Plane Projection Tensor

With the definitions of planes in plain view, we can address the task of projecting vectors into planes. Your simulation may require the velocity vector of an aircraft be projected on the ground or the silhouette of a missile be imaged on a charge – coupled device (CCD) planar array. For all of these situations, the plane projection tensor will be a useful tool.

We met the line projection tensor P earlier. It is formed by the dyadic product of the unit vector и, P = и m. According to Eq. (2.21), P produces the vector r from t by projecting it onto u: r = Pt. Now, и does not only establish the direction of a line, but also the unit normal of a plane (see Fig. 2.19).

The challenge is to find the projection tensor N that projects vector t onto the plane given by и. The projected vector is labeled s. From the vector triangle we derive and substituting Eq. (2.21) forr

s = t — Pt = (E — P)t = (E — uU)t

We define the plane projection tensor with E the unit tensor and и normal of the plane

N ~ E — ий (2.25)

Just like the line projection tensor, the plane projection tensor is symmetric.

Example 2.8 Focal Plane Imaging

Problem. An aircraft is imaged on a focal plane array. To simulate that process, we need to develop the equations that project the aircraft’s silhouette on the focal plane. We keep it simple by modeling the perspective of the aircraft with the displacement vectors of the tip, stem, right wing tip, and left wing tip wrt the

geometrical center C, tBlc, 1в2сЛвъс, and te„c – The displacement of the aircraft center C wrt the focal plane center F is given by tCF and the orientation of the planar array by the unit normal vector u. Separation distance and optics reduce the scale of the projections on the focal plane by a factor /. Determine the aircraft attitude vectors sBlc, $в2с, sb3c, and Sb4c and the displacement vector sCf in the focal plane. (To practice, make a sketch.)

Solution. Subjecting the displacement vectors to the plane projection tensor N = E — ий and reducing the magnitude by / produces the image

S BiC = fNtBlc, s B2C — fNtg2c, Sb3C = fNtB3C, s IhC = fNtBic

and the displacement of the aircraft from the focal plane center

scf = fNtCF

For building the simulation, the vectors have to be converted to matrices. Most likely, the aircraft data are in geographic coordinates ]G, and the image should be portrayed in focal plane coordinates ]F. Therefore, a transformation between the two coordinate systems [T]GF will enter the formulation.