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. ...
LehrLukas's user avatar
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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 ...
Jean Marie's user avatar
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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 ...
Kholdarbekov's user avatar
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2 answers
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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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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 ...
tiwari_at183's user avatar
3 votes
1 answer
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 ...
Victor's user avatar
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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 ...
lonewolf.py's user avatar
1 vote
1 answer
40 views

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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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 ...
Darmstadtium's user avatar
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How can I show that set is cross-ratio?

We have $4$ points: $P_1 =[0:0:0:1], P_2=[1:0:0:-1], P_3=[1:0:0:1], P_4=[1:0:0:0]$ in $\mathbb{P}^3$. I want to show that this set of points is cross- ratio. I am trying to prove it by use equation: $$...
sss's user avatar
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How to convert a numerical value to a percentage of 100?

So, consider that are three tests a student has taken. The overall score is 100 points. The test has three sections. Section 1 - 30 points Section 2 - 30 points Section 4 - 40 points Student scores ...
Sanchay's user avatar
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Synthetic proof of X(12) - harmonic conjugate of Feuerbach point

I'm looking for the synthetic proof of this interesting fact: https://www.cut-the-knot.org/Curriculum/Geometry/FeuerbachIncidence.shtml In words from https://faculty.evansville.edu/ck6/encyclopedia/...
auntyellow's user avatar
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Prove product of 3 cross-ratios equals to 1 or -1

Here is a problem from a Chinese textbook Advanced Geometry (4th edition, ISBN: 9787040537550): A straight line intersects a triangle $P_{1}P_{2}P_{3}$'s 3 edges $P_{2}P_{3}$, $P_{3}P_{1}$ and $P_{1}...
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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 ...
User7238's user avatar
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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 ...
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What can be said about the coefficients of linear transformations which transforms the real axis into the imaginary axis?

Question: What can be said about the coefficients of linear transformations which transforms the real axis into the imaginary axis? Thoughts: If I wanted to transorm the real axis into itself, then I ...
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Proof that given 4 harmonic lines a,b,c,d if the line a bisects the angle between c and d, b must be perpendicular to a

I'm working on a problem in What is Mathematics? by Richard Courant and Herbert Robbins. The problem statement is : "If in a set of four harmonic lines a,b,c,d, the ray a bisects the angle ...
Katie's user avatar
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If ($ACBD$) $= -1$, show that $DA \cdot DB = DC \cdot DO$, where O is the midpoint of the segment $AB$.

If ($ACBD$) $= -1$, show that $DA \cdot DB = DC \cdot DO$, where O is the midpoint of the segment $AB$. My attempt to proof: Suppose ($ACBD$) $= -1$ then $\frac{AC}{CB} = \frac{AD}{BD}$ then $AC \...
dominic eqrut's user avatar
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What is meant by "cross ratio of four points" in Weyl's discussion of the Klein Disk?

The following is from Hermann Weyl's Space-Time-Matter. Although the structure was thus erected, it was by no means definitely decided whether, in absolute geometry, the axiom of parallels would not ...
Steven Thomas Hatton's user avatar
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Will $S(\infty)$ lie on the same circle as that of $S(\mathbb R)$ for every Möbius transformation $S\ $?

Let $S : \mathbb C_{\infty} \longrightarrow \mathbb C_{\infty}$ be a Möbius transformation. Does $S$ necessarily send $\infty$ to $\infty\ $? I am actually following Conway's book on Complex Analysis....
Fanatics's user avatar
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Show that there exists a line that intersects with all lines $l$.

Consider a line $k$ in the projective space $P(\mathbb{R}^4)=\mathbb{R}P^3$ and 3 points $A_0,A_1,A_2\in k$. Let $B$ be an arbitrary point in $P(\mathbb{R}^4)$ but not on $k$ and $l$ be a "...
TheHunter's user avatar
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3 answers
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Can one prove Pappus hexagon theorem with cross ratios alone

Can one prove Pappus theorem knowing that projections don't change the cross ratio and that $(A,B;C,D) = (A,B;C,E) \iff D=E$? I was reading this in exercise 6 they say we can prove Pappus theorem with ...
hellofriends's user avatar
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2 answers
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Show that the harmonic conjugates are collinear - Menelaus and Ceva's Theorems with homogeneous coordinates

