The intersection of two or more sets, written $A\cap B$ or $\bigcap_{i\in I} A_i$, is the set of all elements contained in *all* given sets.

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20
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2answers
2k views

Divide circle into 9 pieces of equal area

I'd like to divide a unit circle disk into nine parts of equal area, using circle arcs as delimiting lines. The whole setup should be symmetric under the symmetry group of the square, i.e. 4 mirror ...
0
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1answer
966 views

Find the intersection points of the line L with the three coordinate planes Oxy, Oyz, and Ozx

Let L be the line given by the parametric equations x = 1 + 2t, y = −1 − t, z = 3t Find the intersection points of the line L with the three coordinate planes Oxy, Oyz, and Ozx
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1answer
576 views

Intersection of Two Circles

I have two circles as: $C_1: (x-x_1)^2+(y-y_1)^2=r_1^2$ and $C_2: (x-x_2)^2+(y-y_2)^2 =r_2^2$ and these circles have non-empty intersection. In other words $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\leq ...
0
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1answer
132 views

Build equation of a curve with set of coordinates

I need to calculate the intersection of two curves. I do not have the equation of the curves, but I will have a finite set of coordinates. Is there a way to build the equation for this curve based ...
0
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2answers
184 views

Bilinear Interp Surface Intersection

Edit: I rephrased the question to make it clearer, sorry! I'm trying to solve for the intersection of two surfaces in three spatial dimensions and time. Consider each of these surfaces as some ...
0
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0answers
124 views

Intersection of a hyperplane and a quadric surface.

I have $r, d \in \mathbb{R}^n$. I want to find all $y \in \mathbb{R}^n$ such that $r \cdot y = 0$ and $d \cdot \langle y_1^2, \dots, y_n^2 \rangle = 0$. The trivial $y = 0$ solution is not terribly ...
1
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0answers
65 views

Intersection between 2 functions describeing falling objects with air drag?

So I got 2 functions, both describing the y position of an object moving with a certain acceleration and air drag(tough a very simplified one) as a function of the time t. ...
2
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1answer
186 views

Angle of first circle where it intersects second circle

First, some background. I'm writing an application which a bit more mathematically challenging than what I'm used to. I have two circles that overlap (not just touch, I mean there are two intersect ...
1
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1answer
167 views

Intersection of 3D curves parameterised by piecewise defined functions

I need to calculate the intersection of two 3D parametric curves $\vec{C_1}$ and $\vec{C_2}$. link to image Those curves are parameterised by piecewise functions. $\vec{C_1}= ...
0
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1answer
94 views

Find the intersection between point and circle

given a line segment with endpoints P1 and P2 and a Circle with Center C and Radius R where it is known that P1 lies outside the circle and P2 lies inside the circle, what is an efficient way to find ...
3
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2answers
273 views

Intersection surface area of 3 circles with 3 different radius

I am trying to find the equations to calculate: The intersection surface area generated by the intersection of 3 circles (3 circles like a Venn Diagram). The 3 circle's radius could be be different ...
2
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1answer
3k views

Finding the line of intersection of 2 planes by row reducing the linear system matrix

Problem Statement: Find the plane that passes through the point (-1, 2, 1) and contains the line of intersection of the planes: $$x+y-z = 2$$ $$2x - y + 3z = 1$$ I understand there is a means of ...
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2answers
2k views

Calculate intersection of vector subspace by using gauss-algorithm

There are two vector subspaces in $R^4$. $U1 := [(3, 2, 2, 1), (3, 3, 2, 1), (2, 1, 2 ,1)]$, $U2 := [(1, 0, 4, 0), (2, 3, 2, 3), (1, 2, 0, 2)]$ My idea was to calculate the Intersection of those two ...
3
votes
1answer
170 views

Poincaré duality

Let $X$ be a a compact oriented manifold of dimension $n$. Assume that its (co)homologies have no torsion. Then Poincaré duality says that $$ H^{k}(X,\mathbb{Z})\cong H_{n-k}(X,\mathbb{Z}) $$ holds ...
0
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0answers
207 views

Algorithm for intersection between polyline and rectangle?

