# geometry – Transformations of the project: "If all the points are in a plane, the linear mapping is reduced to …"

Page 7 of my computer vision textbook, Multiple view geometry in computer vision, says the following:

When applying projective geometry to the process of generating images, it is usual to model the world as a $$3$$D projective space, equal to $$mathbb {R} ^ 3$$ along with points to infinity. In the same way, the model for the image is the $$2$$Projective plan d $$mathbb {P} ^ 2$$. The central projection is simply a map of $$mathbb {P} ^ 3$$ to $$mathbb {P} ^ 2$$. If we consider points in $$mathbb {P} ^ 3$$ Written in terms of homogeneous coordinates. $$( mathrm {X}, mathrm {Y}, mathrm {Z}, mathrm {T}) ^ T$$ and that the projection center is the origin. $$(0, 0, 0, 1) ^ T$$, then we see that the set of all points. $$( mathrm {X}, mathrm {Y}, mathrm {Z}, mathrm {T}) ^ T$$ for fixed $$mathrm {X}$$, $$mathrm {Y}$$Y $$mathrm {Z}$$, but varying $$mathrm {T}$$ it forms a single ray that passes through the center of the projection point and, therefore, the entire mapping to the same point. Thus, the final coordinates of $$( mathrm {X}, mathrm {Y}, mathrm {Z}, mathrm {T})$$ it is irrelevant to where the point is plotted. In fact, the point of the image is the point in $$mathbb {P} ^ 2$$ with homogeneous coordinates $$( mathrm {X}, mathrm {Y}, mathrm {Z}) ^ T$$. Therefore, the mapping can be represented by a mapping of $$3$$Homogeneous coordinates d, represented by a $$3 times 4$$ matrix $$mathrm {P}$$ with the block structure $$P = [I_{3 times 3} | mathbf{0}_3]$$, where $$I_ {3 times 3}$$ is the $$3 times 3$$ identity matrix and $$mathbf {0} _3$$ a zero 3-vector. By taking into account a different projection center and a different projective coordinate frame in the image, it turns out that the most general image projection is represented by an arbitrary image. $$3 times 4$$ rank array $$3$$, acting on the homogeneous coordinates of the point in $$mathbb {P} ^ 3$$ map it to the image point in $$mathbb {P} ^ 2$$. This matrix $$mathrm {P}$$ It is known as the camera's matrix.

In summary, the action of a projective camera at a point in space can be expressed in terms of a linear mapping of homogeneous coordinates as

$$begin {bmatrix} X \ Y \ w end {bmatrix} = mathrm {P} _ {3 times 4} begin {bmatrix} mathrm {X} \ mathrm {Y} \ mathrm {Z} \ mathrm {T} \ end {bmatrix}$$

Also, if all the points are in a plane (we can choose this as the plane $$mathrm {Z} = 0$$) then the linear mapping is reduced to

$$begin {bmatrix} X \ Y \ w end {bmatrix} = mathrm {H} _ {3 times 3} begin {bmatrix} mathrm {X} \ mathrm {Y} \ mathrm {T} \ end {bmatrix}$$

Which is a projective transformation.

The mentioned section of the textbook is freely available here.

This are my questions:

1. Where it says

Thus, the final coordinates of $$( mathrm {X}, mathrm {Y}, mathrm {Z}, mathrm {T})$$ it is irrelevant to where the point is plotted.

it should not be the vector $$( mathrm {X}, mathrm {Y}, mathrm {Z}, mathrm {T}) ^ T$$?

1. What is it $$mathrm {H} _ {3 times 3}$$ it's supposed to be?

I would really appreciate it if people take the time to clarify them.