# Questions tagged [geometric-invariant]

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16 questions
1answer
84 views

### Do two exponential spirals intersect?

I have lists of complex points: orbit of complex point z under quadratic function f(z) = z*z I know that lists are: z, z^2, z^4, z^8, ... (r,t), (r^2, 2*t), .....
2answers
63 views

### The sum of squares of distances from the vertexes of regular polygon to the any line that passes the center of it. [closed]

To prove that it is geometric invariant I need to find some others. I was thinking about proving it by the Pythagorean theorem, using the fact that in all cases the distance from the vertex to the ...
2answers
472 views

### How can I get better at solving problems using the Invariance Principle?

I have some questions regarding the Invariance Principle commonly used in contest math. It is well known that even though invariants can make problems easier to solve, finding invariants can be really,...
0answers
4k views

### How to find Invariant Lines and Lines of Invariant Points, without utilising Eigenvectors?

this is my first post so I do apologise regarding any formatting issues! I have a question regarding Invariant Lines and Lines of Invariant Points; from what I can gather, an Invariant Line is one of ...
1answer
309 views

### Is dot product the only rotation invariant function?

I am looking for rotation invariant scalar functions $f(x,y): x,y \in R^3$ that are not some scalar function over the dot product (or norm), i.e. $f \neq g(x\cdot y, \Vert x \Vert, \Vert y \Vert )$ ...
1answer
382 views

### Invariance of the second moment of area of a regular polygon

Consider a $n$-sided regular (convex) polygon and its circumscribed circle of radius $r$, centered in $(0,0)$. Fixing $(r,0)$ as the coordinate of the first vertex, the $n$ vertices of the polygon ...
2answers
31 views

### Dimension of the group of all motions in $\mathbb{R}^n$ which leaves a fixed r-plane invariant.

As the title, I would like to ask the dimension of the group of all motions in $\mathbb{R}^n$ which leaves a fixed r-plane $L^0_r$ invariant. Here is my observation, but I don't know if it is useful ...
1answer
54 views

### What is known about rational points on the ideal of relations / syzygy ideal?

What is known about rational points on the ideal of relations / syzygy ideal? Let $G$ be a finite group, with $|G|=n$. Then $G$ acts on $\mathbb{Q}[x_1,\cdots,x_n]$ through the regular representation (...
0answers
91 views

### $G$-invariant functions on Manifolds

In a paper I saw the following statement: Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...
0answers
41 views

1answer
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### How to define a “distance” from point to line in 3D projective space which is projectively invariant?

Since the concept of distance in Euclidean space is not invariant in projective space, that is , distance is invariant under Euclidean transformations but not under projective transformations, is it ...
1answer
110 views

### Why are invariants of Homology 3-Spheres interesting?

I see how invariants (of any kind of mathematical objects) are interesting in general, since classification is interesting, but mostly in case there is a vast variety of such objects that we do not ...