My problem is simple, and probably obvious from the title itself, but I'll still clarify it a bit: I have a rectangle and a polyline (array of N connected points). I need an optimal algorithm that ...
0
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1answer
116 views

Union of system of inequalities

I have a system of inequalities $$|z-a_k|\le R_k$$ where $z=x+iy$ (complex number) and $a_k$ and $R_k$ are real numbers for $k=1, \dots, n$. Basically the inequality above shows circle with center ...
0
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1answer
313 views

How to find the intersection of union of two circle groups

I have two groups of circles. S1 is the union of the first group and S2 is the union of the second group of circles. I know center and radius of all circles. I have to find the equation for the ...
0
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1answer
395 views

Circular Sector to Circle Intersection

Is there a formula for determining if a sector intersects a circle (as well as determining if the circle/sector are inside each other)? Sector definition: A center point $P(x,y)$, a starting angle in ...
2
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0answers
530 views

Submatrix Notation

I'm looking through some computer science papers and I see some notation that I'm just not familiar with. Consider an 5 x 6 matrix $$G = \begin{pmatrix} a_{0,0} & a_{0,1} & ...
1
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0answers
419 views

Solve for the intersection point, given two sets of data

I need to (numerically, specifically in C++) solve for the intersection point of two curves, f(x) and g(x). I am given two sets of data, one for each curve (eg. ...
3
votes
1answer
759 views

Intersection between a cylinder and an axis-aligned bounding box

Given a 3D axis-aligned bounding box (represented as its minimum point and maximum point) and a 3D cylinder of infinite length, what's the best way to test for intersection?
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2answers
280 views

I want to find 3 planes that each contain one and only one line from a set

The three lines intersect in the point $(1, 1, 1)$: $(1 - t, 1 + 2t, 1 + t)$, $(u, 2u - 1, 3u - 2)$, and $(v - 1, 2v - 3, 3 - v)$. How can I find three planes which also intersect in the point $(1, 1, ...
2
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4answers
176 views

How many times do these curves intersect?

When the curves $y=\log_{10}x$ and $y=x-1$ are drawn in the $xy$ plane, how many times do they intersect? To find intersection points eq.1 = eq. 2 $$\begin{align*} \log_{10}x &= x-1\\ 10^{x - 1} ...
0
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1answer
71 views

Finding POI's for the follow two curves

So I need to find the POI (point of intersection) of the following two curves: \begin{align*} r & = 1 + \cos \theta, \\ r & = 2 - 2\cos \theta. \end{align*} What I did was I just set both the ...
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1answer
509 views

3D intersection point between circle and triangle

Given a 3D triangle with vertices $(v0, v1, v2)$ and a 3D circle of radius $r$, centered at $c$, and lying in the plane perpendicular to $axis$, how can I test for intersection points between them? ...
0
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1answer
446 views

Finding the distance between the centre of an arbitrarily rotated cylinder and a point on that cylinder

this is a bit more complicated than the post title suggests because I was running out of words. I suppose the full title would be: "Finding the distance between the centre of an arbitrarily rotated ...
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2answers
209 views

ray - parallelogram intersection in 2d

i'm looking for a fast method to get the intersecting points between a ray and a parallelgram defined by the 4 vertices! till now i've thought to test the intersection point between ray and the 4 ...
0
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1answer
1k views

Find the intersection of a line (segment) and an ellipse (from the center of ellipse)

Here is what I know: The location of the center of the ellipse C (20,10). The Major axis (2a) or 400 (a being 200) - this is on the X axis. The Minor axis (2b) or 200 (b being 100) - on the y axis. ...
0
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1answer
213 views

Intersection between 3D closed contour and 3D plane

I have a set of N 3D points (x,y,z cartesian coordinates) defining a closed contour and a 3D plane defined by a point and a normal. What I want to determine is the point(s) where the closed contour ...
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3answers
374 views

Why doesn't this conditional probability equation hold

This is probably a very stupid question but I can't wrap my head around it. $$ P(B \cap A) = P(A \cap B) = P(B \mid A)\cdot P(A) + P(A \mid B)\cdot P(B) $$ Can someone explain intuitively why the ...
2
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0answers
42 views

elipsoid surface intersection in $\mathbb{R}^3$

Is there an explicit parametric solution describing the curve result of the intersection of two elipsoid surfaces with abitrary position and orientation in $\mathbb{R}^3$?
2
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0answers
125 views

Intersection of a cone $x^2+y^2-z^2$ and a generic plane in $\mathbb{RP}^3$

If we take the zero locus of $x^2+y^2-z^2$ to be our cone, I'd like to know how to go about finding the intersection of the cone and a generic plane $Ax+By+Cz+Dw=0$. The result will be a conic, but ...
5
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2answers
547 views

21 sided regular polygon and its diagonals

In a $21$ sides regular polygon, how many points inside it are intersection of its diagonal? I found that a polygon with $n$ sides has $\dfrac{n(n - 3)}{2}$ diagonals, but I feel this is not so ...
3
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1answer
156 views

proving that four axis-parallel rectangles whose intersection graph is a cycle delimit another rectangle

Working on the 2D plane, I'm looking for an elegant proof of the fact that if four regions $A$, $B$, $C$, $D$ delimited by axis-parallel rectangles are such that: $A$ intersects $B$, $B$ intersects ...
2
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2answers
112 views

Is every invertible rational function of order 0 on a codim 1 subvariety in the local ring of the subvariety?

