• Projective geometry - explains how an eye perceives the “real world”

3.1 Perspective

• Terraced perspective - a technique of drawing people at the back of a group higher up than those in the front

• Vertical perspective - An artist would draw pairs of parallel lines place symmetrically on either side of the scene by lines that meet on the center line of the picture

• Focused perspective - A cross section can give the same impression as the original item due to the corresponding points followed by light rays

3.1.2

• Mathematical perspective - all lines through a given point in R^3

  • Not all figures are preserved under perspectivity

• Diagonal vanishing point of the perspectivity - The point in which the images of all lines in a plane in some direction vanish

3.1.3 Desargues’ Theorem

• The idea that information in 3D can be related to information in 2D

• Deargues’ theorem - A 3D figure has properties that allow the points where the image meets the figure to be collinear

  • Example : Triangle ABC and triangle DEF are in R^2. Lines AD BE and CF meet at point U. Lines BC and EF meet at P. CA and FD meet at Q. AB and DE meet at R. P,Q, and R are collinear

3.2 The Projective Plane RP^2

• The correspondence between points of the screen and rays of light through the origin are used to define a new space of points

• Projective point - A line in R^3 that passes through the origin

• These points are notated as [a, b, c]. It represents a unique line in R^3 that passes through (0,0,0) and (a,b,c)

• Projective figure - a subset of RP^2

3.2.2 Projective Lines

• A plane through the origin is considered a projective figure because it contains all point that lie on the plane, and these points are rays from the origin. This defines the plane as a projective line