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Questions tagged [inversive-geometry]

Questions related to Inversive Geometry and its applications.

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How to determine if three distinct points $a,b,c \in \Bbb c$ are collinear using Mobius Transformation?

Given three points $\frac{3}{2} + i , 2i,-6+6i$. I have the mobius transformation that maps these three points to $0,1,\infty$ respectively as $M(z) = \frac{(-4i+6)(z-(1+2i))}{(3-7i)(z-(10-20i)}$ ...
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How to correctly determine the mobius transformation that maps three distinct points to 0,1,$\infty$

I was trying to find the mobius transformation that maps the point $i,-i,\infty$ to $0,1,\infty$. following the formula. $M(z) = K \frac{z-i}{z+i}$. $M(i) = 1 \implies 1 = K\frac{i-i}{i+i} \implies \...
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1answer
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Inversion in circle

Let $C$ be a circle with the middle point $O$ and the radius $r$, we say that the points $P$ and $P'$ are inverse points with respect to $C$ if: $1.$ $|OP|·|OP'|=r^2$ $2.$ $P$ and $P'$ are on the ...
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inverse point with respect to circle

I'm reading Silverman's Complex variable with application. at page 78, the author says "We say that $s$ and $s^*$ are the inverse points with respect to circle in $\mathbb{C}$ if every line or circle ...
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Circular inversion notes or books

I am preparing a small lesson that I have to do about geometry. The topic of the lesson is circular inversion. Can you suggest me some sources where I can find some nice material about it? Thanks!!
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What is the radius and center of the image of $|z|=1$ under $ f(z) = \frac{3z+2}{4z+3}$?

I would like to compute the image of the circle $|z|=1$ about the fractional linear transformation: $$ f(z) = \frac{3z+2}{4z+3} $$ In particular, I'd like to compute the new center and radius. ...
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Circle whose radius is infinite

I have the intuition that a circle whose radius is infinite is a straight line. Nonetheless, I don’t feel that what I’ve just stated is really scientific as it has some vagueness and lacks precision. ...
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Algebraic intuition behind inversion of circles

I am struggling to understand inversions of shapes (namely, circles and lines) algebraically. For one, I understand how to apply inversions to general equations of circles and lines (by replacing all ...
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Inversion in a sphere preserves circles (proof)

Inversion in the unit sphere, for a vector $x$, is defined by $$\frac1x = \frac x{x^2} = \Big(\frac1{x\cdot x}\Big)x$$ How can we prove that a circle's inversion in the sphere is also a circle? (I ...
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What can we do with inversive geometry?

I've recently been introduced to inversive geometry. This seems like it would be a very pretty area of study. Many sources that I have found seem a little old, however. I have two related questions: ...
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How to prove that the inversion $x \mapsto \frac x {\| x \|^2}$ in $\Bbb R^n$ sends circles to generalised circles?

Does anyone know how to prove (or have a reference for) the fact that the inversion $$I : \Bbb R^n \setminus \{0\} \rightarrow \Bbb R^n \setminus \{0\}\\ x \mapsto \frac x {\|x\|^2}$$ sends circles ...
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What does “Inversion about $C$” mean?

Let $\Delta ABC$ be a triangle with orthocenter $H$ and altitudes $AD, BE, CF$. Perform an inversion around $C$ with radius $\sqrt{CH \cdot CF}$. Where do the six points each go? I'm having trouble ...
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Proof of Miquel's six circle theorem

Theorem Miquel's six circle theorem states that if in the following all cocircularities except the last one are satisfied, then the last one is implied. In words: if $ABCD$ lie on a circle, and $...
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Circle Inversion

I don't understand the concept of circle inversion. $OP \cdot OP' = k^2$ For example, in a circle $x^2+y^2=k^2$. If I set a general point $P(x,y)$, why is its image $P'(\frac{xk^2}{x^2+y^2}, \frac{...
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1answer
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Drawing a Poincelet Porism

How to draw a Poincelet porism? In the above image, for the first case, we can draw the inner circle, then the triangle and then it's circumcircle. But how to draw for the other cases and in ...
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2answers
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Converse inversion

I was puzzling with circle inversions and was wondering about a converse of this: Given 3 points A, B and M Is there only one point R so that $d(R,A) * d(R,B) = d(R,M)^2$ or are there more of them? ...
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equation of inversion of a line [closed]

Say we have a line $L: ax+by+c=0 $ in the plane. Define the inversion of a point $(x,y)$ as $$\frac{(x,y)}{x^2+y^2}= (x',y')$$ with $$(x,y) \cdot (x',y')= 1.$$ What is the equation for the inversion ...
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(Logic behind) Geometric construction for imaginary roots of quadratic.

