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The Fundamental Theorem reveals a set of lines that envelope a curve,
and it has been asserted without proof that this curve is a conic.
That proof is developed here.




.....Is this in perspective....

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....and these are just two views of a “rocking” cone-and-cutting-plane ensemble.

  • The above left view has the line at infinity ( * ) of the cutting plane in sight.
  • The above right view has that line at infinity as far out of sight as it can possibly be.

Click 'n' drag the bold dashed line on the left to rotate the cone and its cutting plane around the “horizontal” axis.

Note how the lines from the cone's apex tangent to the circle (representing the ‘horizons’ of the cone's surface) vanish when the cone's apex appears to pass inside the conic made by, and lying on, the cutting plane—

Because the cone's surface then has no visible ‘horizons’ (sometimes styled, ‘cuspidal edges’) , the tangents are imaginary.

The black line, a, represents the “line-at-infinity” for the planes of rotation.

Click 'n' drag point A to move this line left or right to change the perspective.

and/or start the animation.

                                     Some General Features of Conics
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On the left is an interactive version of the pictures above.

You may "click 'n' drag" various points to adjust the conic, and to change the relationship of the conic's centre (pole) to the "line at infinity" (polar), and you can drag point R on the polar to rotate the whole thing round the pole, or centre. (Is it not curious that R is rotating, even while moving in a straight line? This is characteristic of Polarity.)

You may also start or stop the turning using the animation control. The little red slider adjusts the rate of rotation

On the right we see, in perspective, a plane holding conics rotating on a fixed, thin, vertical blue line, and taking the conics round with it, rather than, as above, lines of the conic turning on a fixed point on a fixed plane. (It is a variation of the rocking cone, above.)

The bold blue line could, but need not, represent the
line at infinity.

For fun, part of a cone has been ‘constructed’ on the bold conic, and given a “transmission rod” on its axis, all to simulate a radar dish in rotation—but also to bring out a serious point, which is that non-physical, non-Eucliean geometry may also apply to physical reality, such as a radar dish.

And we see a condition for it to apply:—
the bold blue line must actually “be” the line at infinity.

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saying this, we encounter a paradox.

For we recall that the axioms of incidence are qualitative, not quantitative, and so do not define distance, or indeed countable units of anything, so all references that depend for their validity on infinity understood as as sum of discrete distances (that is, as a quantity), are "interlopers". They are wrong, and wrongly placed, because they rely for their definitions on both quantity and distance. They are parvenu, and alien, and to be extirpated.

Then, especially, references invoking parallelity cease to have any kind of meaning (if, indeed, they ever did), and the "line at infinity" (the one found to be carrying two pairs of conjugate points, and also referred to in the discussion of the radar dish) is just an ordinary line.

And, most notably, we must recognise that
the two conics at the head of this page are geometrically indistinguishable.

Intermezzo, entitled “Seeming and Being.” (skip↓)

That the two conics appear different may seem just a matter of viewpoint
(we note in passing that viewpoints,
are not themselves parts of the things viewed.
If they were, they would not be viewpoints).

Moreover, we might wish to assert that a view is only a copy (that is, a projection), not the original,
and that it is the copy that varies, according to the conditions of copying, not the original.

But—and is it a big ‘but’—this is to imply that the original is, in some sense,

  • absolute, and
  • invariant.

However, if conics that are projections of each other cannot be distinguished one from another,
then any one of these conics could serve as an ‘original’.

To put it another way,
it would not be possible, by geometric means alone,
to determine which of the conics is the original,
or indeed that there is an original at all.

In fact, we must conclude that
originality is a meaningless notion
in a purely geometric context.

It must follow that, when projecting, instead of copying some absolute, invariant original,
we are instantiating from the general collection of incidences corresponding to “conicness”,
and it is this collection which remains absolute and invariant,
not any one of the instances.

For example, all the properties of incidence with lines and points
found for our two conics at the head of this page are retained,
and apply to all conics, no matter how they "look".

One could (with paradoxical but nonetheless accurate logic) say that these retained properties
determine how all conics look—just as long as they remain unseen!

