In crossword puzzles, it is common to come across clues that require knowledge of geometric shapes. One such shape is the ellipse, which is a closed curve that resembles a flattened circle. The ellipse has two axes, the longer of which is called the major axis and the shorter is called the minor axis. In a crossword clue, you may come across the term “half of an ellipse’s shorter diameter,” which refers to a specific measurement of the ellipse.
The shorter diameter of an ellipse is the distance between the two points on the ellipse that lie on the minor axis. When referring to “half of an ellipse’s shorter diameter,” the clue is asking for half of this distance. To find the answer, you will need to know the length of the minor axis.
To calculate the length of the minor axis, you can use the formula:
b = a * sqrt(1 – e^2)
where “a” is the length of the major axis, “b” is the length of the minor axis, and “e” is the eccentricity of the ellipse. The eccentricity of an ellipse is a measure of how “flattened” it is and is given by the formula:
e = sqrt(1 – (b^2/a^2))
Once you have calculated the length of the minor axis, you can find half of it by dividing the value by two.
For example, let’s say the clue is “Half of an ellipse’s shorter diameter (3)”. The number in parentheses indicates the number of letters in the answer. If the length of the major axis is 6 and the eccentricity is 0.8, we can use the formula above to calculate the length of the minor axis:
b = 6 * sqrt(1 – 0.8^2) = 3.12
Half of the minor axis is 1.56, which would be the answer to the crossword clue.
It is important to note that the formula for calculating the length of the minor axis assumes that the center of the ellipse is at the origin of the coordinate system. If the center is at a different point, the formula must be adjusted accordingly.
Additionally, it is worth noting that half of an ellipse’s shorter diameter is also known as its semi-minor axis. This is because it is half of the length of the entire minor axis.
The concept of half of an ellipse’s shorter diameter is not only useful in crossword puzzles, but also in various fields such as mathematics, engineering, and physics. In these fields, the length of the minor axis is often used to calculate other properties of the ellipse, such as its area, circumference, and foci.
In conclusion, half of an ellipse’s shorter diameter is a measurement that refers to half of the length of the minor axis of an ellipse. In crossword puzzles, it may be used as a clue to help solvers find the answer to a specific word or phrase. Understanding this concept requires knowledge of the formula for calculating the length of the minor axis, as well as the ability to apply it in different contexts.