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Projective Geometry

A geometry which begins with the ordinary points and lines of Euclidean plane geometry, and adds an ideal line consisting of ideal points which are considered the intersections of parallel lines.  Thus each equivalence class of parallel lines contains one of these ideal points, which is defined in projective geometry as the intersection of these parallel lines.  The ordinary and ideal points, together, comprise the real projective plane.  Then, for three dimensions, we add an ideal plane consisting of one ideal line for each equivalence class of parallel planes, each of which is the intersection of those planes.

Axioms of Real Projective Plane Geometry

  1. There exists at least one line.
  2. On each line there exist at least three points.
  3. Not all points lie on the same line.
  4. Two distinct points lie on (and determine) exactly one line.
  5. Two distinct lines meet in (and determine) exactly one point.
  6. There is a one-to-one correspondence between the real numbers and all but one point of a line.

The Principle of Plane Duality

Statement or phrase Dual of the statement or phrase
point line
a point lies on a line a line lies on (i.e. contains) a point
There exists at least one line There exists at least one point
On each line there are at least three points Through each point there are at least three lines
Not all points lie on the same line Not all lines pass through the same point
Two distinct points determine a unique line Two distinct lines determine a unique point
Three (or any number of) non-collinear points Three (or any number of) non-concurrent lines
Self Dual: A polygon is a coplanar multiset (i.e. set whose elements are not necessarily distinct) of points (vertices) and lines (sides, or edges) such that each vertex is at the intersection of exactly two sides, and each side contains exactly two vertices. Self Dual: A polygon is a coplanar multiset of lines (sides, or edges) and points (vertices) such that each side is determined by exactly two vertices, and each vertex is determined by exactly two sides.
A polygon is inscribed in a figure when all of its vertices lie on the figure. A polygon is circumscribed about the figure when all of its sides are tangent to the figure.

The Principle of Space Duality

Statement or phrase Dual of the statement or phrase
point plane
a line lies on (i.e. contains) a point a line lies in a plane
two distinct points determine a unique line two distinct planes determine (meet in) a unique line
three distinct points not all on the same line determine a unique plane three distinct planes not all containing the same line determine (meet in) a unique point
if two lines lie on a unique plane, they determine in a unique point if two lines lie on (i.e. contain) a unique point, they determine a unique plane
a line and a plane not containing the line meet in a unique point a line and a point not on the line determine a unique plane
a triangle is three coplanar lines that are determined by three non-collinear points, which lie in a plane the triangle's space dual is three concurrent lines that are determined by three non-collinear planes, which meet in a point
The platonic solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, which have 4, 6, 8, 12, and 20 faces, respectively. The platonic solids are the tetrahedron (self-dual), octahedron, cube, icosahedron, and dodecahedron, which have 4, 6, 8, 12, and 20 vertices, respectively.

To make the dual terminology of a point lying on a line vs. a line passing through a point more nearly parallel, some authorities use the expression "incident to" to mean either one.  For example, three collinear points are said to be incident to a particular line, and three concurrent lines are said to be incident to a particular point.

Measurements

You win some; you lose some.  With projective geometry we gain the ability to use duals, including statements about duals, and complete proofs about duals to get two proofs in one.  We also gain the simplification of geometry in which parallel lines and intersecting lines are not distinguished; quite the contrary, all pairs of lines in a plane intersect in exactly one point.

But we lose some, too.  Since we added a point to every line -- the so-called "ideal" point -- we don't retain order, hence we lose measurement.  The sides of a triangle are no longer line segments contained "between" a pair of points, because the whole idea of "between" is lost; a triangle is the three lines, not just segments of lines.

It has been pointed out that the origin of the term "geometry" comes from geo (Earth) and meter (measurement), so "projective geometry" is something of an oxymoron, since all measurement is lost.

Conic Sections

Having lost all sense of measurement, the conic sections are indistinguishable from one another in projective geometry.  That's actually a good thing, because it makes every theorem proved purely using projective geometry that applies to, say, a circle much more general.  Without further work, the same theorem applies to an ellipse, a hyperbola, a parabola, and even a pair of intersecting lines, which include parallel lines, since they intersect as well!  For that reason, Pascal's Theorem and its dual, Brianchon's Theorem apply equally to any conic section, and are a generalization of Pappus' theorem, which only applies to a pair of intersecting lines.

Related pages in this website:

A list of theorems in geometry

Platonic solids, with illustrations, and descriptions of their space duals.

References

Cut-the-knot: Projective Geometry 

 

The webmaster and author of this Math Help site is Graeme McRae.