In the question on this site Geometric derivation of the quadratic equation there are geometric constructions for real roots of a quadratic. As a follow-up I found the construction below for ...
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Parallel lines as generalised circles in inversive geometry

Consider the inversion of circles(w1 and w2) w.r.t the circle(c) that gives circles of infinite radius(t1 & t2 respectively). What is the nature of the generalised circles t1 and t2 (are they ...
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Concentric circles in inversive-geometry

Is it possible to obtain two concentric generalised circles(parallel straight lines) through inversion of two circles w.r.t a third circle?
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Second point of intersection of two circles in Möbius geometry

Suppose I have two circles \begin{align*} (x-x_1)^2+(y-y_1)^2 &= r_1^2 \\ (x-x_2)^2+(y-y_2)^2 &= r_2^2 \end{align*} and I also have one point $$p=(x_p,y_p)$$ which is known to lie on both ...
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Find the image of vertical line $x=1$ under Mobius transformation of $\frac{2z}{z+1}$.

Let $f(z) = \dfrac{2z}{z+1}$ and $L_{3}$ = the line $x=1$. I am required to find the image of $L_{3}$ under $f(z)$. I have done this by choosing the three points $1, 1+i,\infty$ and seeing what they ...
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Finding the image of the x axis under a Mobius tranfsformation

Let $f(z) = \frac{2z}{z+1}$ and let $L_{1}$ be the $x$ axis. I am told to show that $f(z)$ maps $L_{1}$ to itself. I have picked three points to determine it image, $0,\ 1$ and $\infty$. I'm ...
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1answer
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Parallel lines in inversive geometry

Please give the definition of parallel lines(generalised circles) in inversive geometry? Is there a unique definition (because i am confused about the duality of line with circle)? If parallel is ...
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Finding the images under a Mobius transformation

Let $L1$ be the $x$-axis, let $L2$ be the $y$-axis and let $L3$ be the vertical line $x = 1$. For each $k ∈ Z$ let $C_{k}$ denote the circle of radius $r = 1/2$ with centre $z = 1/2 + ki$. Let $f(z) =...
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1answer
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Is “square inversion” possible?

So, there exists in geometry circle inversion: Can I perform a similar "inversion" technique through a square? What would, for example, a square look like when inverted through another square?
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2answers
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When does inversion respect vector addition?

A recent question inquired as to how one could characterize the solutions of the equation $\frac{1}{z_1}+\frac{1}{z_2}=\frac{1}{z_1+z_2}$ for complex $z_1,z_2$. This is trivially valid whenever ...
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1answer
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Interesting circles hidden in Poncelet's porism configuration

This question is an investigation starting here, with a straightedge and compass construction of $ABC$ given $(R,r,h_A)$. The key lemma is the following one: Let $\Gamma$ be a circle with centre $O$...
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Geometry - Inversion/Cross Ratios

Problem 5. Let ABCD be a given convex quadrilateral with sides BC and AD equal in length and not parallel. Let E and F be interior points of the sides BC and AD respectively such that BE = DF. The ...
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2answers
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Circle inversion of a circle

Given is a circle K with radius r and centre M1. K' is a second circle with radius r' and centre M2 that cuts K in two points A and B so that $[M1A]$ is orthogonal to $[M2A]$ and also $[M1B]$ is ...
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Centre of Invariant Circle under Inversion

Given an inversion of the plane, and a circle invariant under this inversion, what information do we know about the inverse of the centre this circle? (I know that an invariant circle must be ...
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1answer
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How is this circle inversion formula calculated?

I know about the inversion of a point inside a circle. But I was reading Peter Sarnak's paper on the Apollonian gasket, and got to the part where he was trying to prove descartes circle theorem. He ...
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Can someone explain this unit vector calculation for this circle inversion formula derivation?

I'm really stuck. I'm learning about circle inversion. More specifically, I was trying to understand how to derive the inversion formula for a circle, which seems to be explained here. http://...
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Let $I$ be an inversion and let $C$ be a circle such that $I(C)$ is also a circle. When do $C$ and $I(C)$ have equal radii?

