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2D transformation and viewing: The camera projects the 2D world co-ordinate $(x_w, y_w)$ on its projection plane which co-ordinates $(x_v, y_v)$.

3D transformation and viewing: The camera projects the 3D world co-ordinate $(x_w, y_w, z_w)$ on its projection plane which co-ordinates $(x_v, y_v, z_v)$.Then $(x_v, y_v, z_v)$ can be transforms to $(x_v, y_v)$ of display device coordinates.

My questions are:

$A.$ In 2D transformation and viewing projection plane is 2D means it has 2 dimensions height and width. But I am confused projection plane which is a plane (2D) has equation is of the form $ax+by+cz+d=0.$ But here $z$ coordinate is missing, so how will it be plane?

In contrast 3D has projection plane means it has 3 dimensions height, width and depth. But we know that 2D plane $ax+by+cz+d=0.$ But how we're saying 3D? I am struggling to differentiate projection plane and dimensions. What I have misunderstood in the concepts of both cases?

$B.$ In 2D transformation and viewing how world co-ordinate $(x_w, y_w)$ is 2D? Because we know real world is 3D.

$C. $ In perspective projection how we will represents vanishing point coordinates in projection plane? For example I catch one image of railway tracks where two parallel lines intersects. During projection what could be it's world co-ordinate and projection plane coordinates respectively? It's normal form of projection from $(x_w, y_w, z_w)$ to $(x_p, y_p,z_{vp})$ in projection plane?

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  • $\begingroup$ 2d to 2d is not a projection. Theres no reason to do anything. A) Only one plane is possible in 2d. Its the entire 2d world. B. is not defined. Its teh application that defines how to do this, in many cases its just a plane but it can also be a origami etc etc. $\endgroup$
    – joojaa
    Oct 28, 2021 at 13:51
  • $\begingroup$ "In perspective projection how we will represents vanishing point coordinates in projection plane?" Why do you need to? $\endgroup$ Oct 28, 2021 at 16:53
  • $\begingroup$ @nicol I just want to know the coordinates of vanishing point.. $\endgroup$
    – user17337
    Oct 28, 2021 at 16:55
  • $\begingroup$ @User4567: The vanishing point in what space? And again, why? I've been doing 3D graphics for a while now, and I've never needed to know where the "vanishing point" is. $\endgroup$ Oct 28, 2021 at 16:56
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    $\begingroup$ @User4567 because the objects at infinity get scaled to infinitely small space. But in reality as a mathematical point it does not exsist even in 2D. Since we have nothing that extends to infinity. Its just that the continuation of said lines conceptually intersect. It is not a literal concept but a imaginary one. Its more like if Y then we would have X. But since Y does not actually exsist X does not either, but since Y nearly exsists its still in some cases useful to pretend that X exists. If you want to find said point you need to either do it in 2D or pretend that it exists $\endgroup$
    – joojaa
    Oct 29, 2021 at 5:34

1 Answer 1

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There is a great deal of literalism in your understanding of concepts.

A "world coordinate" is a term for a coordinate that is in the space you have designated to represent "the world". Whatever that means to your application. Different objects may have different native coordinate systems, but they will be transformed into a common coordinate system, of which you will render some subsection of. We call that coordinate system "world coordinates".

That's it. That's what it means. It doesn't really have anything to do with "the world" in any literal sense.

It can be whatever numerical coordinate system works best for your needs. It can have however many dimensions you feel comfortable giving it.

Similarly, look at your 2D/3D confusion. What is "2D"? If all of your 3D coordinates and transformations and the like all have a Z of zero, is that not exactly equivalent to "2D"?

Any X-dimension is a special-case of the X+1-dimension where the +1 coordinate is 0 and is unchanged by any transformations. So if it makes you feel more comfortable to think of 2D as 3D with a zero Z, where you're projecting these "3D" coordinates onto a plane at zero Z perpendicular to the Z axis, then you can do that and the math still works out.

But really, you should just think of it as 2D. If the source and destination dimensions of your transformation are the same, then it's not a projection. It's just a generic transformation.

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  • $\begingroup$ What about ax+by+cz+d=0 in2D and 3D? $\endgroup$
    – user17337
    Oct 28, 2021 at 17:10
  • $\begingroup$ @User4567: What about it? What happens to that equation when Z is zero? $\endgroup$ Oct 28, 2021 at 17:34
  • $\begingroup$ "If the source and destination dimensions of your transformation are the same, then it's not a projection."---why it's not projection? During projection dimensions also remains same? $\endgroup$
    – user17337
    Oct 28, 2021 at 17:37
  • $\begingroup$ please say when take photo by camera in 2D , it is transformation? But doing same thing in 3D is called projection?? What's the intuition? Please explain? $\endgroup$
    – user17337
    Oct 28, 2021 at 17:53
  • $\begingroup$ @User4567: "why it's not projection?" Because that's the definition of "projection". As previously discussed. If the transformation doesn't decrease the dimensionality, then it's not a projection. $\endgroup$ Oct 28, 2021 at 18:54

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