# 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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### Does duality mapping preserve cross ratio?

I'm new to projective geometry. I learned the definition of cross ratio of 4 colinear points and 4 concurrent lines. The question is, by duality we can map 4 colinear points into 4 concurrent lines. ...
219 views

### 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 ...
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### Cross ratio of 5 coplanar points

I need to calculate cross ratio of 5 coplanar points. I have formula for one of the invariants. The problem is I don't know how can this be implemented in programming (in python). I've used dot ...
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### Determine the involution when given two pairs of point on a line

I'm studying about involution on a projective line (line with point at infinity). An involution is a map from a projective line $l$ to itself that satisfied $f \circ f =$ is the identity map. ...
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1 vote
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### If the consecutive vertices $z_1,z_2,z_3,z_4$ of a quadrilateral lie on a circle prove that...

If the consecutive vertices $z_1, z_2, z_3, z_4$ of a quadrilateral lie on a circle, prove that $${|z_1-z_3|}\times{|z_2-z_4|}={|z_1-z_2|}\times{|z_3-z_4|} +{|z_2-z_3|}\times{|z_1-z_4|}$$ This problem ...
190 views

### 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 ...
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1 vote
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### Vanishing points are collinear

I am studying projective geometry where I came across a definition of vanishing line, "The line containing the vanishing point of 2 or more sets of parallel lines on a plane form the vanishing ...
1 vote
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### Constructing a point in a projective line given other seven points and two cross-ratios

Let's suppose we have eight points $A,B,C,X,A',B',C',X'$ on the same projective line such that $(ABCX) = (A'B'C'X')$. If the first seven points are known, how could I find the eighth point $X'$?
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1 vote
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### Is there any way of explaining the Cayley–Klein metric to undergrads?

How to explain the Cayley-Klein or sometimes called Beltrami–Klein metric concept to find the distance between two points in a hyperbolic space to an audience with no higher education than maybe a ...
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1 vote
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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 ...
457 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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### 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 ...