Projective geometry begins with a simple repair to Euclidean intuition. Parallel lines are an artifact of refusing to look far enough away. If you enlarge the plane by adding “directions” as legitimate points, then parallel lines meet, and many case distinctions disappear. This enlargement is not a trick; it is the natural setting for incidence, perspective, and the geometry of linear projection. The benefit is that the right objects are the ones stable under projection, and the right quantities are the ones that survive that change of viewpoint.
A projective space is built from a vector space by forgetting scale. For a field $k$, the projective space $\mathbf P^n(k)$ is the set of one-dimensional linear subspaces of $k^{n+1}$. A nonzero vector $v\in k^{n+1}$ represents the point $[v]$, and $[v]=[\lambda v]$ for every nonzero scalar $\lambda\in k^\times$. In coordinates one writes $[x_0:x_1:\dots:x_n]$ for a nonzero $(x_0,\dots,x_n)$, where scaling by $\lambda\neq 0$ leaves the point unchanged. A projective line $\mathbf P^1$ is the set of lines through the origin in $k^2$, and it is the stage on which the central invariant of the subject appears.
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The projective line as “affine line plus a point at infinity”
Inside $\mathbf P^n$ one recovers affine space by choosing a hyperplane at infinity. In $\mathbf P^1$, take the chart where $x_0\neq 0$. Then $[x_0:x_1]=[1:x_1/x_0]$, so this chart identifies with $k$ via the affine coordinate $t=x_1/x_0$. The missing point is $[0:1]$, which plays the role of the point at infinity. In $\mathbf P^2$, the chart $x_0\neq 0$ identifies $[x_0:x_1:x_2]=[1:x_1/x_0:x_2/x_0]$ with $k^2$; the hyperplane $x_0=0$ is the line at infinity, whose points represent directions of parallelism in the affine chart.
This shift of viewpoint turns familiar transformations into the correct symmetry group. A linear automorphism $A\in \mathrm{GL}_{n+1}(k)$ acts on nonzero vectors and hence on their one-dimensional spans, giving a bijection of $\mathbf P^n$. Two matrices that differ by a nonzero scalar induce the same projective map, because scaling does not change the image line. The resulting group is $\mathrm{PGL}_{n+1}(k)=\mathrm{GL}_{n+1}(k)/k^\times$, the group of projective linear transformations.
On $\mathbf P^1$, these are the fractional linear maps
together with the appropriate conventions at points where the denominator vanishes and at infinity. These are also the Möbius transformations when $k=\mathbf C$, but the projective perspective emphasizes what they preserve: incidence and a specific four-point invariant.
The cross ratio
Given four distinct points $A,B,C,D$ on the projective line, their cross ratio is a scalar that is invariant under projective transformations. On the affine chart $k\subset \mathbf P^1$, if the points correspond to affine coordinates $a,b,c,d\in k$, define
The expression is unchanged by simultaneously translating and scaling the affine coordinate, and it extends naturally to configurations involving infinity. For example, if $D=\infty$ in affine coordinates, the cross ratio reduces \to
The cross ratio is not symmetric; permuting the points produces related values such as $\lambda$, $1-\lambda$, $1/\lambda$, and so on, reflecting the action of the permutation group on ordered quadruples.
A key point is that the cross ratio is not merely a convenient formula. It is the complete projective invariant of four ordered points on $\mathbf P^1$: two ordered quadruples are related by a projective transformation if and only if they have the same cross ratio.
Why the cross ratio is projectively invariant
The invariance can be proved in several ways, and each method reveals something structural.
One proof uses the fundamental fact that $\mathrm{PGL}_2(k)$ acts 3-transitively on $\mathbf P^1$: given three distinct points $A,B,C$ and three distinct points $A’,B’,C’$, there is a unique projective transformation sending $A\mapsto A'$, $B\mapsto B’$, $C\mapsto C'$. In affine coordinates, uniqueness comes from the fact that a fractional linear map is determined by its values on three points, because the condition
is a linear relation among $a,b,c,d$ once three pairs $(t,u)$ are prescribed, up to common scaling.
Using 3-transitivity, reduce to the normal form where $A=0$, $B=1$, $C=\infty$. For any fourth point $D=t\in k$, a direct computation gives
Thus, in this normalization, the cross ratio records the affine coordinate of the fourth point. Because projective transformations preserve the ability to normalize triples, the value extracted from $D$ this way must be invariant.
Another proof is computational and is useful when manipulating explicit transformations. Check invariance under generators of $\mathrm{PGL}_2$, such as translations $t\mapsto t+\beta$, scalings $t\mapsto \alpha t$ with $\alpha\neq 0$, and inversion $t\mapsto 1/t$. The formula for $(A,B;C,D)$ is clearly invariant under translation and scaling, since differences scale uniformly. For inversion, substitute $t\mapsto 1/t$ into the formula and simplify; cancellations show the value is unchanged. Since these maps generate the projective group over many fields, invariance follows.
Either way, the outcome is the same: projective geometry has a canonical scalar that survives the main symmetry group, and it is attached to four points, not three. Three points can always be moved \to a standard position; four points contain intrinsic data.
Determination by cross ratio
The “four points determine the map” principle can be stated cleanly:
* If $A,B,C$ are distinct, the map that sends them \to $0,1,\infty$ is unique.
