title a day - Projection

starting up a new project --- a title a day.

just for any words, explain, give a lesson. completely from myself only. hope can last for 1 year.
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Projection
projection is..... projection. in physic, a projection of A on B is an image of A on B -- A'. just like a sun shining on an object (A), that will give a shadow (A') on behind. e.g. a point project on a line, a point project on a plane, a cylinder. for human, we can only imagine a 3-D object project on a 2-D object.

the most easiest projection is the dot product between 2 vectors.

in mathematics, a projection can be presented as a transformation, or matrix (in linear case)

a point can be represented as a vector. the projection for a point on x-y plane is simply taken the first 2 component. on x-z plane is taken the 1st and 3rd component. we can write the transformation matrix as 2X3 :

1 0 0
0 1 0 for x-y plane projection.

from this, we know that for a general plane projection, we can write a matrix to represent it.
the way i do is first transform the space such that the general plane to be x'-y' plane.

to do so, we need to know 2 new matrix. translation and rotation. but this is far away from today topic.

to extend this idea. although we can not imagine higher-D projection, but we can calculate it.

for example. the matrix :

1 0 0 0
0 1 0 0
0 0 1 0

is a projection matrix from 4-D to 3-D. so, in every point of 3-D space, like the points on a pencil, we can thing that the "real" pencil is a 4-D object, but now, be projected on a 3-D space. the funny thing is, by using an inverse transform, we can construct the 4-D space!

D = T P // D is the 3-D vector representing a pencil, T is the 4-D -> 3-D transform, P is the 4-D vector of the pencil.

by solving above equation, we can know how the 4-D pencil look like!~ HA~

however, the inverse transform of T does not exist. that led to multiple solution for P. which mean, for 1 point in 3-D space, there are multiple points in 4-D space can be projected onto it. imagine a cube project on a sheet of paper. the shadow on the the paper is not formed by single point of the cube.

there are many other application for projection matrix. like computer game, they need to project a "3-D" world on a screen. in optics experiment, projection matrix is wildly used, to calculate how a light beam will project on. in civil engineering, they need to calculate how side wind move the building.

the concept of projection is even used out of it original meaning in other field, like business. projected profit -- which mean the expected profit. this kind of projection is mean the projection on time, not on space.