Cube and Circles

A circle is inscribed in a face of a cube of side a. Another circle is circumscribed about a neighboring face of the cube. Find the least distance between points of the circles?

Source: Mekh-Mat Entrance Examinations Problems, Ilan Vardi, Institut des Hautes Études Scientifiques, January 2000, which, in turn cites Smurov, Balsanov, 1986, and A. Shen, personal communications, 1999.

Center the cube about the origin, and let b=a/2 be the distance of each face from the origin. Let the inscribed circle, Circle A, be given by the equations

Circle A: x²+z²=b², y=b

Let the circumscribed circle, Circle B, be given by the equations

Circle B: x²+y²=2b², z=b

Now consider the spheres, Sphere A and Sphere B, which contain, respectively, Circle A and Circle B

Sphere A: x²+y²+z²=2b²
Sphere B: x²+y²+z²=3b²

Circle A is the intersection of plane y=b (parallel to the x-z plane) with Sphere A, which you can see clearly by substituting b in place of y into the equation of Sphere A.  Similarly, Circle B is the intersection of plane z=b (parallel to the x-y plane) with Sphere B.

No two points A and B, with Point A on Sphere A and Point B on Sphere B are closer to one another than the difference in the spheres' radii,
and this minimum distance is obtained only when points A and B both lie on a ray originating at the center of the spheres, which, here, is the origin of the coordinate system.

If such a ray contains a point on Circle A and a point on Circle B, then the answer to the question is (a/2)(sqrt(3)-sqrt(2))

To see that such a ray exists, consider the set of all rays originating at the origin that also contain a point of Circle A.
This is a half-cone whose apex is at the origin, and whose intersection with the plane of Circle B (z=b) is half a hyperbola -- the part whose y coordinates are positive (see fig. 1)

Fig 1: Intersection of Hyperbola A
with Circle B, where b=1

This half hyperbola contains point (0,b,b), which is in the interior of Circle B, and extends infinitely far outside the circle.
On the basis of this information alone, we can conclude with certainty that the answer to the question is (a/2)(sqrt(3)-sqrt(2))
If this were a test, we would write this answer down, and proceed with the next question.

Problem Solved!

But this is somehow a bit unsatisfying. I would want to know which points of Circle A are closest to which other points of Circle B. Let's explore it a bit further…

The cone defined by the origin and Circle A has the equation

Cone A: x² + z² = y²

Its intersection with the plane z=b is the hyperbola given by the two equations

Hyperbola A: z=b; x² + b² = y²

To find the intersection of Hyperbola A and Circle B, solve their equations simultaneously for x and y in terms of b:

Hyperbola A: x² + b² = y²
Circle B: x² + y² = 2b²

x² = b²/2
y² = 3b²/2

x = ±b sqrt(2)/2
y = b sqrt(6)/2

(y is positive, because only the "upper" half of Hyperbola A contains rays passing through circle A)

So the Point(s) B=(±b sqrt(2)/2, b sqrt(6)/2, b) on Circle B are closest to some points of Circle A.  What are these two points of Circle A?  The answer is simple -- the points that are on the same ray from the origin -- by simply shrinking the magnitude of each dimension of Point(s) B by a factor of sqrt(2)/sqrt(3), we find the corresponding Point(s) A.

Point(s) B = (±b sqrt(2)/2, b sqrt(6)/2, b)
Point(s) A = (±b sqrt(3)/3, b, b sqrt(6)/3)

Numerically, these points are approximately

Point(s) B = (±0.70711b, 1.2247b, b)
Point(s) A = (±0.57735b, b, 0.81650b)

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