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Questions tagged [cross-ratio]

Use this tag for questions about the ratio AC ⋅ BD / (BC ⋅ AD) where A, B, C, D are colinear points.

In geometry, cross-ratio is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C, and D on a line, their cross ratio is defined as (A, B; C, D) = AC ⋅ BD / (BC ⋅ AD) where an orientation of the line determines the sign of each distance, and distance is measured as projected into Euclidean space. If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.

D is the harmonic conjugate of C with respect to A and B precisely if the cross-ratio of the quadruple is –1, called the harmonic ratio. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from that ratio; hence, the cross-ratio is also called the anharmonic ratio.

Fractional linear transformations preserve the cross-ratio. It is essentially the only projective invariant of a quadruple of collinear points, which underlies the cross-ratio's importance for projective geometry.

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Geometric interpretation of the Logarithm (in $\mathbb{R}$)

(Note: limited to $\mathbb{R}$) (Note: Geometric here means with straightedge and compass) Standard approaches to introducing the concept of Logarithm rely on a previous exposition of the exponential or simply on that of a power. It then receives…
MASL
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On a Geometric Proof (Ahlfors) that the Cross ratio is real if and only if four points lie on a circle or straight line

The cross-ratio $(z_1,z_2,z_3,z_4)$ is real if and only if the four points lie on a circle or on a straight line. I know this question has been asked numerous times on MSE, but I have a specific question with respect to how Ahlfors proves this on…
User7238
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How to find explicitly the Klein j-invariant

Let $K$ be a field, and let $K(x)$ be the associated field of rational functions. I want to find the subfield $L$ of $K(x)$ of the rational functions that are invariant under this set of transformations: $$x\mapsto x$$ $$x\mapsto…
user558035
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References for a formula generalizing the "cross ratio = -1" characterization of harmonic division.

It is well known that : $$\ \ \ \ \ \ \binom{(A,C;B,D)}{\text{harmonic division}}\ \iff \ \binom{\text{cross-ratio}}{ [A,C;B,D]=-1}.$$ (definition of cross-ratio here). A non-classical formula in the case where $A,B,C,D$ are aligned in this order,…
Jean Marie
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Two complex numbers are symmetric with respect to a circle iff a certain equation is satisfied

Let $ \gamma = ${$z \in \mathbb{C} : |z-a| = R$}. Two complex numbers $z_1,z_2$ are said to be symmetric with respect to $\gamma$ iff $$ (z_1-a)\overline{(z_2-a)} = R^2. $$ I am trying to prove that if $\forall \alpha,\beta \in \gamma$, $$ \Bigg |…
Juan
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mapping the circle $|z|=3$ into $|z-1|=1$, the point $3+3i$ into $1$ and the point $3$ into $0$.

Question: Find the bilinear transformation which carries the circle $|z|=3$ into $|z-1|=1$, the point $3+3i$ into $1$ and the point $3$ into $0$. My Attempt: First, I've done problems like this before, but I ran into something at the beginning, so…
User7238
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Points of intersection of pole lines with the sides of the triangle are collinear in $\mathbb{R}P^2$.

Consider the triangle $\Delta PQR$ in $\mathbb{R}P^2$ and a point $S$ outside the triangle. If $l$ is harmonically added to $PS$ with respect to $\{PQ,PR\}$, $m$ is harmonically added to $QS$ with respect to $\{PR,QR\}$ and $n$ is harmonically…
Belgium_Physics
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Cross-ratio of points in the real projective plane

I would like to compute the cross-ratio of the points $A,B,C,D \in \mathbb{RP}^2$, in the projective plane, given by: $$ A=(0:1:2) \quad B=(1:2:3) \quad C=(2:3:4) \quad D=(3:4:5) $$ First I want to demonstrate that they are all collinear. I believe…
Heinrich Wagner
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What are all the functions that preserve the cross ratio?

Suppose a function $f:\mathbb {RP}^1\to \mathbb {RP}^1$ satisfy: $$ \left[f(a),f(b);f(c),f(d)\right]=\left[a,b;c,d\right] $$ for all $a,b,c,d \in \mathbb {RP}^1$. What can the function be in general? Möbius transformations are certainly one type,…
Trebor
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Does duality mapping preserve cross ratio?

I'm new to projective geometry. I learned the definition of cross ratio of 4 collinear points and that of 4 concurrent lines in $\mathit{P}\mathbb{R}^{2}$. The question is, by duality we can map 4 collinear points into 4 concurrent lines. Does such…
LehrLukas
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If a line through the centroid G of triangle ABC meets AB in M and AC in N then prove that AN.MB +AM.NC = AM.AN both in magnitude and sign.

If a line through the centroid $G$ of $\triangle ABC$ meets $AB$ in $M$ and $AC$ in $N$ then prove that $$AN.MB +AM.NC=AM.AN$$ both in magnitude and sign.
Sinjan Dinda
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Determine lines intersecting four skew lines in $\mathbb{P}^3$

Let $l_1, l_2, l_3, l_4$ be four skew lines in a projective space $\mathbb{P}^3$ (meaning $l_i \cap l_j = \varnothing \;\forall i≠j$). Let $R = \{ r : r \cap l_i ≠ \varnothing,\;i=1,...,4 \}$ be the set of all lines that intersect each one of the…
user526159
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Find the bi-linear transformation for the following data.

My Attempt:- Let $z_1=z_0,z_2=\overline{z_0},z_3=0$ and $w_1=0,w_2=\infty, w_3=\frac{z_0}{\overline{z_0}}$. Applying this in Result in the box, we get…
user464147
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Prove that the cross ratio of four distinct points is real iff the four points lie on single Euclidean line or circle

I have started this proof by rewriting the formula for the cross ratio in terms of the polar decomposition of complex…
Christie
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Geometry : Prove that $PE=PC$

Let $l$ be a line not intersecting circle $\omega$ that has center $O$. Draw $OP$ perpendicular to $l$ at point $P$ and draw $PA$ tangent to $\omega$ at point $A$. Extend $OA$ to cut $\omega$ again at point $B$ and cut $l$ at point $C$. $PB$ cuts…
user403160
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