Intereting Posts

Solving recurrence relation: Product form
Lack of understanding of the proof of the existence of an irreducible polynomial of any degree $n \geq 2$ in $\mathbb{Z}_p$
Kolmogorov Backward Equation for Itô diffusion
Find an orthonormal basis of a plane
what is the definition of Mathematics ?
$f:\mathbb R \to \mathbb R$ be differentiable such that $f(0)=0$ and $f'(x)>f(x),\forall x \in \mathbb R$ ; then is $f(x)>0,\forall x>0$?
When was Regularity/Foundation universally adopted?
Product of one minus the tenth roots of unity
How to show this sequence is a delta sequence?
How can I prove the formula for calculating successive entries in a given row of Pascal's triangle?
Prove that a holomorphic function with postive real part is constant
Relationship between Continuum Hypothesis and Special Aleph Hypothesis under ZF
Precise definition of epsilon-ball
Definition of a measurable function?
Homogeneous riemannian manifolds are complete. Trouble understanding proof.

*This is a spin-off from a comment on Stack Overflow.*

How can I find a homography between two ellipses in the plane?

- Plot $|z - i| + |z + i| = 16$ on the complex plane
- Convert this equation into the standard form of an ellipse
- 2D point projection on an ellipse
- How to geometrically prove the focal property of ellipse?
- Parabola in parametric form
- 538.com's Puzzle of the Overflowing Martini Glass - How to compute the minor and major axis of an elliptical cross-section of a cone

- What is the angle $<(BDE,ADH)$?
- Rigorous synthetic geometry without Hilbert axiomatics
- What Does Homogenisation Of An Equation Actually Mean?
- Is it possible to partition a real Banach space into closed half-lines?
- Minimum operations to find tangent to circle
- Probability that $n$ random points on a circle, divided into $m$ fixed and equal sized slices are contained in less than $m/2$ adjacent slices.
- Different ways finding the derivative of $\sin$ and $\cos$.
- Finding the radius of a third tangent circle
- find the point which has shortest sum of distance from all points?
- Tarski-like axiomatization of spherical or elliptic geometry

You can map any non-degenerate conic (i.e. it doesn’t factor into two lines) and find a homography to any other non-degenerate conic. So you can even map ellipses to hyperbolas and the likes. The mapping won’t be unique, but leave you three real degrees of freedom even after both conics have been defined.

A projective transformation (i.e. a homography) of $\mathbb{RP}^2$ is uniquely determined by four points and their images. You can start by choosing three points $A,B,C$ on the first conic, and corresponding image points $A’,B’,C’$ on the second. Choosing the preimage points by themselves tells you nothing about the mapping. Associating image points gives you one degree of freedom each, for choosing a point on a conic.

The fourth point $D$ might be chosen arbitrarily, but $D’$ has to be in a specific position in order to map the conic as a whole correctly. This is becuse four points don’t define a conic yet, in general you need five there. The quantity which determines the correct choice for $D’$ is the cross ratio of the lines connecting $A,B,C,D$ to another point on the first conic. This cross ratio has to be the same as the cross ratio for $A’,B’,C’,D’$ connected to a point on the second conic. So one way would be computing the cross ratio for $A,B,C,D$ then constructing $D’$ to match that value.

A more geometric approach does not compute the cross ratio, but instead construct a specific cross ratio, namely $-1$ so that you obtain a harmonic range. Here is a method to construct this: Construct tangents to the first conic in $A$ and $B$. Intersect these tangents, and connect the point of intersection with $C$. That line will intersect the conic in $C$ itself and one other point. Call this other point $D$. Do the same for the second conic to obtain $D’$. Then the uniquely determined homography from $A,B,C,D$ to $A’,B’,C’,D’$ will map the first conic to the second.

- One root of an irreducible polynomial in an extension field, so is the other.
- What are $x$, $y$ and $z$ if $\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y} = 4$ and $x$, $y$ and $z$ are whole numbers?
- Show that $S\cap(T\,\cup \,T') = (S\,\cap\,T)\cup(S\,\cap\,T')$
- Is the product of three positive semidefinite matrices positive semidefinite
- Least Impossible Subset Sum
- Proof of the relation $\int^1_0 \frac{\log^n x}{1-x}dx=(-1)^n~ n!~ \zeta(n+1)$
- In how many words the letter of word RAINBOW be arranged so that only 2 vowels always remain together?
- Diophantine equation: $7^x=3^y-2$
- Is there a reason why the number of non-isomorphic graphs with $v=4$ is odd?
- Asymptotic behaviour of sums of consecutive powers
- If we would have a perfect random decimal number generator, what would the chances be for the occurence of the numbers?
- Maximum likelihood with Bernoulli trials: what to do if there are no successes?
- How to check if a 8-puzzle is solvable?
- Using tan(x), show that open interval is diffeomorphic with the real line
- Expected number of intersection points when $n$ random chords are drawn in a circle