Let $I$ be an inversion and let $C$ be a circle such that $I(C)$ is also a circle. When do $C$ and $I(C)$ have equal radii? When it comes to inversion in a circle, I only know two cases: a circle ...
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1answer
723 views

undefined angles with arcsin

I have this problem but I couldn't solve it. In a paper I'm reading for controlling a device, I need to generate the following angle $$ \theta = \tan^{-1}\left( \frac{Y_{2} - Y_{1}}{ X_{2} - X_{1}} \...
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Prove special case of Brianchon's theorem using inversion

Brianchon's theorem says: When a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. From interactive demo: (...
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1answer
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Decide whether there exists an inversion that transforms one onto the other.

Question: For the following pairs of curves, Decide whether there exists an inversion that transforms one onto the other. Identify the inversion if it exists. The circles $x^2+y^2=16$ and $(...
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Inversion: Find the transformation of a circle

To what point or curve does the inversion $I_{0,5}$ transform the circle centered at (15,0) with radius 13? ($I_{0,5}$ means inverting on a circle of radius $5$ centered at $(0,0)$.) I was looking ...
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If three circles have two common points, prove that every circle that is orthogonal to two circles is also orthogonal to third.

Three circles are given $k_1$,$k_2$,$k_3$ that have two common points A and B. Prove that every circle $k$ that is orthogonal to circles $k_1$,$k_2$, is also orthogonal to $k_3$. Here is my proof ...
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2answers
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inversion of the circle $t \mapsto (3 + is) + e^{it} $ around the unit circle.

We know that inversion interchanges lines and circles, but it's very hard to The inversion map about the unit circle is just $\displaystyle z \mapsto \frac{1}{\overline{z}}$. As a Möbius ...
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3answers
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On inversive geometry

I am given the following problem set: Observe the circle $K$ with center $0$ and radius $r$ in the complex plane $ \mathbb{C} \simeq \mathbb{R}^2$. Show that the inversion on $K$ is given by the ...
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The radius of image of a circle under mobius transformation

A Mobius transformation of the plane takes $z \mapsto \frac{az+b}{cz+d}$. These are known to take circles to circles, but given an explicit circle, how do we compute the radius. Let's parameterize ...
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Image of a locus via stereographic projections

Yesterday evening I was playing around in my head with stereographic projections and I've come up with this idea. Let $\gamma(t)=(x(t),y(t))$ be a certain curve on a plane. Define a new curve \begin{...
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Finding the polar line of the intersection of a polar line and a tangent

Let $K$ be an inversion circle with center $O$ and let $C$ be the point of intersection of two lines tangent to $K$ in $A$ and $B$. Then let $E$ be the intersection of the line $AB$ and the line ...
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Inverse with respect to a given circle

Determine the inverse with respect to a given circle $g:\mathbb{R}^{2} \to \mathbb{R}^{+}, g(x,y)=x^{2}+y^{2}$. I have looked around for non geometric derivations without finding any of value. Anyone ...
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Task “Inversion” (geometry with many circles)

Incircle $\omega$ of triangle $ABC$ with center in point $I$ touches $AB, BC, CA$ in points $C_{1}, A_{1}, B_{1}$. Сircumcircle of triangle $AB_{1}C_{1}$ intersects second time circumcircle of $ABC$ ...
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Given the circumcircle, the 9-point circle, and the angular measures for a triangle, construct the triangle?

This is similar to some questions that have been asked (e.g. construct-triangle-given-inradius-and-circumradius), but I don't see the exact same question. It arose out of an inversive geometry formula ...
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Draw three congruent circles all touching one another, and a second set of three such circles, each touching also two of the first set.

This corresponds to a Steiner's Porism configuration with n = 4, however the trouble I'm having is that while it is easy to construct an n = 4 Steiner's Porism configuration (see second image below), ...
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Is this usage of the term 'Jacobian' related to the other uses e.g. matrix or elliptic function?

In the inversive geometry exercise below (from Geometry Revisited by Coxeter and Greitzer), the solution refers to the point pair $(L, O)$ as the Jacobian of the two point pairs $AC$ and $BD$. I was (...
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545 views

Why is inversion in a straight line the same as reflection?

In the inversive plane points $P$ and $P'$ are defined to be inverses with respect to a circle $\omega$ of radius $k$ and center $O$ when they are distinct from $O$, on the same ray emanating from $O$,...