# Tagged Questions

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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### How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
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### How can I tell when two cubic Bézier curves intersect?

I'm working a little program that converges on vector-based approximations of raster images, inspired by Roger Alsing's genetic Mona Lisa. (I started on this after his first blog post two years ago, ...
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### Minimum number of vertex moves to un-intersect a polygon with itself

In my game, I have n points that form a self-intersecting polygon. The points can be moved by dragging them. How can I form a non-intersecting polygon this way, in ...
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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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### 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 $\mathbb{R}^3$?
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### 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 ...
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### How do I use k-dimensional planes as bounds for generating k-dimensional vectors?

I am an essentially self-trained programmer with little mathematical background. I do not quite know where to start for this problem, and do not know the terminology to help me get conclusions ...
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### How to find all intersection points of two splines?

2D-Cubic splines are given in parametric form (X(t), Y(t) and X(s), Y(s)). Every segment has it's own X and Y expression. And I want to find all intersection points. Some segments are intersecting ...
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### 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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### 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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### 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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### 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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### Intersection between sphere and ellipsoid

I am failing since two days to compute and to plot the intersection of an ellipsoid in parametric notation ...
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### Is there a relation for when a circle intersects more than half the perimeter/circumference of another circle?

Is there some nice formula or algoritm for determining when a circle "hides"/intersects more than half of the perimeter of another circle, in a circle-circle interaction. Example image: Two example ...
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### Finding intersections points of pairs of polar curves?

Find all intersections of the curves $r=3^{(1/3)}\cos(\theta) , r=\sin(\theta)$ What I have done so far is to just put them equal to each other like this: $3^{(1/3)}\cos(\theta)=\sin(\theta)$, but ...
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### Find the basis for the intersection of two subspaces in infinite-dimensional vector spaces

For an infinite-dimensional vector space U, and two subspaces W and V, we assume at least one of the two subspaces (W and V) is also infinite-dimensional. How can I find the basis for the ...
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### Odly phrased big-oh question, have I done what is required? (because it's not really big-oh, it's more graph sketching)

I've encountered these before, but never phrased or defined as follows, I'd like to know if I've done whatever the question wants to draw attention to (if it didn't want to draw attention to ...
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### Finding the intersection of two three-dimensional functions

I have two large equations, both of the same form which I am trying to find the intersections of. The equations are:  f(x, y) = \frac{\frac{x ^ 2}{r_1} + \frac{y ^ 2}{r_2}} {1 + \sqrt{1 - \frac{p_1 ...
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### Arc To Plane Intersection

I have a plane and an arc with center, radius, start and end angles. All i need to find if if the arc hits on the chosen plane or not. Line to plane intersection is much easier as the surface is ...
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### Probability on intersection

If I have two sub-areas confined in a larger area, and I have the intersection of these two sub-areas. If I replace one of the sub-areas with a new sub-area, which is the same size. What is the ...
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### Probability for intersection of sets

(It is an easier - hopefully - formulation of a question I asked previously). Assume that we are given $J$ sets, each set with $n_j$ elements ($J$ and $n_j$ for each $j \le J$ are known). The ...
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### Vector Tangent to Curve of Intersection

I am having problems solving this. Find a vector tangent to the curve of intersection of $z = 4x^2 + y^2$ and $z=(27-x^2-y^2)^{1/2}$ at the point $(1,1,5)$. I'm able to do this kind of thing using ...
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### Finding the equation of a plane

I need to find the equation for the plane which goes through the intersection line of the planes $x-z=1$ and $y+2z=3$, and perpendicular to the plane $x+y-2z=1$. What I got so far - to find the ...
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### Equation to calculate intersected cells in a grid, given a selection rectangle

I am looking for an equation which will calculate which cells (returned either as a pure index or as a row/col) are intersected by a selection rectangle when provided with the box coordinates of each ...
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### 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. ...
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### 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. ...
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### Segre classes of singular projective varieties

Corollary 4.2.4 from Fulton's Intersection Theory gives a method for computing the Segre class of varieties, but in particular it allows computation of the Segre class for singular varieties. Let $X$ ...
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### Intersection and union of 2 variable intervals with integers and indexes

I have a set of indexes of integer $\mathbb{I} =\{i_0, i_1, ..., i_n\}$, and an environment $f: \mathbb{I} \rightarrow \mathbb{Z} \times \mathbb{Z}$ which assigns a pair of integer to each index. For ...
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### Intersection theorems for a certain type of subsets of integers modulo $N$

I've been working on something with integers modulo $N$ and have sort of hit a roadblock where I'd like to have some references. The particular problem goes as follows. We have a system $\mathcal{S}$ ...
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### Conical frustum tangent to two spheres

I've two spheres in cartesian space: $(x_1, y_1, z_1, r_1)$ and $(x_2, y_2, z_2, r_2)$. They don't intersect each other. I want to calculate the conical frustum tangent to these two spheres. In ...
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### 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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### 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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### 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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### Area covered by multiple (possibly intersecting) circles on surface of sphere

I have a number of circles of same radius on surface of sphere (Google Maps API). I'm trying to calculate the total area covered by these possibly intersecting circles. My current solution is ...
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### Geodesics intersection on a cylinder

My problem is the following: I have a cylinder, and a couple of geodesic segments on its surface. The segments are defined by the coordinates of their start and end points. I have to obtain the ...
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### Are all subbasis subsets of basis?

In topology, each element of basis $\{B_k\}$ can be expressed as finite intersections of elements of subbasis, i.e. $B_k=S_{n_1}\cap ...\cap S_{n_m}$ Does the meaning of "finite intersections" also ...
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### Transversal Intersection

If $X, Z$ are manifolds of complentary dimension in $Y$, where $X$ compact, $Z$ is closed. Then show that in $X \times Z, I_2(X \times \{0\},\{0\} \times Z) = 1$. Is this obvious because they ...
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### Detect Regions Described By Lines in Rectangular Coordinates

Need some help from the superior math minds here. This problem is part of a software project. Essentially, I have a Cartesian grid. The user can create lines by plotting points (every 2 points ...
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### Conic equation from cone/plane intersection

In an orthonormal cartesian frame $(O; \vec{x}, \vec{y}, \vec{z})$ consider: an infinite plane $P$ defined by: a point $p = (p_x, p_y, pz)$ an normal vector $\vec{n} = (n_x, n_y, n_z)$ a cone $C$ ...
I am solving quartic equations by finding the intersections of two quadratics. The given quartic function is of the form $x^4 + px^2 + qx + r = 0$. I know one of the quadratic functions to be $y=x^2$. ...