Suppose we have a triangle $\triangle ABC$ such that $D$ is an arbitrary point on $BC$, $E$ is an arbitrary point on $AB$ and $F$ is an arbitrary point on $AC$. Let $G$ be the harmonic conjugate of $E$...
mandella's user avatar
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prove cross ratio of 4 point on a circle is invariant

The text which I was reading said that if we assume points $\displaystyle O_{1} ,O_{2} ,A,B,C,D$ on a circle then it's cross ratio is $\displaystyle R( O_{1} A,O_{1} B;O_{1} C,O_{1} D)=R( O_{2} A,O_{2}...
Daniel's user avatar
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Most general linear transformation of $|z|=r$ into itself using cross ratio

This question (without the cross ratio part) was asked earlier today, as well as a few times before. Here was the question that was asked earlier today: Find all Möbius transformations that map ...
User7238's user avatar
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Mapping $|z|=1$ and $|z-\frac{1}{4}|=\frac{1}{4}$ into concentric circles using the cross ratio

The question: Find a linear transformation which carries $|z|=1$ and $|z-\frac{1}{4}|=\frac{1}{4}$ into concentric circles and find the ratio of the radii. This question is already answered here: Find ...
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cross ratio definition

I'm stuck with this exercise: I have 4 points in the projective plane $\Bbb{P}^2(\Bbb{R})$ $$P_0=\{0,0,1\},\quad P_1=\{0,1,-1\},\quad P_2=\{1,-1,0\},\quad P_3=\{1,1,-3\}$$ and I have to compute ...
Arianna's user avatar
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1 answer
188 views

Why does this cross ratio equal infinity?

I'm currently studying linear fractional transformations and cross ratios and came across this in a book (this is translated from Korean, so I apologize if there are any errors or ambiguities): We ...
Sean's user avatar
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51 views

Cross ratio on $T(z)=(z,2i;2,-2)$ and $S(w)=(w,-1;2i,1+4i)$

Cross ratio on $T(z)=(z,2i;2,-2)$ and $S(w)=(w,-1;2i,1+4i)$. I am using these to find a Mobius transformation that takes the circle $|z|=2$ to the line $2x-y=2$. I want to map the points $u=2i, v=2$, ...
user1990's user avatar
1 vote
0 answers
70 views

Finding a Möbius transformation from $\{z=x+iy\in\mathbb{C}:x+y>0\}$ to $D(1,4)$.

Find a Möbius transformation $T$ from $\{z=x+iy:x+y>0\}$ onto the disk $D(1,4)$, such that $T(1)=2$ and $T(0)=-3$. The proof given is as follows: Since $-i$ is symmetric to $1$ with respect to ...
Representation's user avatar
5 votes
0 answers
84 views

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 ...
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computation of the cross ratio

I was troubled by the definition of cross ratio,give four ordered points $X,A,B,Y$ the cross ratio should be $\frac{XB}{XY}/\frac{AB}{AY}$.But according to the bove definition in the screenshot,the ...
math112358's user avatar
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3 votes
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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 ...
Belgium_Physics's user avatar
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Prove that angles in the given projective geometry setting are equal

We are given triangle $DIV$. On the side $IV$ a point $U$ is chosen, such that $UDI$ is the right angle. Point $P$ on $IV$ is such that $(I,U;P,V)=-1$ (harmonic conjugates that is). Prove that angles $...
user75619's user avatar
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1 answer
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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 ...
Heinrich Wagner's user avatar
17 votes
2 answers
2k views

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 ...
MASL's user avatar
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2 votes
3 answers
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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. [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.
Sinjan Dinda's user avatar
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1 answer
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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 ...
user75619's user avatar
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3 votes
1 answer
330 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? ...
Trebor's user avatar
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2 votes
1 answer
487 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 the ...
user526159's user avatar
1 vote
1 answer
123 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 $l_1, l_2, l_3, l_4 \in \mathbb{RP}^3$. ...
user526159's user avatar
0 votes
1 answer
128 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 ...
user526159's user avatar
1 vote
1 answer
209 views

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 ...
user526159's user avatar
2 votes
1 answer
146 views

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}{\...
user avatar
2 votes
1 answer
1k 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(\dfrac{z_1-z_3}{z_1-z_4}\Big)\Big(\dfrac{z_2-z_4}{z_2-z_3}\Big)=\...
Christie's user avatar
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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 $\...
Akatsuki's user avatar
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1 vote
1 answer
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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 ...
user514961's user avatar
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1 answer
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 ...
yagod's user avatar
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2 votes
5 answers
150 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 ...
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