How to determine the coordinates of a point inside a sphere after tilting the sphere

Question

I'm a neuroscientist PhD student working on a localization problem in the brain. Rather than explain the actual details of the problem, I'll give an abstract version of it.

Imagine a clear glass marble. A small bead of red metal is embedded inside, not at the center nor at the surface. Two circles are draw around the marble at the equator and meridian. There is also a small star drawn on the marble's surface, somewhere along the meridian between the equator and upper pole.

You tilt the sphere downward 24 degrees relative to the star (the star is at the vertex of the angle formed by tilting downward). After doing so, the red bead is located at the following coordinates relative to the star, as measured by a rigid ruler: 1.5 mm away from star along a line perpendicular to the meridian; 3.5 forward from the star along a line parallel to the meridian; and then 4 mm straight down into the marble.

Knowing these coordinates, what formula would you use to determine the new coordinates of the bead, again relative to the star, if you were to tilt the marble up or down from 24 degrees? For instance, what would the coordinates be if you tilted the marble down another 21 degrees, or tilted it back up to 0 degrees?

Note that there is no azimuthal movement, only tilt, like a head nodding.

In the diagram below, the orange arrows show the measurements made outside the marble using a rigid ruler. They are perpendicular and on the plane horizontal to the ground. The 1.5 mm arrow was difficult to draw — if you had your head against the wall, the arrow would be drawn on the wall, and the 3.5 mm arrow would be perpendicular to the wall. The blue arrow points down into the marble. The view is from the side, so the meridian is the circle, and the tilt is a clockwise turn of the circle. Degrees are for the angle of the dashed lines, the upper line being parallel to the ground. The exact radius is unknown, but it is approximately 5 mm.

Answer 1

You have the following situation

The $Y$'s are related to the $X's$ as follows:

$Y_1 = \dfrac{X_1}{\cos \theta} $

$Y_2 = X_2 \cos \theta - (X_1 + X_3) \sin \theta $

$Y_3 = X_3 \cos \theta + X_2 \sin \theta - \dfrac{X_1 }{\cos \theta} + X_1 \cos \theta $

This is a linear system in the three unknowns $X_1, X_2 , X_3$, while the knowns are $Y_1, Y_2, Y_3$, as well as $\theta = 24^\circ$, so it can be solved using standard techniques (Gaussian elimination, or any other method). Once $X_1 , X_2 , X_3 $ are determined the coordinates at any other angle $\theta$ can be determined from the above equations.

Answer 2

Lets assign 4mm direction as positive X-axis, 3.5mm direction as positive Y-axis and 1.5mm direction (coming out of computer screen) as positive Z-axis - with the star mark considered as origin (0,0,0)

• Because you mention that embedded red ball is not at surface of sphere, hence, we consider x,y,z coordinate system

• Also, you intend to rotate the big clear sphere downward (clockwise from your drawings perspective - about the z-axis) - hence, the z-coordinate does not really change even after rotation

So considering only the x and y axes in 2d for simplicity, we get the diagonal distance from origin to red ball as SQ.RT. X² + Y² = SQ.RT. 4² + 3.5² = 5.315MM and the CURRENT 2d-angle of this diagonal w.r.t. x-axis = acos (4/5.315) = 41.185°.

Now rotating the sphere in xy-plane in any direction (about the z-axis) by a given angle can be easily used to calculate the new x and y coordinates in 2d - while the z-coordinate remains the same as was originally.

Also, the diagonal length too remains constant (5.315mm) as per desired rotation about z-axis - hence, calculating any one coordinate (x or y) - would yield the other too.

Hope the above is clear and helps.