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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.

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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 ...
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2answers
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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 ...
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3answers
61 views

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. [closed]

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.
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A composition of projections with three fixed points — is it necessarily the identity?

We are given a line $l$. The line is mapped onto itself through a series of projections that involve other lines and -- importantly! -- conics. In the end, points $A$, $B$, and $C$ on $l$ appear to be ...
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61 views

The cross ratio $ (z_1,z_2,z_3,z_4)$ is real iff the four points lie on a circle or a straight line

It's written in Alfors Complex Analysis that, for a proof of the above, "we need only show that the image of the real axis under any linear transformation us either a circle or a straight line. Indeed,...
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1answer
63 views

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? ...
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1answer
55 views

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 ...
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1answer
41 views

Can skew lines preserve cross ratio?

I am currently trying to understand the cross ratio in projective geometry more. I wondered about the following and appreciate any answers: Assume four lines $l_1, l_2, l_3, l_4 \in \mathbb{RP}^3$. ...
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1answer
24 views

Cross Ratio of line through tetrahedron same as of planes of vertices with line

Given a line $l$ through a tetrahedron $ABCD$ (not intersecting any of its edges), take the four points $P_1, P_2, P_3, P_4$ of intersection of the line with the faces of the tetrahedron. Also take ...
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1answer
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Problem: Connection between cross ratio and collinearity

In $ \mathbb{RP}^2$ given a triangle $A_1A_2A_3$ and a point $P$ not lying on either one of the edges , set points $$B_i = PA_i \cap A_kA_l,\ k,l \not= i \ \forall i$$ next choose points $C_i \in ...
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0answers
103 views

Bilinear transformation at infinity

I have been asked to find the bilinear tranformation which maps the points $z_1=i\sqrt3$, $z_2=-i\sqrt3$, $z_3=1$ into $w_1=\infty$, $w_2=0$, $w_3=1$ Using the formula for finding the cross ratio of ...
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1answer
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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 $$\frac{w.(1-\frac{w_3}{w_2})}{(w-\frac{z_0}{\...
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1answer
310 views

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 numbers: $r=\Big(\frac{z_1-z_3}{z_1-z_4}\Big)\Big(\frac{z_2-z_4}{z_2-z_3}\Big)=\...
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J-invariants, Elliptic Curves, Cross Ratio

Let $E$ be an elliptic curve in $\mathbb P^2$ and $p$ be any point on $E$. From $p$ we can draw four tangent lines to $E$ and let $\lambda$ be the cross ratio of their slopes. How can we prove that $\...
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What is the analogue of the cross ratio in higher dimensions and what role does it play in n-dimensional geometry?

I know the cross ratio is defined for four real collinear points and for four points in the complex plane. This is an important projective invariant for linear transformations. Is there an analog for ...
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1answer
130 views

Showing connection of cross-ratio and Schwarzian derivative for a holomorphic function $f$.

For some holomorphic function $f$ show that $$\lim_{t\to0} \frac{cr(f(ta),f(tb),f(tc),f(td))-cr(a,b,c,d)}{t^2cr(a,b,c,d)}=\frac{(a-b)(c-d)}{6}S(f)(0)$$ where $cr(a,b,c,d)$ is the cross-ratio of ...
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5answers
105 views

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 ...