Henry Ernest Dudeney/Modern Puzzles/126 - Drawing an Oval/Solution

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Modern Puzzles by Henry Ernest Dudeney: $126$

Drawing an Oval
It is well-known that you can draw an ellipse by sticking two pins into the paper, enclosing them with a loop of thread,
and keeping the loop taut, running a pencil all the way round till you get back to the starting point.
Suppose you want an ellipse with a given major axis and minor axis.
How do you arrange the position of the pins, and what would be the length of the thread?


Solution

Place the pins $2 \sqrt {a^2 - b^2}$ apart, where $2 a$ and $2 b$ are the major axis and minor axis respectively.

Then make the string $2 a + 2 \sqrt {a^2 - b^2}$ long.


Proof

Let $L$ denote the length of the thread.

Let $E$ denote the ellipse being drawn.

Let $f_1$ and $f_2$ denote the foci of $E$.

Let $2 a$ and $2 b$ be the major axis and minor axis respectively of $E$.

Let $2 c$ be the distance between $f_1$ and $f_2$.

From the equidistance property of the ellipse, the pins are to be placed at $f_1$ and $f_2$.

From Equidistance of Ellipse equals Major Axis, the sum of the lengths of the sections of thread between the pins and the perimeter of the ellipse equals $2 a$.

From Focus of Ellipse from Major and Minor Axis:

$c^2 = a^2 - b^2$

Hence the required length of string needed to draw $E$ is given by:

$L = 2 a + 2 \sqrt {a^2 - b^2}$

and the pins are to be placed a distance $2 \sqrt {a^2 - b^2}$ apart.

$\blacksquare$


Dudeney's solution focuses on the special case where the major axis is $12$ inches and the minor axis is $8$ inches, and uses an ad hoc semi-geometrical method of solution.


Sources

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