfmathsetmacro{\a }{0} \addplot [very thick, penColor, smooth] ({3*cos(deg(x))},{3*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{5*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{3*sin(deg(x))}); \par \addplot [] plot coordinates{(0,0) ({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \par \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({3*cos(deg(\a ))},{3*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{3*sin(deg(\a ))})}; \end{axis} \end{tikzpicture} & \begin{tikzpicture} \begin{axis}[ clip=false, width=2.5in, domain=(0:2*pi), axis lines=center, unit vector ratio*=1 1 1, xlabel=$x$, ylabel=$y$, every axis y label/.style={at=(current axis.above origin),anchor=south}, every axis x label/.style={at=(current axis.right of origin),anchor=west}, ] \pgfmathsetmacro{\a }{pi/10} \addplot [very thick, penColor, smooth] ({3*cos(deg(x))},{3*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{5*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{3*sin(deg(x))}); \par \addplot [] plot coordinates{(0,0) ({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \par \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({3*cos(deg(\a ))},{3*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{3*sin(deg(\a ))})}; \end{axis} \end{tikzpicture} & \begin{tikzpicture} \begin{axis}[ width=2.5in, clip=false, domain=(0:2*pi), axis lines=center, unit vector ratio*=1 1 1, xlabel=$x$, ylabel=$y$, every axis y label/.style={at=(current axis.above origin),anchor=south}, every axis x label/.style={at=(current axis.right of origin),anchor=west}, ] \pgfmathsetmacro{\a }{2*pi/10} \addplot [very thick, penColor, smooth] ({3*cos(deg(x))},{3*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{5*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{3*sin(deg(x))}); \par \addplot [] plot coordinates{(0,0) ({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \par \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({3*cos(deg(\a ))},{3*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{3*sin(deg(\a ))})}; \end{axis} \end{tikzpicture} \\ \begin{tikzpicture} \begin{axis}[ width=2.5in, clip=false, domain=(0:2*pi), axis lines=center, unit vector ratio*=1 1 1, xlabel=$x$, ylabel=$y$, every axis y label/.style={at=(current axis.above origin),anchor=south}, every axis x label/.style={at=(current axis.right of origin),anchor=west}, ] \pgfmathsetmacro{\a }{3*pi/10} \addplot [very thick, penColor, smooth] ({3*cos(deg(x))},{3*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{5*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{3*sin(deg(x))}); \par \addplot [] plot coordinates{(0,0) ({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \par \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({3*cos(deg(\a ))},{3*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{3*sin(deg(\a ))})}; \end{axis} \end{tikzpicture} & \begin{tikzpicture} \begin{axis}[ width=2.5in, clip=false, domain=(0:2*pi), axis lines=center, unit vector ratio*=1 1 1, xlabel=$x$, ylabel=$y$, every axis y label/.style={at=(current axis.above origin),anchor=south}, every axis x label/.style={at=(current axis.right of origin),anchor=west}, ] \pgfmathsetmacro{\a }{4*pi/10} \addplot [very thick, penColor, smooth] ({3*cos(deg(x))},{3*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{5*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{3*sin(deg(x))}); \par \addplot [] plot coordinates{(0,0) ({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \par \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({3*cos(deg(\a ))},{3*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{3*sin(deg(\a ))})}; \end{axis} \end{tikzpicture} & \begin{tikzpicture} \begin{axis}[ width=2.5in, clip=false, domain=(0:2*pi), axis lines=center, unit vector ratio*=1 1 1, xlabel=$x$, ylabel=$y$, every axis y label/.style={at=(current axis.above origin),anchor=south}, every axis x label/.style={at=(current axis.right of origin),anchor=west}, ] \pgfmathsetmacro{\a }{5*pi/10} \addplot [very thick, penColor, smooth] ({3*cos(deg(x))},{3*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{5*sin(deg(x))}); \addplot [very thick, penColor, smooth] ({5*cos(deg(x))},{3*sin(deg(x))}); \par \addplot [] plot coordinates{(0,0) ({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \par \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({3*cos(deg(\a ))},{3*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{5*sin(deg(\a ))})}; \addplot [color=penColor,fill=penColor,only marks,mark=*] coordinates{({5*cos(deg(\a ))},{3*sin(deg(\a ))})}; \end{axis} \end{tikzpicture} \end{tabular} \]

(a): Can you guess parametric formulas for the circles and for the ellipse?
(b): Do you notice anything about the dots in the pictures? Can you explain why this happens?
(c): Can you give a compass and straightedge construction that will give you as many points on a given ellipse as you desire? Give a detailed explanation.

Can you give a parametric formula for this cool spiral? \[ \begin{tikzpicture} \begin{axis}[ width=3in, clip=false, axis lines=center, unit vector ratio*=1 1 1, xlabel=$x$, ylabel=$y$, every axis y label/.style={at=(current axis.above origin),anchor=south}, every axis x label/.style={at=(current axis.right of origin),anchor=west}, ] \addplot [very thick, penColor, smooth,samples=100,domain=(0:8*pi)] ({x*cos(deg(x))},{x*sin(deg(x))}); \end{axis} \end{tikzpicture} \]

Remind me once more, do the formulas that produce these plots define functions? Discuss. Clearly identify the domain and range as part of your discussion.

2024-10-10 13:45:28