A and B are two points on a unit sphere. We know the space distance between A and B is sqrt3 What is distance from A to B along the (minor) arc of a great circle?

itsash Oct 30, 2023

#1**0 **

The distance between two points on a sphere along the minor arc of a great circle is equal to the central angle between the two points in radians. The central angle between two points on a sphere is equal to the arccosine of the dot product of the unit vectors pointing to the two points.

Therefore, to find the distance from A to B along the minor arc of a great circle, we first need to find the unit vectors pointing to A and B. We can do this by dividing the coordinates of A and B by the radius of the sphere, which is 1.

Let a and b be the unit vectors pointing to A and B, respectively. Then the distance from A to B along the minor arc of a great circle is equal to the central angle between a and b, which is given by:

theta = arccos(\mathbf{a} \cdot \mathbf{b})

We know that the space distance between A and B is 3. We can use this information to find the dot product of a and b.

\mathbf{a} \cdot \mathbf{b} = \cos(theta) = \frac{\sqrt{3}}{2}

Therefore, the distance from A to B along the minor arc of a great circle is given by:

theta = arccos(\mathbf{a} \cdot \mathbf{b}) = arccos(\frac{\sqrt{3}}{2})

theta = \frac{\pi}{3}

Therefore, the distance from A to B along the minor arc of a great circle is pi/3.

parmen Oct 30, 2023

#2**+8 **

thanks for trying, but the correct answer I found out on my own. I guess ill post an explanation here

We take a cross-section along the great circle. Let O be the center of the sphere, and let M be the midpoint of line AB

Since OA = OB = 1, angle OMA = 90, Am is sqrt3/2 so triangle AOM is a 30 60 90 triangle

So AOM is 60 degrees, BOM is also 60 degrees, therefore AOB is 120 degrees, therefore arc AB has length

120/360 *2pi=2/3pi

itsash
Oct 30, 2023