Details

If a drone (represented as a particle) can move with equal velocity in any direction, the set of possible 3D coordinates after one time step is a 3D sphere. The image plane of a camera projects all 3D points into a 2D camera frame [1]. If the sphere is centered at with radius and camera focal length , then the optimal goal location for the drone is parameterized by the (latitude, longitude) pair :

,

.

If is , there are infinite solutions with .

The goal location is:

This projects onto the image plane at the point . Because is always negative, the drone decreases the distance in . Interestingly, the sphere projects into a 2D ellipse in the image plane. With equations given by [2], the ellipse is centered at

,

with

,

and

.

The focal length is 5 in this Demonstration.

Snapshot 1: , , and

Snapshot 2: 2D view when , , and

Snapshot 3: , , and with "show drone" turned off

Snapshot 4: 2D view when , , and with "show sphere" turned off

Snapshot 5: , , and

Snapshot 6: , , and with "show drone" turned off

Snapshot 7: 2D view when , , and

References

[1] M. W. Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control, Hoboken, NJ: John Wiley and Sons, 2006.

[2] D. Eberly. "Perspective Projection of an Ellipsoid." (Oct 30, 2018) www.geometrictools.com/Documentation/PerspectiveProjectionEllipsoid.pdf.