* Under that map, any fourth point $D$ is sent \to $\lambda=(A,B;C,D)$.
* Two ordered quadruples $(A,B,C,D)$ and $(A’,B’,C’,D’)$ are related by a projective transformation if and only if $(A,B;C,D)=(A’,B’;C’,D’)$.
This makes the cross ratio a practical coordinate on the moduli of ordered quadruples of points on the line: the space of distinct ordered quadruples modulo projective equivalence is essentially $k\setminus\{0,1\}$.
A useful special case is the harmonic cross ratio. If $(A,B;C,D)=-1$, then $C$ and $D$ are said to be harmonic conjugates with respect \to $A,B$. Harmonic division arises naturally from complete quadrilaterals and from involutions on $\mathbf P^1$. The value $-1$ is stable under all projective transformations, so harmonicity is a purely projective notion, not an affine accident.
Incidence, duality, and the fundamental theorems
Projective geometry is often summarized as “incidence geometry with a linear model.” Two theorems make that slogan precise in low dimension.
In $\mathbf P^2$, incidence means statements of the form “these three points are collinear” or “these three lines concur.” The duality principle reflects the symmetry between points and lines: the statement “two distinct points determine a unique line” has a dual statement “two distinct lines meet in a unique point.” Many projective arguments proceed by designing a configuration and then applying duality to infer a companion configuration.
The fundamental theorem of projective geometry in dimension at least two says that any bijection of $\mathbf P^n(k)$ that sends lines to lines is induced by a semilinear transformation of $k^{n+1}$. Over fields with no nontrivial automorphisms, this says “line-preserving bijections come from linear algebra.” In the plane, this theorem explains why coordinate methods are not a betrayal of synthetic geometry. Incidence plus mild regularity forces linearity.
On the line $\mathbf P^1$, the corresponding theorem is sharper: any bijection preserving cross ratios is a projective transformation. Since the cross ratio can be recovered from incidence in many synthetic settings, it functions as a bridge between pure incidence and analytic formulas.
Worked example: recovering a fractional linear map
Suppose a projective transformation $f$ sends $A\mapsto A'$, $B\mapsto B’$, $C\mapsto C'$, where all are distinct. To compute $f(D)$ for a given $D$, use the invariant
If we choose affine coordinates in which $A’=0$, $B’=1$, $C’=\infty$, then $(A',B';C',x)=x$. In that chart,
In an arbitrary chart, this identity becomes a concrete formula: $f$ is the unique fractional linear map matching the three prescribed values, and the cross ratio supplies the fourth.
This procedure is more than a computational trick. It encodes the geometric idea that specifying three points fixes the projective frame, and the position of any other point is measured relative to that frame by a cross ratio.
Conics and projective equivalence
Conic sections are the next place where projective invariants clarify structure. Over an algebraically closed field of characteristic not two, any smooth conic in $\mathbf P^2$ is projectively equivalent \to a standard quadratic such as $x_0x_2-x_1^2=0$. The reason is linear algebra: a conic is the zero locus of a homogeneous quadratic form, and changes of homogeneous coordinates act by congruence on that form.
This equivalence is not merely classificatory. It explains why many “metric” statements about ellipses, hyperbolas, and parabolas are not intrinsic to projective geometry, while incidence statements are. Tangency, intersection multiplicity, and the behavior of pencils of lines are projective concepts. Angles and lengths are not.
A particularly instructive viewpoint is to parametrize a smooth conic by $\mathbf P^1$. A line through a fixed point on the conic meets it in exactly one other point (counting multiplicity), and this gives a rational parametrization. Under such a parametrization, cross ratios on $\mathbf P^1$ translate into projective invariants of quadruples of points on the conic.
What survives projection
A clean way to keep track of the subject is to separate what is genuinely projective from what is metric or affine.
| Notion | Preserved by projective maps | Typical measurement |
| — | — | — |
| Collinearity and concurrency | Yes | incidence conditions |
| Cross ratio on $\mathbf P^1$ | Yes | scalar in the base field |
| Tangency and intersection multiplicity | Yes | local algebra of curves |
| Parallelism in an affine chart | No (becomes meeting at infinity) | direction classes |
| Angles and lengths | No | inner product data |
| Midpoints and ratios of segments | No (unless encoded by cross ratio choices) | affine structure |
The table is not a prohibition. It is a guide to what kind of hypotheses are meaningful. If a statement is meant to be stable under projection, it must be phrased in terms of incidence and cross ratios, not in terms of Euclidean measurements.
A projective habit of mind
Projective geometry trains a specific mathematical reflex: when a configuration seems cluttered by special cases, look for the ambient symmetry group and the invariants it allows. On $\mathbf P^1$, the symmetry group is large enough to normalize any triple, and the cross ratio captures what remains. That fact turns many arguments into a two-step move:
* Use a projective change of coordinates to place three points in a convenient position.
* Express the remaining data as a cross ratio and read off the result from the normalized picture.
In higher dimension the invariants change, but the principle persists. Projective geometry is not primarily about drawing pictures with points at infinity. It is about understanding which features of a configuration are real and which are artifacts of a coordinate choice. The cross ratio is the canonical example of a feature that is real.
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