For it would very much seem
that the process of manifestation specifies
things not needing to be specified
until they are required to manifest

Here we have the issue, I believe, that so exercised Einstein
when, with Podolsky and Rosen, he framed his famous objection to the emerging Quantum Theory.
He spoke then of “real” properties,
by which he meant properties that must exist before observation
in order to be altogether available for observation,
and that are in fact largely unaffected by observation,
possibly to the point of remaining hidden from observation.

In our case, we know that the base elements of geometry are forever invisible,
and therefore hidden, because, having no sizes,
they cannot reflect, refract or in any other way modify light;
- or, as to that, any other physical thing -

but we are, notwithstanding, entirely sure that the axioms of incidence,
for all that they are incorporeal,
pre-exist, and pertain!

It is therefore a vitally interesting puzzle
as to just how manifestations of geometric property
actually occur in the so-called real world.


The axioms of incidence
pertain to primitive elements that are just

  1. somewhere, or
  2. straight, or
  3. flat,

So there is no "niche" within the axioms for curved items

such as non-degenerate conics, or the surfaces of (for example) cones, or eggs, or buds,

so Curvature is not axiomatic.

If Curvature must instead be specified by those axioms that are actually available,
then this must be by a combination of two or more of the base axioms.

And what is more, Motion,
because it implies an inconstant connection to Time,
is neither a geometric element, nor covered by a geometric axiom.

Nevertheless, if we allow the intuitive notion of Motion to stand for now, a tangent and its contact point can be said to represent or specify a curve in a single place as a combination of two axiomatic elements—a line (wanted for its orientation), and a point (wanted for its position).

By the axioms of incidence, these elements can combine (that is, be mutually incident) in only the following two ways—

  • the point can only be in the line, and
  • the line can only pass through the point.

Then, to represent another place on the curve, the whole combination must move, and, by the axioms again, while moving,

  • the line of the combination is limited to rotation around the point, and
  • the point of the combination is limited to translation along the line.

Now, as there is no axiom of curvature, on a move the point is free to go anywhere in that point's line, and that line is free to rotate at random in that line's point. Let us look in some detail at how we want it to move.


Suppose we draw an arbitrary curve. Drawing it, we know that it is continuous, because we never lift our pen from the paper, so there is ink—points—to be found everywhere and anywhere on the curve3. We see that it is the act of drawing, and not the axioms, that selects the points that lie on such a curve. This is what makes it "arbitrary".

On this curve, we choose two distinct points, T1 and T2, and draw tangents, t1 and t2 respectively, to the curve at these points. They meet in a point, X. Also, we connect points, T1 and T2, with line, x.

We think of these tangents and their points as two, successive (the first and the second) dispositions of a combination-in-motion, and understand that, in the course of that motion, the tangent-of-the-combination will turn continuously around the point-of-the-combination in contact with the curve, while that point moves along that tangent, taking point T from position T1 to position T2, and rotating line t from orientation t1 to orientation t2.

"The Anatomy of a Plane Curve"
(using Lawrence Edwards' apt phrase)

frame frame frame

We note that the kind of the curve must depend greatly on how this line and its point move, or are moved, especially in regard to their relative rates of motion.
How will we measure these motions, and their respective rates?

We have here a major “paradigm shift”, for, up to now, we have not needed to measure, because Projective Geometry, as an "incidence-preserving" system, does not define sizes of anything.

But our arbitrary curve did not arise from the axioms of incidence. It arose from a free "act of drawing."

Measurement, then...

We need sizes, and units, after all...

Projective Geometry cannot provide absolute size, but it can provide regular intervals; that is to say, intervals which are equal with respect to the manner of their making ("regulation"). These intervals are geometrically indistinguishable, one from another, and therefore can be identical and countable units, and so be the basis of a metric, or system of measurement.

Two such measures working together on the plane define a co-ordinate-system: the so-called Cartesian system (the familiar, squared “graph-paper”, or chart, with axes at right angles) is an instance. That same graph-paper viewed in perspective is another. Both of these styles are illustrated on the right.


The old, familiar Cartesian Co-ordinates -
standard "Graph Paper."

