Projective geometry was developed prior to the 20th Century and there are two main approaches: algebraic and synthetic.  The former method maps projective objects and processes onto linear algebra,  using matrices and vectors, which at one time led to the view that the subject was "dead" as all that could be said was said by that algebra.  The synthetic method uses the axioms of incidence to study the subject directly in terms of points, lines and planes and is essentially pictorial in nature.  The latter approach will be adopted here, and it will be seen that far from exhausting the subject, the algebraic approach may even obscure important developments if care is not taken.  That is not intended, however, to detract from the importance of the latter.

We will not attempt to develop the ideas with full rigour as that has been done elsewhere and need not take the excessive amount of space required on a page such as this, so an informal approach will be presented to give some intuitive insight.  Proofs may be found in the references cited. 

Essentially the synthetic approach is based on the notion of incidence, so that two distinct lines in space are incident if they intersect one another, in which case they do so in exactly one point and one plane.  A line and plane in space are always incident in one point, assuming the line does not lie wholly in the plane.  By line we mean a straight line, and by plane a flat plane, the concepts straight and flat being fundamental and undefined.  The two most fundamental operations are projection and section, illustrated below:

FIGURE 1

On the left we start with the red line, choose a centre of projection S, and project its points onto the blue line.  It is clear that for any point such as A on the red line there is a unique corresponding point A on the blue line and vice versa.   We may instead, on the right, start with a red point P and a line s, and then project the lines through P into lines through another blue point Q using s, as shown.  The process of section is the generating of points on s by its intersections with the red lines.  The systematic naming of the points and lines illustrates the principle of duality, which says that anything you can do with points and lines can equally well be done with the roles of the lines and points interchanged.  We could even label the black lines on the left as a b c d, and the points of intersection of the blue and red lines on s on the right as A B C D to further reinforce this symmetry.  In three dimensions we find that points and planes are dual, while lines are self-dual.  This is illustrated below:

FIGURE 2

On the left we start with an axial pencil of blue planes (i.e. they share a green line) and a black plane sigma.   The blue planes meet the plane sigma in four black lines forming a flat pencil (lines in a plane through a common point).  A is the centre of this pencil.  We take a second green line through A as the axis of a second axial pencil, the planes corresponding to the blue ones being shown in red.  Now we dualise this on the right:- first take the black point S dual to sigma and another plane alpha dual to the point A, which must contain S as A lies in sigma.  We now start with a blue range of points i.e. a set of points on a green line dual to the axis of the initial blue axial pencil.  This line lies in alpha because that axis contains A.   We then project those points, using a black flat pencil in S, onto another range of red points necessarily lying on a green line in alpha.

This illustrates the principle of duality in three dimensions in respect both of dual elements (points and planes) and dual operations: namely putting two axial pencils of planes in perspective, and putting two ranges of points in perspective.  We say two axial pencils are in perspective if they contain a common flat pencil, and similarly for two ranges if they determine a flat pencil when corresponding points are joined.  The same applies for ranges in two dimensions, but then the dual is that two flat pencils are in perspective if the points of intersection of corresponding lines lie on another line (see Figure 1).

Generally we speak of perspective pencils and ranges.   If one such perspectivity is followed by another so that the second range or pencil is projected onto a third one, then the first and third are said to be projective.   In general they are not in perspective, and in the plane two projective ranges determine the tangents to a conic section when corresponding points are joined.  This is shown below:

FIGURE 3

Essential to the concerns of this page is the process of transformation.   This refers to the systematic re-arrangement of the elements of a point, line or plane (or indeed of other objects in more advanced applications).  Thus we may move all the points on a line 1 cm to the left, or rotate all the lines of an axial pencil through 10 degrees, or we may use the operations of projection and section.  We now illustrate this for the transformation of a line, where nearly all its points are systematically moved to new positions:

FIGURE 4

We start with a point on the horizontal black line and project it (first red line) from a centre on the blue line.  We then take a section using a fixed green line to give a point which we then project back onto the black line using a second centre on the blue line.  This process can be carried out for every point on the black line, and all its points except two are moved to a new position.  However they are not all moved through the same distance, so this is a genuinely projective transformation in contrast to the metric transformation which moved all points through 1 cm.  Following the history of a point as it is repeatedly transformed (as above) shows this clearly.  It will be observed that there are two points which are self-corresponding known as double-points.  A fundamental theorem states that there can be no more than two such points on a line, and if a third is forced to be self-corresponding then all are.  Those two points may coincide in which case we have step-measure instead of our initial breathing-measure (terminology due to George Adams).  This is illustrated below, the case when the double-point is at infinity on the black line being shown on the right which illustrates why it is called step-measure, for there we are back with a metric transformation of equal steps:

Figure 5

It is important to note that making a point at infinity fixed ushers in the metric case.  There is just one point at infinity on a line, sometimes known as an ideal point because in ordinary geometry it does not exist, but in projective geometry it is added to the line.  Generally two distinct lines meet in one point unless they are parallel, but in projective geometry two such lines always meet in just one point, the latter being the ideal point if they are parallel.  In three dimensions two distinct planes always intersect in a line, which is an ideal line if they are parallel.

An intermediate case occurs when one of the two double-points is at infinity:

FIGURE 6

Again the use of infinity enables measurement, as successive steps form a geometric series i.e. successive pairs of points determine distances from the accessible double-point that are always in the same ratio.  This is known as a growth-measure.

Another possibility is that there are no real double-points - known as circling measure - which is most easily constructed using a circle instead of two lines:

FIGURE 7

Just as a quadratic equation may have two roots which are complex numbers e.g. the equation for the intersection of the line and circle above, so in this case we have two so-called imaginary points for the double-points as they cannot be located on the line in the customary sense - even at infinity.  If the circle intersects the line we recover breathing measure, while if it touches it we have step measure.

We now move on to transformations of a whole plane, so that all of its points but three move to new positions.  It can be shown that generally three non-collinear points are self-corresponding, exceptions being the trivial case when all points are fixed (the identity transformation), or when all the points on a line are self-corresponding together with one other point not in the line (known as a homology).  Even in that case the isolated double-point may lie on the fixed line, giving an elation.   The three double-points form an invariant triangle and by using the measures induced by the transformation on two of its sides we may construct such a transformation:

FIGURE 8

We start with two red lines in the plane, and wish to find where they go when the transformation is applied.  The two black lines are two sides of the invariant triangle, and a breathing measure is constructed on each.  This suffices to determine the whole transformation, and by finding where the points go in which the red lines meet the black we can find the brown and dark-cyan lines which are the transforms of the red ones.  To illustrate another point, we took the red lines parallel to show that in general parallelism is not conserved by such a transformation.  We may transform points in the plane similarly, which will be illustrated below.

Just as we followed the history of one point on a line as it is repeatedly transformed, we may do the same thing in the plane:

FIGURE 9

We see that a curve arises as a result, known as a path curve.  For the purposes of research two special cases are of importance: firstly when two sides of the invariant triangle are parallel, which results in growth measures on them and the path curve becomes egg-shaped:

FIGURE 10

We show the dual condstruction where a red line is moved repeatedly to give the tangents to the curve.  This egg form has been shown by Lawrence Edwards to fit the profiles of plant buds, birds eggs, fir cones and more (see the Research Page).   We will look at the quantitative aspect later.

The second important special case occurs when two of the double-points of the invariant triangle are conjugate imaginary, because this helps lead into the three-dimensional case of full egg shells.  It is easiest, and also what we will require, to take them as lying on the line at infinity in the plane.  If in Figure 7 we move the black line to infinity we get the following situation:

FIGURE 11

We have marked the centres of the pencils as P and Q, and we start with the line A' through Q.  This meets the circle in a point through which we draw the line B of the pencil centred on P.  The line through Q meeting this line at infinity is the line B' parallel to it, and the next line C through P meets it on the circle, and so on (c.f. Figure 7 where the process is illustrated for many more lines).  The two angles theta are equal because they are in the same segment, and hence the angles between B and C, B' and C' also equal theta i.e. the transformation causes the lines of the pencils to move successively through equal angles.  This angle theta will prove to be most important when we come to quantitative calculations.  But we could have placed the circle and the points P and Q anywhere in the plane, with the same result!   It follows that a line through the real double-point of the transformation we seek will rotate through equal angles as the transformation is repeated, as this must happen to all pencils in the plane.  The angle theta is the argument of the complex numbers which are the coordinates of the double points.

