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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linear algebra find a line that intersects another line

question: Let L be the line with parametric equations x = 3+2t y = −5 z = −6−3t Find the vector equation for a line that passes through the point P=(−5, 5, −6) and intersects L at a point that is ...
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Explanation of solving intersection of two planes

I understand that in order to solve for intersection line of two planes, you must find the cross product of the normal vectors of each plane which will be parallel to the line of intersection. That ...
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1answer
17 views

intersection of an ellipsoid and cylindrical plane.

I need to understand if an ellipsoid and a cylindrical arc intersect, what will be the general equation of the cutted ellipse? How can I solve for that equation? I know in 3D, the equation of an ...
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1answer
19 views

Intersection of Level Curves and a Ellipse at a given angle

I am preparing for an exam and I'm going over previous administered tests. I have come across the following problem and have little idea how to tackle it. It goes as follows: Let ...
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1answer
19 views

Calculating number of students who don't study any language

According to a survey of 100 students, there are 40 students studying English, 30 studying French, and 25 studying Spanish. Inaddition, 8 students are studying English and French, 6 are ...
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1answer
32 views

What does it mean for the infinite intersection of nested sets to be empty?

$\bigcap I_i=\emptyset$ where $I_i=(0, \frac{1}{i})$ What does it mean that the infinite intersection is empty?
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1answer
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line segment intersection strange results

I'm using this formula. I am getting very strange results with (1,3) to (29,17) and (6,19) to (7,8). I got an X* value of 7. When I plugged this into my intercept calculator it said they intercept at ...
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1answer
20 views

Probability Intersections

$A$ and $B$ are two events from certain probability space $\Omega$. Knowing that: $P(A)=0.6$, $P(B)=0.7 $ and $P(A\cup B)-P(A\cap B) = 0.2$ Determine $P(A\cap B)$ It says in this sheet that the ...
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23 views

Intersection points of two trignometric equations

I am studying for a SAT II Math2C and I came across this question in Barron's book. Solve $2 \sin(x) + \cos2(x) = 2 \sin^2(x) - 1$ [0<= x <= 2pi] The solution says put the equations in a ...
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29 views

Unit sphere x axis intersections

This is a problem from a vector calculus textbook, Higher Order Derivatives Consider the unit sphere S given by x^2+y^2+z^2=1. S intersects the x-axis at 2 points. Which variables can we solve for at ...
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How do I define a three space?

I'd like to take a hierarchically-modeled (vertices, lines, faces, etc) 4D object and find its intersection with three-space. In Paul Isaacson's thesis, Computer Graphic Presentation of Hypothesized ...
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2answers
84 views

Proving that the set of real numbers is a topological space.

I recently finished an activity provided by a professor where one of the questions was to prove that the set of real numbers is a topological space. The hint provided was to "consider the union of ...
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2answers
12 views

Points of intersection for two polar equations question

Why is it that when I try to find the points of intersection for $r=2$ and $r=4*\cos(2\theta)$, I only get the $\theta$ where the reference angle is $\pi/6$? There is clearly another solution between ...
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0answers
26 views

Volume of intersection between two equal cones with parallel axes

The two infinite cones (nappes) (each 45-degree wide) have parallel axes. They are oriented in opposite directions, and the top of one is inside the other, so that the common volume V is finite. How ...
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13 views

Construction of concurrent or parallel lines from a parallelogram (proof by vectors)

I have a problem with this probably easy exercise on vectors. Any help would be great. Let ABCD be a parallelogram. The line parallel to AB intersect BC and AD in points Q and S, respectively. The ...
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26 views

Find points that defines the intersection of an ellipse with a plane.

