08 May 2021

*Tags: geometry
,
vectors
,
cartography
*

An earlier article mentioned how to find the intersection of two great circles on the sphere using vectors. Another common problem is to find the intersection of small circles, or equivalently, the point at a certain distance from a set of other points. This is called multilateration in the general case, or bi- or trilateration in the case of 2 or 3 distances. Sometimes the circles do not meet exactly, and an approximate solution is required.

Circles on the unit sphere are the intersection of the plane \(\mathbf{\hat{c}} \cdot \mathbf{\hat{v}} = \cos r\) with the sphere, where \(\mathbf{\hat{c}}\) is the center of the circle and \(r\) is its radius. Temporarily ignoring the restriction to the sphere, a set of two or more planes forms a linear system.

In the following, \(\begin{bmatrix} \mathbf{c}_1 & \mathbf{c}_2 & \ldots \end{bmatrix}\) denotes a matrix whose columns are \(\mathbf{c}_1\), \(\mathbf{c}_2\), etc.

Two planes intersect in a line, unless they are parallel.

\[\begin{bmatrix} \mathbf{\hat{c}}_1 & \mathbf{\hat{c}}_2 \end{bmatrix} \mathbf{v} = \begin{bmatrix} \cos r_1 \\ \cos r_2 \end{bmatrix}\]This has general solution (as per Wikipedia):

\[\mathbf{v} = h_1 \mathbf{\hat{c}}_1 + h_2 \mathbf{\hat{c}}_2 + t (\mathbf{\hat{c}}_1 \times \mathbf{\hat{c}}_2)\]where

\[h_1 = \frac{\cos r_1 - \cos r_2 \left(\mathbf{\hat{c}}_1 \cdot \mathbf{\hat{c}}_2\right)}{1 -\left(\mathbf{\hat{c}}_1 \cdot \mathbf{\hat{c}}_2\right)^2},\]\(h_2\) is the same with \(r_1\) and \(r_2\) exchanged, and \(t\) is a real number. For the point to be on the sphere, \(t\) must be such that \(\|\mathbf{v}\| = 1\). That is given by:

\[t = \pm \frac{\sqrt{1 - h_1^2 - h_2^2 - 2 h_1 h_2 \mathbf{\hat{c}}_1 \cdot \mathbf{\hat{c}}_2}} {\|\mathbf{\hat{c}}_1 \times \mathbf{\hat{c}}_2\|}.\]\(t\) is zero if the two circles just touch in a single point. If the two circles do not intersect, \(t\) is not defined. An approximate single solution, whether or not the circles intersect, is given by letting \(t=0\) and normalizing \(\mathbf{v}\) so that it is a unit vector. In the images below, the points of intersection are marked with a black dot, and the approximate solution is marked with an x.

If both circles are great circles, then \(r_1 = r_2 = \pi/2\), \(h_1 = h_2 = 0\), and \(t = \pm \frac{1}{\|\mathbf{\hat{c}}_1 \times \mathbf{\hat{c}}_2\|}\), as expected.

If the second circle is a great circle, then \(r_2 = \pi/2\) and \(\cos r_2 = 0\). Then \(h_1\) simplifies to

\[h_1 = \frac{\cos r_1}{1 -\left(\mathbf{\hat{c}}_1 \cdot \mathbf{\hat{c}}_2\right)^2},\]and \(h_2 = - \left(\mathbf{\hat{c}}_1 \cdot \mathbf{\hat{c}}_2\right) h_1\). \(t\) then simplifies as so:

\[t = \pm \frac{\sqrt{\sin^2 r_1 - \left(\mathbf{\hat{c}}_1 \cdot \mathbf{\hat{c}}_2\right)^2 }}{1 -\left(\mathbf{\hat{c}}_1 \cdot \mathbf{\hat{c}}_2\right)^2}\]Three planes intersect at a point, unless any pair are parallel to each other. This point may or may not lie on the surface of the sphere. If it does not – if the circles do not intersect exactly in a single point – this gives an approximate solution that does lie on the sphere.