In the first of these two instances, the two “end-points” or “invariants 5” belonging to each of the two scales lie together and at the infinite. The second case is the same, except that that line at infinity, linking these end-points, is in plain sight, perhaps the "sight" of a camera, such as might be used to take a photograph of the above graph-paper.

(We know by now that we need to be geometrically circumspect about this "infinity" - because geometry is blind to it - but let us, provisionally, define it physically—that is, call it a place at which, say, the effect of a sun's gravity, or the intensity of the electric field due to a point charge, falls to zero.

I think we can agree there might actually be such places, that we could probably detect or infer them, and that they may be, for corporeal beings like us, physically unreachable. Given that the sun, or the charge, is reachable with finite effort, we could then decide, provisionally, that

  • geometric lines from reachable places converging on such an unreachable place are parallel
  • a line joining two such unreachable places can do duty as a line "at infinity".)

But there is no absolute fiat concerning the orientation of the axes with respect to each other. They need not be at right angles—indeed, the pure geometry cannot specify this or any other angle. To serve as axes, the axes need only be distinct (apart from their meeting place), and bear scales. Now, the numbering of intervals on a scale may begin at any of its intervals. Where there are two or more axes with scales, it is often convenient (though not absolutely required) to begin the counting for them all at the meeting place of their axes, as the "Origin" of co-ordinates. The zero-marks of all the scales lie at this origin.

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These systems assign numbers to places, at the rate of one number (or "ordinate") per axis per place (so that the members of a number-ordering are thus "co-ordinates"), so we might be able to describe our arbitrary curve in terms of some "ordered" series, or set, of assignments—and if we can do this, then

we have "measured (metricised) " the curve,

and it is no longer entirely arbitrary!

Moreover, if we specify a continuous curve, then we must have that to every number we choose from one axis, there will correspond a definite number, or numbers, determined from the curve, on the other axes4.

And each of these numbers can take any value (including infinite), and have either sign, or no sign (zero and infinity are unsigned, as each marks a transition from a segment of one sign to a segment of the other).

A Question of Appearance

One often reads that an ordered assignment, supplied by an equation such as y = 1/x (for which there is exactly one y for each x), is discontinuous at x = 0, because it appears that, for this x, y has two values of opposite sign; +∞, and -∞.

But it is just that—appearance.

In fact, as noted above, numerical infinity is unsigned. There is therefore just one value of y for x = 0, namely ∞, not two values, and so y = 1/x is everywhere continuous.

When plotted on a standard cartesian chart, this equation produces a hyperbola (which has the chart axes as its asymptotes), that indeed appears to be doubly discontinuous, once for when x = 0, and again when y = 0. But if the curve is plotted on a perspective version of the cartesian chart, then it can instead appear as an ellipse crossing the line joining the points (0,∞) and (∞,0) [the "line at infinity"] without discontinuity, twice, and we see why the axes are asymptotes!

y = 1/x is an algebraic equation, or "function". Algebraic functions must have at least one solution (never none), and may have many, but they are never discontinuous.

Below the perspective plot of y = 1/x, there is a perspective plot of y = x2, which, on a "standard" Cartesian chart, gives a parabola, but, as here, can appear as an ellipse.

In any case, it is a closed conic, tangent to the line y = 0, and tangent to the line y = ∞, when x = 0, and once again, though appearance might suggest that a parabola is a curve closed only at one "end", now we see that it is entirely closed, "begins" at zero, and "ends" precisely at infinity.



We might say, very loosely, that—

  • a parabola, reaching exactly to infinity, is a conic of exactly infinite length
  • an ellipse (or a circle), not reaching infinity, has less than infinite length
  • a hyperbola, reaching over infinity, has more than infinite length
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This interactive shows that a parabola, with equation y = x2, can appear as an ellipse with a major axis of precisely infinite length!

Enclosure, Regions, Relationships and the Imaginary

We see that all non-degenerate conics, including the hyperbola and the parabola, are closed, and so divide their planes into two regions, both of which are enclosed by the conic.

This "enclosing" property of the conic
is really rather significant

since it admits the possibility of a line travelling entirely in only one of the regions, such that it appears to "miss" the conic.

