Projective geometry focuses on relationships between points and lines.[br]It projects an object based on your view from a particular angle or perspective.

Some properties never change despite how the object is projected.[br]They are called [b]invariant [/b]- does not vary or change.[br][br]Invariants in projective geometry:[br][list=1][*][b]Infinity[/b]: In projective geometry, we add an "ideal point" to every line. Parallel lines are simply lines that intersect at this point at infinity.[/*][*][b]Collinearity[/b]: If three points lie on a single line, they will always lie on a single line, no matter how you project them.[/*][*][b]Cross-Ratio[/b]: This is the "Golden Rule" of projective geometry. For four collinear points [i]A[/i], [i]B[/i], [i]C[/i], and [i]D[/i], the cross-ratio is defined as:[/*][/list][center][math]Cr\left(A,B,C,D\right)=\frac{AC\cdot BD}{BC\cdot AD}[/math][br]No matter how you stretch or tilt the line, this value stays the same[/center]Observe the above invariants using the applet below.

[list=1][*]Axiom 1: For every two distinct points there is one distinct line that passes through them.[/*][*]Axiom 2: Any two distinct lines intersect with at least one point in common (i.e., no parallel lines).[/*][*]Axiom 3: Three points that do not lie on one line are contained in a unique plane.[/*][*]Axiom 4: Three planes that do not contain a common line contain a unique common point.[/*][/list]

The "magic trick" of projective geometry: "[b]points and lines are interchangeable[/b]"[br][br]Select a true statement about points and lines. Try to swap the word "point" with "line". [br]You also have to swap their actions. For example,[br]"lies on" becomes "passes through,"[br]"collinear" becomes "concurrent,"[br]"intersection of lines" becomes "join of points"[br]Check if the new statement is also true.[br][br]Example 1:[br]Original statement: Two points define a line.[br]Dual statement: Two lines define a point.[br][br]Example 2:[br]Original Statement: A triangle is a set of three points not on the same straight line, but connected by lines.[br]Dual Statement: A triangle is a set of three lines not passing through the same point, but connected by points.

A Greek mathematician, Pappus of Alexandria, found that three intersection points will always lie on a single straight line (collinear).[br]This became the starting point for projective geometry.[br][br]No matter how you angled the lines or where you placed the points, the three intersection points (X, Y, Z) will always lie on a perfectly straight line.[br][br]Observe the applet below.

A French engineer, Girard Desargues, discovered about perspective triangles:[br]If two triangles are in perspective from a point, then their corresponding sides intersect at three collinear points.[br][br]Suppose triangles [math]\bigtriangleup ABC[/math] and [math]\bigtriangleup A'B'C'[/math] are perspective from point P. If the corresponding sides of the triangles intersect (AC meet A’C’ at point X, BC meet B’C’ at point Y, and BA meet B’A’ at point Z),[br]then the points of intersection X, Y, and Z are collinear.[br][br]Observe the applet below.

Projective Conic Sections: In Euclidean geometry, a circle, an ellipse, a parabola, and a hyperbola are all different shapes.[br][br]In projective geometry, they are all the same thing. If you shine a flashlight (a projection) on a circle at different angles, the shadow can become an ellipse, a parabola, or a hyperbola. Because they can all be projected into one another, they are considered projectively equivalent.