\[\begin{bmatrix} \mathbf{\hat{c}}_1 & \mathbf{\hat{c}}_2 & \mathbf{\hat{c}}_3 \end{bmatrix} \mathbf{\hat{v}} = \begin{bmatrix} \cos r_1 \\ \cos r_2 \\ \cos r_3 \end{bmatrix}\]Assume that \(\mathbf{\hat{c}}_1\), \(\mathbf{\hat{c}}_2\), and \(\mathbf{\hat{c}}_3\) are in counterclockwise order on the sphere. Let \(\mathbf{r}\) be the column vector with entries \(\cos r_i\), then the solution is

\[\mathbf{\hat{v}} = \frac{\begin{bmatrix} \mathbf{\hat{c}}_2 \times \mathbf{\hat{c}}_3 & \mathbf{\hat{c}}_3 \times \mathbf{\hat{c}}_1 & \mathbf{\hat{c}}_1 \times \mathbf{\hat{c}}_2 \end{bmatrix} \mathbf{r}}{\|\ldots\|}\]Note that the solution may lie outside the small triangle formed by the circles. This makes sense if you think about the intersection of planes, but can feel wrong if you’re thinking about “triangulation” and expect that triangle to be involved somehow. For an alternative, take each pair of circles and calculate the points of intersection, and then use the spherical centroid of that small triangle. (Assuming the circles all intersect, of course.)

Four or more planes do not in general meet at a single point, but the least squares method can be applied to find an approximate solution.

\[\begin{bmatrix} \mathbf{\hat{c}}_1 & \mathbf{\hat{c}}_2 & \ldots \end{bmatrix} \mathbf{\hat{v}} = \begin{bmatrix} \cos r_1 \\ \cos r_2 \\ \vdots \end{bmatrix}\]Let \(\mathbf{\hat{c}}_i = [x_i, y_i, z_i]\). Let \(\mathbf{x}\) be the vector having \(x_i\) as its \(i\)th component, and so on for \(\mathbf{y}\) and \(\mathbf{z}\). Also let \(\mathbf{r}\) be the column vector with entries \(\cos r_i\) as before. (It is not necessary to assume that \(\mathbf{\hat{c}}_i\) are in any particular order.) Then the least squares solution to the system above is:

\[\begin{bmatrix} \mathbf{x} \cdot \mathbf{x} & \mathbf{y} \cdot \mathbf{x} & \mathbf{z} \cdot \mathbf{x} \\ \mathbf{x} \cdot \mathbf{y} & \mathbf{y} \cdot \mathbf{y} & \mathbf{z} \cdot \mathbf{y} \\ \mathbf{x} \cdot \mathbf{z} & \mathbf{y} \cdot \mathbf{z} & \mathbf{z} \cdot \mathbf{z} \end{bmatrix} \mathbf{\hat{v}} = \begin{bmatrix} \mathbf{r} \cdot \mathbf{x} \\ \mathbf{r} \cdot \mathbf{y} \\ \mathbf{r} \cdot \mathbf{z} \end{bmatrix}\]Given this system, proceed as in the three-plane case. Although the least squares method is used here, the solution does not minimize the squared error in \(r\), since the elements of \(\mathbf{r}\) are \(\cos r_i\), not \(r_i\). The effect of bringing the vector back onto the sphere also causes deviations from the least-squares solution. However, it’s a good approximation.

Does your application have angles of bearing instead of distances, or a mix of bearings and distances? Convert the bearings to great circles using the method described in the solving the “forward problem” section of this earlier post, and then apply the methods decribed above.

*A Python script to create the figures in this post is located here.*

- 2023-11-04: Spherical area coordinates, and a derived triangle center
- 2023-07-29: Homogeneous coordinates on the sphere with vectors
- 2022-08-30: Scaling the Schwarz triangle function
- 2022-08-27: Edges in the image of the Schwarz triangle function
- 2022-07-20: Area-preserving swirling of the disk
- 2022-03-03: Some dubious ways to calculate conformal polyhedral projections
- 2021-10-02: A variation on the Chamberlin trimetric map projection
- 2021-08-31: Snyder's equal-area projection
- 2021-06-12: Perspective projections with vectors
- 2021-05-29: A (sort of) Euclidean triangle on a non-Euclidean surface