Equations for three points on a fixed plane

Question

I have a mathematical problem and I'am asking for help with derivation of equations.

Here is the problem definition: We have three points (A, B and C) on a fixed plane, so when one moves in z direction other two also move. Center point F (0,0,0) is fixed so it can't move and is also located on this plane. All three points are located on a circle of radius R and are vertexes of equilateral triangle as shown on sketch below.

I need to derive equations for coordinates x, y, and z for all three points A, B and C, because for example I'm interesed in how much points A and B move in x, y and z if point C is moved in direction z for some value. I also know that C also changes y coordnate in this example.

EDIT: I went through my problem again and I can "simplify" it a little bit. First thing that changes is that point F is no longer on location (0,0,0), but can move away from it. It is located on the same plane as points A, B and C. Second thing is that I can fix the z coordinates of points A, B and C, because they are a known information, which will be used as input. All three can be different. Third thing is that the coordinates x and y of points A, B and C can move only a certain radius limit $\Delta$r from their inital position.

To summarize: Triangle is first located in plane x-y as shown in sketch. Then we move z coordinates of points A, B and C for known value, which can be different for all three values, so triangle can then make an angle with plane x-y. Location of x and y coordinates also change, but we limit them to certain radius value $\Delta$r around initial location of a point. We could also try and fix a x coordinate of point C, so that the triangle can't rotate around z axis.

Answer 1

If I understand the question properly all three points can move spherically around point F as long as they stay co-planer, remain equidistant from each and equidistant from point F (describing an equilateral triangle about point F)

My first thought would be to convert to spherical coordinates. Lets start with 2D (polar coordinates) then consider 3D.

Polar Coordinates (2D): the position of a point is described by it's distance from the origin (R), and the distance rotated from a horizontal axis ($\theta$): (R, $\theta$). To convert back to the cartesian system simple trig gets you (R cos$\theta$, R sin$\theta$)

For the triangle in question the three points would be:

• A:(R, $\theta$) or (R cos$\theta$, R sin$\theta$)
• B:(R, $\theta$+120$\unicode{xB0}$) or (R cos($\theta$+120$\unicode{xB0}$), R sin($\theta$+120$\unicode{xB0}$))
• C:(R, $\theta$+240$\unicode{xB0}$) or (R cos($\theta$+240$\unicode{xB0}$), R sin($\theta$+240$\unicode{xB0}$))

Note: We must know the position of at lease one point to describe the other two.

Spherical coordinates (3D): the position of a point is described by it's distance from the origin (R), the distance rotated down from the vertical axis ($\theta$), and the distance rotated from a horizontal axis ($\phi$): (R, $\theta$, $\phi$).

from: https://en.wikipedia.org/wiki/Spherical_coordinate_system

To get to cartesian coordinates gets a little more drawn out. First, we need to know the distance the point is away from the vertical axis (r).

r = R sin$\theta$

Next we shift our point of reference to the position on the XY plane, where x = R cos$\phi$ and y = R sin$\phi$. Including the vertical component's influence:

x = R sin$\theta$ cos$\phi$

y = R sin$\theta$ sin$\phi$

z = R cos$\theta$

This is where we hit a snag. To describe the next point on the triangle we need to know which direction to move in. Unlike with polar coordinates, we need to know the location of two points to be able to describe the triangle. Imagine the case where we know the position of one point, then we rotate the triangle around the line described by the known point and the origin. The other two points could fall anywhere within the cone.

TL:DR In 2d space we need to know the position of two points to define an equilateral triangle centered about the origin. In 3d space we need to know the position of three points to define an equilateral triangle centered about the origin. The origin counts as one of these point, but without more information we cannot describe the triangle.

Update

With the updated problem statement we are now assuming a fixed $\phi$, defining the coordinate system such that C$_x$ = 0, and z position is known we can determine the coordinates:

$\phi_{(A,B,C)} = (30\unicode{xB0},150\unicode{xB0},270\unicode{xB0})$ (from our polar coordinate equations)

$\theta_{(A,B,C)} = \left(cos^{-1}\left(\frac{A_z}{R}\right),cos^{-1}\left(\frac{B_z}{R}\right),cos^{-1}\left(\frac{C_z}{R}\right)\right)$ (remember: z = R cos$\theta$)

Now just pop those values back into the spherical to cartesian coordinate equations above, and Bob's your uncle.

...but, if you are looking for the allowable angle between the jackscrew and the plane described by A, B, and C, the angle $\theta$ should be a good approximation (for a relatively small $\theta$).

Also, remember that if you raise all three jackscrews an equal amount you are effectively raising your z$_0$ since you are only interested in the angle between the plane and the jackscrew. Therefore, your z position is really the difference in the position from one point to another. That is why I was using $\Delta$z in my clarifying comment.