Now we recall that when one double-point is at infinity (c.f. Figure 6) points move away from the other in geometric progression.  Due to the symmetry of the situation this means that circles centred on the accessible double-point must expand with their radii in geometric progression, so combining this with the rotation of lines shown in Figure 7 we finally have the following transformation:

FIGURE 12

We have shown it superimposed on a Nautilus Pomp. shell; the path curve is a logarithmic spiral.  This is because the the radii of such a spiral increase in geometric progression with equal angles turned.  The first few points on the curve have been related to the construction by the small white circles.

Next we observe that the spiral intersects any radius in points forming a geometric progression, so the top and bottom black lines of Figure 10 may be regarded as planes seen edge on, with a spiral in each plane determining the points of the construction.  We will now use this for a full three dimensional transformation of space.

The logic continues quite simply: in one dimension (i.e. on a line) there are two double-points, in a plane three, and in space four.  This gives rise to an invariant tetrahedron in space, with four invariant planes, four invariant points and six invariant lines.  We show below the progression from a fully real tetrahedron to a semi-imaginary one which will suit our purposes:

FIGURE 13

On the left we have a tetrahedron with a vertical red edge and a horizontal green one.  We move the green edge outwards to infinity as in the centre.   Note that the double-points W and Z at infinity are in both parallel faces as indicated by the arrows.  Finally we replace W and Z by two conjugate imaginary double-points on the green line at infinity, leaving two real double-lines (the red and infinitely distant green), two real blue double-planes, and two real double-points X and Y where the red line meets the blue planes on the right.  Our three-dimensional transformation of space can now be constructed using the spirals in the the blue double planes as in Figure 12, and in each vertical plane containing the red axis we may either have egg profiles as in Figure 10, or vortex profiles as we shall see in a moment.  The result is shown in the following animation:

FIGURE 14

The red plane turns through an angle theta each time the transformation is applied (c.f. explanation of Figure 11), and the points in the top and bottom planes follow round their spirals as the plane rotates.  In each red plane we join the upper double-point X (not marked) to the point on the lower spiral, and Y to that of the upper spiral, and where those two red lines meet is the point in space that we are transforming.  The result is an egg-shaped spiral winding about the vertical axis.  If we took a horizontal circle centred on the axis and passing through the first red point, then each of its points would follow a similar spiral, the aggregate of which would  yield an egg shell surface with vertical profiles as in Figure 10.  This is shown below superimposed on a pine cone:

Notice in Figure 14 that the spirals wind in the opposite sense in the top and bottom planes.  This gives eggs.  If the spirals wind in the same sense then instead we obtain a vortex form:

There are two special cases, obtained if either the top or bottom invariant plane is moved to infinity.  The above picture shows the case with the bottom plane at infinity, giving a profile that accurately fits water vortices when the parameters are chosen suitably.  The other case looks more like a tornado.

PARAMETERS

We have already come across one parameter of the egg-transform, namely theta for the rotation of the axial planes.  Referring to Figure 10, notice that an egg profile may be more or less "blunt" at one end and "sharp" at the other.  This is entirely controlled by the way the two geometric series in the top and bottom lines are related.  If they are equal we obtain an ellipse.  They are characterised by their multilipliers i.e. the ratios of successive distances from the vertical axis.  For a given value of the following ratio we always obtain the same type of egg:

lambda = - log x / log y

where x is the multiplier in the top plane and y that in the bottom one.  The parameter lambda is defined to be positive for eggs, and the fractional value for one of the multipliers then implied by the negative sign corresponds to the opposite sense of the spirals in Figure 14.  Note that lambda does not control the maximum width of the egg in proportion to its height, but it does determine the relative height at which the radius is a maximum.  The effect of lambda on the shape is shown on the Research page.  For vortices lambda is negative and its effect is also shown on the Research page.

For a given lambda the shape of the egg shell surface is determined as such, but the spiral may ascend more or less steeply as it winds around it.   This is controlled by another parameter epsilon which is such that for a given value of the following relationship between the multipliers the same relative steepness of ascent is always obtained:

epsilon = (log x + log y)/2

The following document may be downloaded which derives some useful expressions for working practically with these parameters:  Practical Path Curve Calculations

For further information and examples you may refer to the "Basic" and "Path Curve" pages in Projective Geometry, as well of course to the pioneering work of Lawrence Edwards.

                                                                                                                        Nick Thomas