Stefan-Liebscher.DESte-Lie.DE Meine kleine «Ich-AG»

Projective Geometry


La géométrie est une espèce de hochet que la nature nous a jeté pour nous consoler et nous amuser dans les ténèbres.

(Jean-Baptiste le Rond d'Alembert to Friedrich II., 1764)

The Geometry is a kind of toy which nature has given to us to console and amuse us in times of darkness.


Stefan Liebscher: Projektive Geometrie der Ebene

Meanwhile, my textbook «Projektive Geometrie der Ebene» (in German) has been published by Springer.

JavaScript application to interactively visualise the real projective plane

This program started as a small JavaScript exercise, just a little more than the infamous «Hello World»... It has grown into an interactive visualization of the real projective plane. It provides elementary constructions of points, lines and conic sections. The selection of an absolute conic — as the set of points at infinity — defines a metric and embeds the Cayley-Klein geometries into the projective plane. This defines midpoints, angular bisectors, perpendiculars, reflections, circles, and more.

The projective transformation which is used to visualize the projective plane can be changed by mouse or touch gestures:

left-button drag
shift the drawing plane
middle-button drag
rotate the drawing plane
right-button drag
tilt the drawing plane
mouse wheel
scale the drawing plane
double-touch drag
shift/rotate/scale the drawing plane
triple-touch drag
tilt the drawing plane

There is more projective geometry with a different Java applet on the web page of my father.


Conic sections
Conic sections can be identified by their points as well as their tangent bundles. This provides natural unfoldings of the singular conic sections.
Conic pencils and Dual conic pencils
Relations of pairs (or pencils) of conics can be classified by their diagonal triangles. This animation shows all possible cases.
Round trip through the geometries
Using the example of the common tangents and focal points of a pair of conic sections, all Cayley-Klein geometries are traversed

Projective constructions (without metric)

The complete quadrangle
The complete quadrangle (and its dual counterpart, the complete quadrilateral) constitutes the fundamental object on the projective plane and defines harmonic relations.
Conic sections and more conic sections
Five points (or 5 tangent lines) in general position uniquely define a conic section. One tangent lines and 4 points define 2 conic sections, two tangent lines and 3 points define 4 conic sections. Two conic sections have 4 common points.
Conic pencil, dual conic pencil, pencil of circles and pencil of horocycles
The conic sections through 4 given points form a linear family — a pencil of conics. The dual conic family is given by its 4 common tangent lines. Two conics are circles to each other if they touch in two points, i.e. if two pairs of intersections and two pairs of tangents collide. Finally, horocycles are circles whose two touching points collide.
Theorems of Desargues, Pappus, Pascal, Brianchon and the three conics
The projective plane is crowded with collinear points and concurrent lines.
The focal points of two conic sections as the vertices of the complete quadrilateral of their common tangents
Two conic sections have 4 points of intersection and 4 common tangent lines. They define a complete quadrangle and a complete quadrilateral. The six focal points are given as the 3 pairs of opposite vertices of the tangent quadrilateral. The diagonal points of the quadrangle of intersection and the diagonal lines of the tangent quadrilateral form a (common) triangle. This diagonal triangle is self-polar to both conics.

Metric constructions (depending on the absolute conic)

Circumferences as well as in- and ex-circles of a triangle
A triangle possesses four circumferences (circles through the vertices) as well as four in- and ex-circles (tangential to the edges). Some of them may degenerate or become complex. On the Euclidean plane, for example, three of the four circumferences degenerate to pairs of parallel lines.
Circles given by two points and a tangent line and two tangent lines and a point
Two peripheral points and a tangent line as well as two tangent lines and one peripheral point define four circles, respectively. Again, circles might degenerate or become complex.
The nine-point conic
In the flat geometries (Euclidean, Minkowskian and Galilean), the six midpoints and the three diagonal points of every complete quadrilateral lie on a common conic section.
The Feuerbach circle
In the flat geometries (Euclidean, Minkowskian and Galilean), the nine-point conic of a triangle augmented by its orthocenter becomes a circle. It touches all 16 in- and ex-circles of the 4 individual triangles.
The gardener's construction of an ellipse
The gardener's construction of a (planar) ellipse uses the constant sum of the distances of peripheral points to a pair of focal points. This is in fact a general property of conic sections or, more precisely, of the pair of conic and absolute conic:
  • The pole of every secant through a focal point lies on the orthogonal to the secant through the focal point.
  • Each secant through a focal point in reflected in the tangents through its intersections with the conic onto a secant through the opposite focal point.
  • The images of a focal point, reflected in the tangents of the conic, form a circle around the the opposite focal point.
The last two properties indeed imply the constant sum (or difference) of the distances to a pair of focal points.
Three focus-sharing ellipses
Let a triangle on the Eucliden plan be given, such that each pair of vertices are the focal points of an ellipse. Let each pair of ellipses intersect in two points. Then, the three common secants are concurrent, i.e. go through a common point! In fact, this is not the full truth: Each two conic sections have four intersections, which may become complex. However, secants through conjugate complex points remain real. The three (suitable chosen) pairs of common secants of the three ellipses form a complete quadrilateral. We find four triple-points of common secants. This general fact holds true for arbitrary conic sections sharing their focal points with respect to an arbitrary absolute conic.