Conics in Projective Geometry

The study of conics begun in the 430 B.C. as a result of Athenians in order to put a stop to a plague by trying to duplicate the size of Apollo's cubical alter instructed by the oracle at Delos. Hence, the study began whilst trying to duplicate the cube which is one of the three famous problems from the antiquity that includes also squaring the circle and trisection of the angle. Attempts that were made included placing an end cube right to the original one or doubling the edge length that actually produced a cube of eight times the value of the original one that made pestilence worse than ever.

Hippocrates of Chios (460 BC – 370 BC) first explained that to solve this problem one needed to multiply the edge length by the 2^(1/3). Archytas (428–347 BC) round 400 BC using twisted curve gave the geometrical solution. However round 340 BC Menaechmus (380–320 BC) gave a simpler solution using conics that triggered exploration of the conics for the next six or seven centuries. Apollonius of Perga (262 BC–190 BC) contributed first to the base theory of all three conic sections of one circular cone, right or oblique. He also gave the names ellipse, hyperbola and parabola. After that the conics were not explored until the 17th century when Johann Kepler (1571 – 1630) in his exploration of planetary notion showed that parabola can be considered as a limiting case of an ellipse and of a hyperbola. Another name comes out and that is the name of Blaise Pascal (1623–1662) who discovered a projective property of a circle that holds as well for any type of conic.

Around the 15th century during Renaissance, artists developed a theory of perspective in order to realistically paint on a paper two-dimensional representation of three-dimensional scenes. This theory described the projection of points in the scene onto the artist's canvas by lines from those points to a fixed viewing point in the artist's eye. The mathematical formulation of this theory was named the projective geometry. In this technique of projection, parallel lines that lie in a plane are painted as meeting on a point of a horizon. That suggested a difference from the Euclidean geometry in which parallel lines meet at infinity at the infinity point.

Hence, projective geometry is a branch of geometry dealing with the properties and invariants of geometric figures under projection. In older literature, projective geometry is sometimes called "higher geometry," "geometry of position," or "descriptive geometry".

As every mathematical theory, this one is also built on axioms. The axioms of projective geometry are:

1. If A and B are distinct points on a plane, there is at least one line containing both A and B.

2. If A and B are distinct points on a plane, there is not more than one line containing both A and B.

3. Any two lines in a plane have at least one point of the plane (which may be the point at infinity) in common.

The most amazing result arising in projective geometry is the duality principle. The two-dimensional principle of duality asserts that every definition remains significant, and every theorem remains true, when interchanging the words point and line. One of the beauties of the projective geometry therefore lies in the fact when prove a theorem on some property we can immediately assert the dual theorem. This one can be written down automatically by dualizing each step in the proof of the original theorem. For instance, duality exists between theorems such as Pascal's theorem and Brianchon's theorem which allows one to be instantly transformed into the other that are mentioned here afterwards.

Here the goals is to explore conic in the projective geometry. First let us assert conics in the Euclidean geometry.

As mentioned in the introduction, the conic sections are the non degenerate curves generated by the intersections of a plane with a cone. When a plane is perpendicular to the axis of a cone, we get a circle. When a plane is not perpendicular to the axis we get either an ellipse or a parabola or a hyperbola. Parabola or ellipse are obtained when the plane intersects a single cone, and hyperbola when it intersect a double-cone. In absence of a cone one can demonstrate this easily using a green board and light source.

However, conic sections may be more formally defined as the locus of a point T in the plane P that moves in it of a focus F(fixed point) and a directrix d (fixed line) such that the ratio of d(T, F):d(d, F)=e. Hence, the ratio is the constant e called the eccentricity. The classification of conic sections depends on e. When e=0, the conic is a circle. For 0<e<1, the conic is an ellipse. For e=1 the conic is a parabola, whereas when e>1 the conic is a hyperbola.

A point conic is the set of points of intersection of corresponding lines of two projectively, but not perspectively, related pencils of lines with distinct centers.

Dually, a line conic is the set of lines that join corresponding points of two projectively, but not perspectively, related pencils of points with distinct axes.

Hence, from here we can see one of the differences. In projective geometry there is only one kind of conic. We shall see afterwards that distinction between ellipse, parabola and hyperbola can be made by assigning a special role to the line at infinity.

A pencil of lines usually means a set of straight lines that are incident with one point. Dually, a pencil of points usually means a set of points that are concurrent with one line.

A perspectivity is a correspondence between two line segment ranges that are sections of one pencil by two distinct lines. The product of any number of perspectivities.

Fundamental theorem of projective geometry states that the projectivity is determined when three collinear points and the corresponding three collinear points are given. But because of the duality the three collinear points can be replaced by three concurrent lines.

Using the just stated dual of the Fundamental Theorem, we know that a projectivity between two pencils of lines is uniquely determined by three pairs of corresponding lines. Thus, the definition of a point conic implies that given any three pairs of corresponding lines a unique projectivity is determined. Further, three points of a point conic are also determined.

Looking at the picture above we can conclude that if P and P' are centers of the two pencils, then both points lie in the point conic.

Let P and P' be the centers of two pencils of lines defining a point conic. Let p be the line connecting points P and P'. We can now conclude that p is a line in the pencil with center P. Properties of projectivity assure us that since the two pencils of lines are projectively related, there is a line p' corresponding to p in the pencil of lines with center P'. Since the pencils of lines are not perspectively related, p and p' are distinct. Hence, by definition of point conic mentioned above, P' = p · p' is a point of the point conic. The argument is the similar for P. Now let p' = PP' be a line in the pencil with center P'. Thus, there is a distinct line p corresponding to p' in the pencil of lines with center P. Therefore, P = p · p' is also a point of the point conic. Therefore, P and P' are both points in the point conic. (*)

Hence, we can easily obtain five points of a point conic. But one can ask whether those five points no three collinear determine a point conic?