I want to test for the intersection of two ellipses $E_1$ and $E_2$ in $\mathbb{R}^3$ represented on a computer. In some sense, this isn't a hard problem: ...
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1answer
22 views

Measure on intersections of unions

Let $(X,\mathcal{A},μ)$ a measurable space and let $A_1,A_2,...∈\mathcal{A}$, assume that $\sum\limits_{j=1}^{\infty}=\mu (A_j)<\infty$ We have ...
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14 views

Measure of intersections [duplicate]

Let $(X,\mathcal{A},μ)$ a measurable space and let $A_1,A_2,...\in \mathcal{A}$, assume that $\sum^{\infty}_{j=1}\mu(A_j)<\infty$ I want to show that ...
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68 views

Prove that the two polynomials intersect each other only at a single point

Here are the polynomials: $$D^K_1(\theta)=\sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i(1-\theta)^{K-i}$$ and $$D^K_2(\theta)=\frac{1}{2}\sum_{i=\lceil{K/2}\rceil}^K ...
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1answer
26 views

Vector Parametrization of a Hyperbolic Paraboloid and a Plane

So I need to find the intersection between a hyperboloid ($z=\frac {y^2}{b^2}-\frac{x^2}{a^2}$) and some related plane ($bx+ay-z=0$). I have tried solving for $z$ and equating the two: ...
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1answer
25 views

Determine Circle of Intersection of Plane and Sphere

How can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the circle? At a minimum, how can the radius and center of the circle be determined? ...
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1answer
30 views

Find the equation of line and finding a point in given example

The outer circle is $x^2+y^2=1$ and the smaller circle is $x^2+(y+1-r)^2=r^2$. The arclength is parameterised anticlockwise with $s=0$ at the bottom as shown. If we know $s_n$ and $s_{n+1}$ can we ...
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1answer
36 views

Longest chord inside the intersection area of three circles

I am currently working on my masters thesis in computer science and I stumbled onto a geometry problem. My goal is to compute the length of the longest possible chord inside the intersection area of ...
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1answer
18 views

Intersection problem

I am given an interval $A=(0,1)$ and $B=(1,2)$. How can I show that $A \cap B$ is an empty set? I tried to prove by contradiction by saying an arbitrary element $k\in A\cap B$ so $k \in A$ and $k \in ...
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Finding the limit of sets

I have a problem, where I have difficulties with solving these two exercises. Here is the problem If $C_1$, $C_2$ ... are sets where $C_k \supset C_{k+1}$, where k=1,2,3. Then the limit of $C_k$, ...
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1answer
42 views

How to compute coordinates of a point that intersects an sphere

Hi all. Is there a way to compute the S(x,y,z), given the following information: A(x,y,z) e = elevation (from the line AS) Az = azimuth (over A). Perpendicular to x axis. Can vary from 0 tp 360. ...
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1answer
22 views

How to determine a general arithmetic sequence formula for two intersecting trig function

I have equations out of two trigonometric functions. For example $\cos(4\alpha$) = -$\sin(5\alpha)$ $\tan(0.5\alpha$) = 2 $\sin(\alpha)$ How can I determine a general arithmetic sequence formula ...
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5answers
83 views

How to find the point of intersection with three equations?

Given the following equations with three variables $a, b, c$ $a-5b+4c=-3$ $2a-7b+3c=-2$ $-2a+b+7c=-1$ How can I determine the point (if it exists) at which all three lines intersect?
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4answers
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The intersection of $3$ set is empty, would the intersection of $4$ sets be empty?

Let me clarify some more. Let's say we have four sets $A,B,C,$ and $D$. If the intersection of any three sets is empty, by default is the intersection of all four sets empty?
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Find circles that completely cover a polygon minimizing the amount of space covered outside the polygon

I have an arbitrary polygon that I need to roughly represent using circles. Any point inside the polygon must lie inside a circle. There will be points outside the polygon that will fall under a ...
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1answer
14 views

calculate the coordinates of the intersection between a bisector and a sector

I have a sector and would like a formula that gives the intersection between the bisector and the arc. here's a graph of the situation: the point B is the center of a circle of radius AB and the BD ...
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Probability of infinite intersections

While I was studying Probability and random processes I came across the following question. Say I have $A_1,A_2, \ldots, A_n$ events such that $A_i$ is in $E$ but not equal to $E$. What is: ...
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1answer
37 views

length of secant line.

