All Questions
Tagged with geometric-probability euclidean-geometry
17
questions
0
votes
2
answers
116
views
Probability that a line intersect two other lines inside the unit disk.
Consider the unit disk: $D=\{(x,y)\in\mathbb{R}^{2}:x^2+y^2\leq 1\}$ and the lines $x=0$ and $y=0$. We want to find the probability to draw a line which intersect the first two with intersection ...
- 888
0
votes
0
answers
91
views
Average Minimum Distance between Curve and Circle
I aim to formalize the average minimum distance between any point on a circle with a radius $r$ and an infinitely long curve.
I know the location of the curve and circle, but there is no formula for ...
3
votes
1
answer
93
views
Distribution of distances to a hyperplane
Suppose I have the unit sphere in $R^3$ and WLOG have a hyperplane be the $x-z$ plane. I uniformly at random choose points within the sphere. I know that the distribution of the coordinates of those ...
user1013993
0
votes
1
answer
197
views
Similar parallelogram within parallelogram, calculating ratio of areas
Here are two questions from my probability textbook:
481. A floor is paved with tiles, each tile being a rhomboid whose breadth measured perpendicularly between two opposite sides is $a$, and ...
- 1,415
2
votes
2
answers
203
views
Probability stick I drop parallel to diagonal of rectangle fits within the rectangle
Here's a question from my probability textbook:
A floor is paved with rectangular bricks each $a$ inches long and $b$ inches wide. A stick $c$ inches long is thrown upon the floor so as to fall ...
- 1,415
1
vote
1
answer
183
views
Probability chord of bigger circle intersects smaller circle
You are given two concentric circles $C_1$ and $C_2$ of radius $r$ and $r/2$ respectively. What is the probability that a randomly chosen chord of $C_1$ will intersect $C_2$?
Answer: $1/2, 1/3$ or $1/...
- 1,107
0
votes
1
answer
38
views
How to introduce 'noise' to an N-Ball boundaries while keeping it balanced?
Given an N-dimensional space and a set of randomly distributed points in it, I define an N-Ball and I classify as "1" all points within the N-Ball, and "0" for all the rest.
I'm ...
- 1
0
votes
1
answer
461
views
Calculating Average shortest distance between random points in a rectangle
My question is quite similar to this one. But what would be the solution in case of a rectangle with width W and length L??
Q1
I would like to find the average shortest distance between randomly ...
5
votes
1
answer
389
views
Expected projected length of radial vectors of n-sphere
Situation
In $n$-dimensional Euclidean space rests a unit $(n-1)$-dimensional sphere that is orthographically projected onto a $(n-1)$-dimensional plane. The topological definition of a sphere is used,...
- 523
1
vote
1
answer
92
views
Prove that probability of choosing an isosceles traingle in Set of traingles is $0$.
$S$ is set of triangles of unit area. All members of $S$ are uniformly distributed. Let $A$ be the event that a randomly chosen member of $S$ is an isosceles triangle. Prove that the probability of $A$...
7
votes
1
answer
396
views
Expected triangle area of normal distributed vertices with colinear expectations
Situation
Given are 3 independent multinormal distributions $X_i=\mathcal{N}(\vec\mu_i,\Sigma)_{i=1,2,3}$ in $\mathbb{R^3}$.
For simplification the expectations are colinear:
$\vec\mu_1=\begin{pmatrix}...
- 523
5
votes
2
answers
2k
views
Average shortest distance between some random points in a box
Suppose there is a square box with side length $m$ (measured in pixels). Let there be $n$ points in this box, distributed uniformly within the box (with integer coordinates, aligned to a pixel grid). ...
- 51
1
vote
1
answer
107
views
How likely is it that a random plane through the origin will intersect positive space?
In an n-dimensional hyperspace, how likely is it that a randomly chosen plane passing through the origin will intersect "all-positive co-ordinate space"?
(By "all-positive co-ordinate space" I mean ...
- 113
2
votes
1
answer
156
views
How can I uniformly draw points from an ellipsoid?
Specifically, given a positive definite matrix $A \in \mathbb{R}^{n \times n}$, how can I efficiently generate points $x \in \mathbb{R}^n$ that satisfy $x^TAx \leq 1$? I know how to do this when the ...
- 144
12
votes
0
answers
415
views
Expected area of an inscribed triangle in a sphere
On the surface of a unit sphere, three points $A$, $B$ and $C$ are
chosen in the following way:
Points $A$ and $B$ are chosen randomly and independently on the whole surface
After $A$ and $...
- 22.5k