Home Site Map Tutorial Material Elements (1) Elements (2) Proscribed Ops Absolutes(1) Absolutes(2) Absolutes(3) Projection
Seeing uses Local Central Projection. PerspectiveThinking uses Central Projection from Infinity. 

You may left click and drag the blue dashed line to swing the building around one corner. The roof apex here is in a vertical plane that exactly bisects the building longitudinally. 
For “Accurate” Perspective –
If the conics “on the ground” in the scene on the left are,
“in objective, physical fact”, concentric circles, with radii r, 2r, 3r, ... ,
and the bold blue line with the “vanishing” points VP_{1} and VP_{2} is at the infinite,
and is the polar of the common centre of the circles,
these points are conjugates in involution, and,
if the third vanishing point, VP_{3},“below” the glass house also lies in the infinite, the following metric statements are held to be correct –
A fanciful Cartesian Coordinate system — on legs! 

Making a ScenePerspective derives from the Axioms of Incidence

Just Elementary Incidence
Formally
Any two lines, such as a and b, in a point, such as E, have a plane in common, in this case π_{ab}, which plane is also common to lines u_{1} and u_{2}, So we have three planes, π_{1}, π_{2} and π_{ab} Their pointincommon is C, which lies in the line c, common to π_{1} and π_{2}. 

The General Picture
IFF (‘IF and only iF’) the three conics
(here red, green and blue) enclosed in the “box” above and tangent to its faces are all “circles in
absolute/physical reality”, and IFF they have a “common centre”, then the box is a cube, and each of the three planes of the circles
stands at right angles to the other two.
Notice that, purely projectively, the common centre is the common (degenerate) pole of the three lines (here, the red, green and blue “dotdash” lines) in which the three conics’ planes cut a fourth plane, and these three lines are the three polars in that plane of this degenerate pole. These projective things are the case (by the axioms of elementary incidence), whether or not the conics are “absolute” circles, and whether or not the fourth plane is “absolutely” at infinity. So, IFF the fourth plane lies at infinity, the three polar lines in it meet by twos in the three vanishing points of the three pairs of parallel cuboid faces. IFF the conics are absolute circles, then the cuboid is a regular cube, and the circles are sections through the centre of the same absolute sphere, so the thing is isotropic, and we appear to have the basis of a coordinate system, in perspective.
I stress again that all these things about absolutes and
infinity I reiterate that they do not arise from elementary incidence,


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