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Functions and linearity (1)

Defined the function

decomposing each variable with a linear combination :

where is the coefficient for the combination,

we get

and finally

Functions and linearity (2)

Once defined the linear combination


applying an affine transformation

we get

Perspective correct mapping

In the system of coordinates where the observer O is in the point
and the projection screen has equation Y=d:

Given the real points  we have their perspective projections ;
taking a linear combinations of them  we can locate the corresponding projected real point on the real line joining  .

Using the notation for each coordinates of the new real point
and for the original real points  ,
we have the Perspective correct mapping :

Wich is very different from the simple Affine texture mapping :