# Shape Transformation Via Medial Axis Transform

Mathematics Asked by fweth on December 22, 2020

Let $$P_1,dots,P_n$$ be simple polygons which don’t intersect each other and $$Ssubseteqmathbf{R}^2$$ the set of points lying in the interior of an odd number of the $$P_i$$, so $$S$$ can be thought as the interior of a finite number of simple polygons, each with a finite number of holes, also described by simple polygons. I want to perform the following transformation to $$S$$:

• Apply medial axis transform, i.e. get the set $$textsf{MAT}(S)$$ of triples $$(x,y,r)$$ such that the open ball $$B_r(x,y)$$ is contained in $$S$$ but not strictly contained in another open ball $$B_{r’}(x’,y’)subseteq S$$.
• Given a bijective linear map $$f:mathbf{R}^2rightarrowmathbf{R}^2$$, construct the shape $$S_f:=f^{-1}(cup_{(x,y,r)intextsf{MAT}(S)}B_r(f(x,y)).$$ In other words, for each $$(x,y,r)intextsf{MAT}(S)$$, draw the squeezed ball $$f^{-1}(B_1(0,0))$$ centered at $$(x,y)$$ and scaled by $$r$$ on the plane.

The lines in the medial axis transform don’t have to be straight, but I have the hunch that $$S_f$$ can again be described via polygons like $$S$$. If that’s correct, is there an easy way to directly compute the line segments of $$S_f$$, without computing the medial axis transform first? What if we look at balls in another $$p$$-norm?

[EDIT] No, $$S_f$$ can’t be described via polygons in general. But maybe for certain $$f$$?

## Related Questions

### How to prove that there is not a monomorphism from Klein 4-group to $Z_6$(or a epimorphism from $Z_6$ to $V_4$)?

1  Asked on January 25, 2021 by tota

### Homomorphisms between fields are injective.

4  Asked on January 25, 2021

### We have an integer n. We have n boxes where each box contains a non-negative amount of balls. Find all the permutations which satisfy some criteria

1  Asked on January 25, 2021 by michael-blane

### Approximation of the sum of a series $S(t)=-frac{2}{pi t} cos(frac{pi t}{2}) sum_{m odd}^{infty}frac{m^2alpha_m}{t^2-m^2}$ as $tto +infty$

0  Asked on January 25, 2021 by yolbarsop

### Homogenous space of elliptic curve E/$Bbb Q$

1  Asked on January 25, 2021 by bellow

### On the functional square root of $x^2+1$

9  Asked on January 25, 2021 by user1551

### Is there a simple way to find all the solutions of $x_1 + x_2 + dots + x_k + dots + x_K = N$ when $x_k$s and $N$ are all non-negative integers?

2  Asked on January 25, 2021 by cardinal

### Minimizing the Schatten 1-norm over symmetric matrices.

2  Asked on January 25, 2021 by gene

### The frontier of a set

1  Asked on January 25, 2021 by gal-ben-ayun

### express a matrix using Kronecker product

0  Asked on January 25, 2021 by jyothi-jain

### Approximating multiples of reals with integers

1  Asked on January 25, 2021

### How is this sentence written correctly in maths? (sets)

0  Asked on January 25, 2021 by eyesima

### Let $n$ be an integer. If the tens digit of $n^2$ is 7, what is the units digit of $n^2$?

6  Asked on January 24, 2021 by user713999

### Partitioning $mathbb{R}$ for given functions.

0  Asked on January 24, 2021

### How to count the number of symmetries of a 3-d object?

1  Asked on January 24, 2021

### Is checking really needed?

3  Asked on January 24, 2021 by 1b3b

### Struggling to understand proof, there is no simple group of order $525$

1  Asked on January 24, 2021 by scott-frazier

### prove the following compact result about $GL(n,mathbb{R})$

1  Asked on January 24, 2021

### i need to proove that An∪B→A∪B

1  Asked on January 24, 2021 by methodcl

### How can I show that $T(omega) = T(overline{omega})$ when $X_{t}(omega)=X_{t}(overline{omega})$ for all $t in [0,T(omega)]cap [0,infty)$

1  Asked on January 24, 2021 by minathuma