Homogeneous coordinates have one dimension more than the corresponding Euclidean coordinates. The Euclidean origin can be described with projective coordinates as (0,0,0,1). So, geometrically, what would be an interpretation of the homogeneous coordinates (0,0,0,8) ? This is clearly a point in the Euclidean space, but how does it relate to the origin? What I am trying to say is: How does the last component of homogeneous coordinates relate to Euclidean coordinates?
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1$\begingroup$ It's still the origin! Note: $v$ and $\lambda v$ represent the same point if $\lambda\ne 0$. $\endgroup$– BerciMar 10, 2019 at 17:14
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$\begingroup$ Thx, Berci! Don't we need to distinguish tho if we are talking about a point or a plane? Say (0,0,0,1) is a point in projective coordinates. This point corresponds to the Euclidean origin. Now, let's say (0,0,0,1) is a plane in projective coordinates. This corresponds to a plane at infinity in Euclidean space, doesn't it? What would a plane (0,4,0,8) correspond to? $\endgroup$– LukMar 11, 2019 at 8:36
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$\begingroup$ Well, there's indeed a duality between points and planes in $\Bbb RP^3$, given by orthogonality of $\Bbb R^4$. $\endgroup$– BerciMar 11, 2019 at 10:55
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