I have been trying to read Fulton's Intersection Theory, and the following puzzles me. All schemes below are algebraic over some field $k$ in the sense that they come together with a morphism of ...
2
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1answer
2k views

determine if 2 line segments are intersecting

currently I write a program where finding out whether 2 line segments intersect is an essential part of the algorithm. Could anyone tell me if there's a way to determine if two segments are ...
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4answers
1k views

Given two parallel line segments, how do I tell if and where they overlap?

To find if two line segments intersect I am this code The problem is this code: ...
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2answers
137 views

Partition of a Nonempty Set $X$

Let $X$ be a nonempty set, and $\{A_\alpha : \alpha\in I\}$ be a partition of $X$. If $B\subseteq X$ such that $A_\alpha\cap B\neq\emptyset$ for every $\alpha\in I$, is $\{A_\alpha\cap B : \alpha\in ...
0
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1answer
1k views

Find intersection of two functions in Maple

I have two functions $f$ and $g$ defined as: $$f := \operatorname{Spline}(\operatorname{pointsBW}, t)$$ $$g:=x \rightarrow \frac{1}{x}\int_0^x f\;dt$$ where $f$ is a functional form for a set of ...
2
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1answer
578 views

Find intersection(s) between parametrized parabola and a line

I'm trying to find the value(s) of the parameter $t$ at the intersection point(s) between a 2D general parabola (as a parametric function of $t$) and a line whose equations can be derived from two ...
0
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2answers
425 views

Position of 3 circles intersecting at the centre of bounding box

Here's what I feel is a neat challenge: I'm building a data visualization comprised of 3 circles of dynamic sizes. I want to have them all intersect at the centre of a bounding box that will also be ...
2
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1answer
104 views

Is the intersection of the diagonal with a graph always transverse in characteristic zero

Let X be a projective smooth connected curve over $\mathbf{C}$. Let $f:X\to X$ be a non-constant morphism. Is the intersection of the diagonal $\Delta_X$ and the graph $\Gamma_f$ on $X\times X$ ...
2
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1answer
129 views

how does one intersect the diagonal with a graph on the surface $X\times X$

I want to do a concrete example of an intersection product for myself. Consider the endomorphism $f:\mathbf{P}^1_k\to \mathbf{P}^1_k$ given by $(x:y)\to (y:x)$. It has precisely two fixed points: ...
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0answers
364 views

How to find intersections of hyperbolas by MATLAB

Let's have two hyperbolas given by equations: $$\mathbf{r}^T\cdot \mathbf A_1\cdot\mathbf r+\mathbf b_1^T\cdot\mathbf r+c_1=0$$ $$\mathbf{r}^T\cdot \mathbf A_2\cdot\mathbf r+\mathbf b_2^T\cdot\mathbf ...
2
votes
2answers
150 views

How do I think about solving this sort of problem without having to count intersections?

I have asked this sort of question before, and I have a new similar question here. The latter question got me thinking about this specific question. When throwing 5d20 (rolling five twenty sided ...
2
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1answer
303 views

points of intersection on a randomly situated plane and ellipsoid (spherical) in 3d space

if i have an ellipsoid and a plane oriented in any way in a 3 dimensional coordinate system, and they intersect; is there a way to find an equation that describes (or at least approximates) all points ...
0
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1answer
84 views

Is this a valid solution to this probability problem?

I have two non-mutually exclusive events with probability $P(A)$ and $P(B)$. In addition, I am given the intersection of both events: $P(A \cap B)$ Is it then valid to say: $$ P(A' \cup B') ...
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2answers
4k views

determine where a vector will intersect a plane

I have a vector with position $O=(o_1,o_2,o_3)$ and direction $D=(d_1,d_2,d_3)$ and a plane determined by 3 points $A=(a_1,a_2,a_3),B=(b_1,b_2,b_3),C=(c_1,c_2,c_3)$. In which point will the vector ...
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1answer
344 views

How to describe the intersection of two sets?

The following is a homework problem: Let $$\begin{align*} W_1 &= \{(a_1, a_2, a_3) \in\mathbb{R}^3 \mid a_1 = 3 a_2\text{ and }a_3 = -a_2\}\\ W_2 &= \{(a_1, a_2, a_3) \in ...
3
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1answer
140 views

Linear Algebra problem: intersection of a subspace with a cone.

In $\mathbb{R}^n$, consider the closed cone $$C^+ = \{ (x_1, \ldots, x_n) : x_i \geq 0,~~i= 1, \ldots, n\}.$$ Let $S \subseteq \mathbb{R}^n$ be a subspace (of any dimension) such that $S \cap C^+ = ...