We investigate the question by letting P, P', Q, R, S be five distinct points, no three of which are collinear. The lines PQ, PR, PS and lines P'Q, P'R, P'Sare two pencils of lines with centers P and P', respectively. By the dual of the Fundamental Theorem, there is a unique projectivity between the two pencils of lines. Since Q, R, and S are intersections of corresponding lines and are non collinear, the projectivity is not a perspectivity.Hence, by the definition of a point conic and (*) P, P', Q, R, share points of a point conic.

Therefore, we can state that any five distinct points, no three collinear, determine a point conic where two of the points are the centers of the respective pencils of lines.

However, we have chosen two arbitrary points P and P' to be the centers of the respective pencils. Does this affect point conics? If so, how? Do we get the same one or it depends on different choice for the centers? Big question arises: Do any five points, no three collinear, determine a unique point conic?

Introducing hexagon and famous Pascal Theorem will help us answer that question.

A simple hexagon ABCDEF is a set of six distinct points, no three collinear, called vertices, and the six lines AB, BC, CD, DE, EF and FA called sides. Two vertices of a hexagon are said to be adjacent, alternate or opposite according as they are separated by one side, two sides, or three sides. For instance F and B are adjacent to A, E and C are alternate to A, and D is opposite to A. The join of two opposite vertices is called a diagonal. Hence, ABCDEF has three diagonals: AD, BE and CF. Also, it has three pairs of opposite sides: AB and DE, BC and EF, and CD and FA.

A given hexagon can be named in twelve ways. Any one of its six vertices can be named A, either of the two adjacent vertices can be named B, and rest are then determined by the alphabetical order.

Six given points, no three collinear can be named A, B, C, D, E and F in 6!=720 ways. However since they have to be distinct then the number of those hexagons is equal to 720/12=60.

Figure bellow shows three out of the sixty hexagons determined by six points on a circle. This results naturally not having only convex hexagons as we are used to.

One of the most important theorems of the plane geometry is called Pascal's Theorem after Blaise Pascal (1623-1662) who made this statement at the age of sixteen. Even though his proof was never found, it was seen and praised by Leibniz and carries his name

Pappus Theorem. If all six vertices of a hexagon lie on a circle and the three pairs of the opposite sides intersect, then the three points of intersection are collinear.

Menelaus's Theorem. If points X, Y, Z on the sides BC, CA, AB of triangle ABC are collinear, then

Proposition. If two lines through a point P meet a circle at points A, A' and B, B', respectively, then PA x PA'=PB x PB'. (**)

We want to prove that G=BC·EF, H=AB·DE and J=CD·FA are collinear. Applying the Menelaus's Theorem to the triads of points HDE, AJF, BGC on the sides of the triangle KIL, we get that

Multiplying those expression and using (**) we get,

From that we get . Hence, points G, H and J are collinear.

Q. E. D.

The line containing points G, H and J is called the Pascal line of the hexagon ABCDEF.

Because of the Principle of Duality we obtain Brianchon's Thereon. If all six sides of a hexagon touch a circle, the three diagonals are concurrent.

The converse of the Pascal's Theorem was proved independently by William Braikenridge and Collin MacLaurin in the 18th century.

Braikenridge-MacLaurin Theorem. If the three pairs of opposite sides of a hexagon meet at three collinear points, then the six vertices lie on a conic, which may degenerate into a pair of lines.

We are given five points on a conic. In order to construct more points on this conic, we will need to construct a projectivity between sets of three concurrent lines. For this, we use the dual of the construction which gave us a projectivity between sets of three collinear points. The diagram below shows how we do this. Link to a Geometer's Sketchpad document that will do it dynamically for you.

Let a, b, c be lines from P to Q, R, S. Let a', b', c' be lines from P' to Q, R, S.

Let y be a line on the point Y=a'·c. Since GSP does not like arbitrary elements I translated point P and obtained line y as a line segment through P' translated and Y.
Let x be the line (a·a')(b·y), and z the line (y·b')(c·c').

Now, given any line o on P, use x to project it onto the point Y. Use z to project this new line f onto the point P', and call the result of this projection line o'. Our conic is the locus of intersections of o and o', as o becomes each line on the point P.

In order to construct our conic, we needed to construct a projectivity that sent a=PQ to a'=P'Q, b=PR to b'=P'R, and c=PS to c'=P'S. Because there is only one such projectivity, any construction giving those three correspondences will yield the same projectivity. In particular, this shows that our conic did not depend on our choice of line y. Our choice of line did affect the particulars of our construction, but it did not affect the projectivity or the conic.

If you use this graphic in Geometer's Sketchpad and show all hidden objects, you may see a circle. This is just a convenient way of generating the set of all lines lying on a point.

Hence, five points in a plane determine a conic. Dually, five tangent lines in a plane determine a conic.

By playing with the script tool you can see that depending on points chosen you get different point conics.

The general equation for a conic is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. D is the determinant of a conic section and is defined as follows:

"The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea." - George Pólya.

Coxeter, H. S. M. (1987), 1907-2003. Projective geometry. New York: Springer-Verlag.

Coxeter, H. S. M., & Greitzer, S. L. (1967). Geometry Revisited. Washington, DC: Math. Assoc. Amer.

Veblen, O., & Young, J. W. (1917). Projective geometry. Boston, MA: Ginn and Co.

Weisstein, Eric W. "Projective Geometry." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ProjectiveGeometry.html

Ellipse	Hyperbola	Parabola
d>0 C·d>0	d<0 C·d<0	d=0

Ellipse	Hyperbola	Parabola

Hyperbola	Data

Parabola	Data