Questions related to Inversive Geometry and its applications.
0
votes
3answers
66 views
How to solve these?
Inverse Trigonometric Functions
They are incomplete and I don't know how to complete them.
Who can help me?
1st
$$
\int\frac 1{ x \sqrt{x^{6} - 4}}dx
$$
I tried with:
$$u = x^3 $$
$$du= 3x^2dx$$
...
2
votes
1answer
14 views
value of this inverse trigonometric expression.
How to evaluate this expression.
$$\sec^2(\tan^{-1} 2)+\csc^2(cot^{-1}(3))$$
I'm stuck on how to process squares, which is on sec and cosec function?.
0
votes
2answers
52 views
Motivation for the definition of the projective line over a ring
Given a ring (with identity) R, the definition for the projective line over a ring I have is as follows:
On $R \times R$ define an equivalence relation as $(u, v) \sim (a, b)$ if and only if there is ...
2
votes
0answers
89 views
A geometry problem uses inversion
Let a triangle $ABC$. $M$ is a point inside triangle. construct the line through M perendicular to $MA, MB, MC$ and intersect $BC, CA, AB $ at $A_0,B_0,C_0$ respectively. Prove that $A_0,B_0,C_0$ are ...
8
votes
3answers
382 views
inverting a cone to a torus
I'm looking at "A Geometric Paradox" by B. H. Brown, in the May--June 1923 issue of The American Mathematical Monthly, pages 193--195. I think people studied advanced Euclidean geometry a lot more ...
1
vote
1answer
166 views
Construction of touching circle
My question is:
Consider five collinear points $D$, $A$, $C$, $B$ and $E$ such that $DA=AC=a$ , $CB=BE=b$.
Let M be the midpoint of $DE$. Let $S_1$ be a circle with center $A$ and radius $a$, Let ...
1
vote
2answers
492 views
construct inverse point with respect to the circle by the use of the compass alone
If the given point P lies inside a circle C ，with center O，the circle of radius OP about P intersects C in two points.
How to construct point P' inverse to point P with respect to the circle C
by ...
2
votes
0answers
352 views
Is it possible to use inversion to solve this USAMO problem in 2007?
I've no previous experience to solve any problems by inversive geometry but I am willing to see how it works. But I think I know some of the basic definition about inversion in geometry. Also I expect ...
8
votes
5answers
1k views
A circle with infinite radius is a line
I am curious about the following diagram:
The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and ...
1
vote
1answer
104 views
Inversion map in higher dimensions (preserving angles' size)
Consider the inversion with respect to the sphere $S^n \subset R^{n+1}$, that is the map
$$ \rho \colon x = (x_1,\dots x_{n+1}) \in R^{n+1}-\mathrm{O} \mapsto \frac{x}{\left \| x \right \|^2} \in ...
1
vote
1answer
125 views
Möbius transformation from the complement of the closure of a disc to the unit disc
Let $D$ be the unit disc. Let $U$ be a disc of radius $R$ centered at some $\alpha \in \mathbb{C}$. I want to show that there is a Möbius transformation from $\mathbb{C_{\infty}}$\ $\bar{U}$ into $D$.
...
6
votes
3answers
2k views
Constructing a circle through a given point, tangent to a given line, and tangent to a given circle
While browsing around about problems similar to the problem of Apollonius, I have found references to constructions of all types of circles. For example, not only is it possible to construct a circle ...
6
votes
2answers
444 views
Constructing the circle inversion inverse of a point with ruler only
I've been reading a bit about inversive geometry, particularly circle inversion. The following is a problem from Hartshorne's classical geometry, which I've been struggling with on and off for a few ...
12
votes
2answers
510 views
Finding the circles passing through two points and touching a circle
Given two points and a circle, construct a/the circle through the two points and
touching the given circle.
I came across this problem in History of Numerical Analysis by H. Goldstein. I
spent some ...
2
votes
1answer
85 views
Is there any invariance under the inversion mapping?
In geometry, there is a transformation called the inversion mapping which maps nonorthogonal circles into nonorthogonal lines and vice versa.(If I make a mistake, inform me, since I am not very ...
3
votes
3answers
486 views
Conformality of Inversion Map
I am trying to show that elements of the general Möbius group generated by an affine transformation $f(z) = az+b$, the inversion map $f(z)=\frac{1}{z}$ and complex conjugation $f(z)=\overline{z}$, ...
11
votes
4answers
738 views
Equation of the complex locus: $|z-1|=2|z +1|$
This question requires finding the Cartesian equation for the locus:
$|z-1| = 2|z+1|$
that is, where the modulus of $z -1$ is twice the modulus of $z+1$
I've solved this problem algebraically ...
8
votes
2answers
249 views
If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear
If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear.
I really can't seem to get anywhere on this problem, ...