I'm looking for way to find the length of a secant line intersecting another line through the center of a circle with a known radius. The intersection point is on the circle and the angle between 2 ...
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2answers
67 views

Circle Line segment intersection

I have a circle with radius r and center $(c_x, c_y)$. I have a line segment $(x_1, y_1)$ and $(x_2, y_2)$ given $(x_2, y_2)$ is always a point inside the circle. I am trying to find the ...
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1answer
50 views

Union of Intersections [closed]

What is the general formula for finding the union of intersections? For example , Consider the sets $A , B , C $ I need to find $$((A \cap B)\cup(A \cap C)\cup(B \cap C))$$ Please extend this as a ...
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1answer
46 views

Projective and affine conic classification

I have a doubt on the classification of non-degenerate conics (parabola, ellipse, hyperbola) in projective geometry (my textbook is "Multiple View Geometry in Computer Vision", which, as the title ...
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1answer
36 views

Outer interval of circle intersection

Is there a consistent way to calculate the outer interval $\left(~\mbox{element of}\ \left[0, 2\pi\right]~\right)$ of a circle created by an intersection ?. I calculated the intersection points and ...
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calculate circle segment area: determine distance

I have a problem calculating the area of a circle segment. I know how to separate this into smaller tasks (triangle and remaining circle segment) that are basically easily solvable, but one distance ...
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Testing hypothesis about window non-overlap

I have a large number (~1.5 million) of protein sequences, each of them of different lengths.There are 6 schematic examples in the attached image. Within each of these sequences, there are >= 0 ...
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38 views

Intersection between 2 lines (3D). This doesn't have a solution does it?

so I was looking through an old exam and this question was given: The teachers answer was the point (9, -9, 21) I tried solving this myself, I got x = x, y = y, but I could not find a point where ...
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intersection of a line and plane on a 3-sphere

Suppose I have two 4D points, $\mathbf{a}=(a_1,a_2,a_3,a_4)$ and $\mathbf{b}=(b_1,b_2,b_3,b_4)$, that both lie on a unit 3-sphere (i.e. unit distance from origin). In addition, I have a 2-D plane that ...
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1answer
55 views

Determining intersecting points between square and circle

I unfortunately have spent too much time trying to solve this question, and have turned to you for help. The corner of my square has intersected some circle, and I need to move it out. I only know one ...
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2answers
64 views

Geometry : find the points of tangency between two lines and two circles [closed]

I have a programming problem. I need to find the intersection points between two lines tangent to two circles and the circles! I have the circles' radiuses and centers. So I need points ...
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2answers
60 views

May a monoid have two disjoint submonoids?

I'm asking this question inspired by the similar question about group and its subgroups. I tried to modify the proof presented there to work for monoids but I failed. I'm also not able to find any ...
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122 views

Check if point lies on a line segment

I know there are shorter solutions that use dot product, but I don't know what the logic behind doing so involves so I came up with something that I understand myself (i will research the dot product ...
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130 views

Parametric equation for intersection of curve

Here's the three part question: A) Find parametric equations for curve which is the intersection of the cylinder $x^2 + z^2 = 1$ and the plane y = -x. B) Show that the curve lies on the surface $x^2 ...
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3answers
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Euclidean “straight line” calculation

Please see image first.. I have as input the following (I presume these are in effect Euclidean coordinates): The angle and the length of the red line. The angle and the length of the green ...
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1answer
78 views

How to determine if two ellipse have at least one intersection point

All of the question are in sequence and related. 1.Given 2 ellipse with the position x1,y1, x2,y2 and the radius a1,b1, a2,b2, construct an equation to determine if both of them has at least one ...
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43 views

Binomial distribution or probability intersection

I flip a biased coin, p = 0.5 for getting heads. What is the probability of getting heads 8 times ? Firstly I used probability intersection $$ P(A \cap B \cap C \cap D \cap E \cap F \cap G \cap H) = ...
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2answers
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internal rectangle area intersected by a circle

I need to compute the internal rectangle area intersected by a circle like (the blue area) on these 3 examples: I know every vertex (x,y) coordinate and then their distance from circle center but ...