Old Friend with a New Haircut

Geometry Level pending

Consider a circle and a sphere, whose definitions are given below:

$\large{x^2 + y^2 = 1 \\ x^2 + y^2 + z^2 = R^2}$

Suppose that the circle is projected upwards (in the $+z$ direction) onto the sphere. Let $L$ be the length of the projected curve, and let $A$ be the surface area enclosed by the projected curve. Determine the following:

$\large{\lim_{R\to \infty} \frac{L^2}{A}}$

The answer is 12.56637.

1 solution

Jason Apostol
Dec 20, 2017

Simply put, as R tends to $\infty$ , the surface of the sphere at which the circle is projected gets infinitely "flatter", meaning that we can disregard any distortion of the circle made by its projection onto the sphere.

So, this is simply the ratio of the circumference of the circle squared over the area of the circle defined by $x^2 + y^2 = 1$ .

As the circle is defined with the form $x^2 + y^2 = r^2$ , it so follows that $r = \sqrt{1} = 1$ and $d = 2r = 2$ . Plugging in,

$\frac{(\pi d)^2}{\pi r^2} = \frac{4{\pi}^2}{\pi} = \boxed{4\pi}$

Old Friend with a New Haircut

The answer is 12.56637.

1 solution

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