Henry Ernest Dudeney/Modern Puzzles/126 - Drawing an Oval/Solution
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
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $126$. -- Drawing an Oval
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $304$. Drawing an Ellipse