Yet again this is appearance only, for the conic's relationships to lines and points extend into both of the regions it encloses, and such a line, though apparently missing the conic, has those relationships (namely, the so-called “imaginary” connections) pertaining to the region in which it happens to be.

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For example, the interactive above depicts the transformation of a line into itself through an interaction of the line with a conic - namely the black conic with the yellow point. Other conics are revealed, generated by this interaction.

Drag either the black or the white line, or drag both, to cut the original black conic - or not. You will see elements of the transformation appear, as they become "real", or disappear, as they become "imaginary".

But the generated conics remain real, whatever the states, real, or imaginary, of their generating elements.

Now, deploying these ideas, let us plot a circle of unit radius,
with centre on the origin,
and equation y2 = 1 - x2,
on both styles of chart.

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It will be appreciated that the "infinity" referred to here in, "line at infinity", is of the numerical variety.

It is worth bringing out that straight lines, such as tangents, with equal slopes (gradients), being "parallel", when plotted on either of these co-ordinate styles, must, by the axioms of incidence, all meet in the same point on the line at infinity.

If in fact they are tangents to curves, then their slopes must be the first derivatives (dy/dx) of the curves they contact at the points of contact. The line at infinity is thus an axis, too, of a kind, but the scale on it denotes rate-of-change, not position.


If two such parallel lines are tangent to a circle (as above at contact points A and B, for example, or as on the right), then the chord through their contact points must always pass through the centre of the circle.

Here we have the fundamental reason why the geometry of pole and polar works.

  • The axioms of incidence allow the construction of co-ordinate systems, such as those above. In terms of those axioms of incidence, all of these systems are identical (though not unique), and provide measure (metrics) identically.
  • A circle plotted according to the algebraic ordering of pairs of ordinates (i.e., enumerated points) of a circle is a circle in all of the co-ordinate systems, appearance notwithstanding, because the circle is specified numerically, by an algebraic function, NOT by the axioms of incidence.
  • Thus, according to the choice of co-ordinate system, the circle may appear to be a circle (as above left), or an ellipse (as above right), or a hyperbola, or a parabola—in fact, any of the members of the conic family.

So, just by "running these findings backwards", we can say:-

  • Any pair of tangents to any conic may legitimately be deemed to meet at infinity(and so be parallel) in a co-ordinate system with respect to which the conic is a circle,
    • and if they do that, then the corresponding chord must pass through the centre of the circle/conic wrt that system.
  • The meet of a second pair of tangents to the same conic may similarly be deemed to be at infinity,
    • in which case the line between the two meets must be the line at infinity wrt the same co-ordinate system, and
    • the corresponding chords must meet in the centre.

With things generalised like this, we speak of the Centre as the Pole, and of the corresponding Line at Infinity as the Polar.


It may come as a surprise that when the line at infinity cuts the conic in two real points, the centre of the conic appears to lie outside the conic, but when the line at infinity misses the conic (and therefore cuts the conic in two imaginary points) the centre appears to lie inside the conic. This might give us pause! See the discussion below.

run / stop animation
In fact, "inside" and "outside" are undefined: all we have is a closed curve, dividing the plane into two regions, both of which are enclosed by the curve. We might wish to adopt the convention that the region containing the centre is "inside" the curve, but, if we do, we should keep in mind that this is an artifact with conventional meaning only, to which geometry is blind.
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So here, at last, is
The Proof!

In the diagram on the left, Point T projects point E on the conic into point D on line l via line c.

The polar of point D (passing through the pole of line l) and the polar of point E must both pass through the same point—the pole of line c—on the tangent to projecting point T.


Please enable Java for an interactive construction (with Cinderella). In the second diagram, exactly this construction is repeated, and stepped forward several times, now using two projecting "points-T" in the conic, giving a range of "points-D" in line l, corresponding to a range of "points-E" in the conic.

The polars of points-D, since they all pass through the same point (i.e., the pole of line l), must produce two ranges of intervals with the same cross-ratio in the two tangents at points-T.

And, as we see,
the tangents to points-E in the conic do the same thing,
with the same ranges!

Thus, Four lines projecting a constant cross ratio from one line to another
always envelop a conic,
whether or not that conic is degenerate.

This is what we set out to show.
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