Questions on the use of algebraic techniques to prove geometric theorems.

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6
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63 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
5
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169 views

Are there eigenvectors, eigenvalues, and characteristic functions for non-linear vector fields?

An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would ...
5
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262 views

Proper mapping theorem

My professor mentioned a proper mapping theorem after the name of Remmert which says: Let $X$ and $Y$ be complex manifolds, $f:X \to Y$ be a proper holomorphic map, and $V \subset X$ be a complex ...
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77 views

Points at Integer Distances in 3-space

With the restriction no three points in a line, no four points on a circle, there is a 7 point configuration of points on the plane such that all pairs of points are at integer distances. [1] For ...
3
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67 views

Clarification of the Jacobian

Well, that was cool if not tedious but I understand the Jacobian and its application to changing coordinate systems. $${J_{POLAR}= \rho}$$ $$ {J_{cyl}= \rho}$$ and $${J_{sphere}=\rho^2\sin\phi}$$ ...
3
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592 views

Turning radius of a vehicle

What's the minimum turning radius of a vehicle, rectangular in shape, with length l units and width w units? One key point to consider, would be that, the inclination of the front wheels can be ...
3
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71 views

packing for the polytope

Let $X=(X, \|\cdot\|)$ be some normed space. Let $C=[-1,1]^n$ and $H$ be a plane with equation $\sum_{i=1}^nr_i=s, 1\le s\le n.$ (Here $r_i$ are such that $Proba(r_i=1)=Proba(r_i=-1)=1/2$). The ...
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121 views

Illustrations of a line and a curve intersecting for complex field

Are there nice illustrations on the Net of say $y=a·x+b$ and $y=x^2$ intersecting where x and y are complex? I'm thinking of the amplitude of y being depicted as height above the complex plane with ...
2
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42 views

Cauchy-Schwarz on a Euclidian Space

I was thinking about this proof of the cauchy-schwarz inequality, I wanna show that $$|\langle u,v\rangle|\leq|u||v|$$. We know that, $$|\langle u,v\rangle| = ||u||v|\cos{\theta}|$$ where $\theta$ ...
2
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55 views

A sphere packing problem

Suppose there is a large sphere of radius $R$. We want to pack it with smaller spheres. The volume of the smaller spheres change depending on where they are situated in the larger sphere. A smaller ...
2
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47 views

points possible on a circle

Let $A, B, C, D$ and $E$ be five points marked in clockwise order, on the unit circle in the plane (with centre at origin). Let $\alpha$ and $\beta$ be real numbers and set $f(p)=\alpha x+\beta y$ ...
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24 views

Finding the point of concurrence of a family of straight lines

If $6a^2-3b^2-c^2+7ab-ac+4bc=0$, then the family of lines $ax+by+c=0$ is concurrent at $(-2,-3)$ $(3,-1)$ $(2,3)$ $(-3,1)$ Multiple answers are possible. I am not able to group terms of the ...
2
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124 views

Ellipses given focus and two points

I would like to find all ellipses which contain 2 given points and has one focus at origin (zero). All in 2D plane. There are several possible approaches but I'm not sure which is the best - both ...
2
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0answers
106 views

Find $k$ such that the intersection of $x+ky=1$ and $y^2 - x^2 - z^2 = 1$ is an ellipse or a hyperbola

Find the values of $k$ such that the intersection of the plane $x+ky=1$ with the two-sheeted elliptic hyperboloid $y^2 - x^2 - z^2 = 1$ is (a) an ellipse and (b) a hyperbola. My attempt is the ...
2
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315 views

Proof of coarea formula

I want to prove the coarea formula $\operatorname{Vol}(M) = \int_M d\operatorname{Vol}_M = \int_{-\infty}^\infty \frac{1}{|\nabla f|} \operatorname{area }(f^{-1}(t)) dt$ where $f: M \rightarrow ...
2
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71 views

Algebraic Geometry studied via Filters

Is there any research relating varieties with filters instead of radical ideals? For example, Suppose we have a variety V in C^n, now consider the fixed filter consisting of all algebraic sets ...
2
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393 views

Equations of branches of a mind map

Sorry for the long question, but it's not so simple to explain. Consider a mind map like this: I want to draw branches in a cartesian coordinate system. I'd like to find two equations which ...
2
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41 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 $\mathbb{R}^3$?
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42 views

degeneracy loci of dimension $2$

Let $X$ be a smooth complex projective variety of dimension $n \ge 4$ and let $F$ and $E$ be two (holomorphic) vector bundles of rank $f$ and $e$ over $X$. Given a morphism $\varphi: F \to E$ of ...
2
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365 views

projection of a sphere onto a plane

Consider you have a sphere centered at the origin.The sphere has a diameter of $\frac{1}{2} \sqrt{\frac{3}{2}}$. This means that the inscribed cube has an edge of 1. Take any point from the plane ...
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21 views

Circle Tangent question

I would like to ask for assiatance on the following: Find the eqation of a circle, with a radius of$\sqrt 2$ , which also has as tangetns the lines: $ y=x+2 $ , $ y=-7x $. It is known that the ...
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22 views

points of intersection of two circles drawn on a sphere

How does one write and thereafter solve the equations to find the points of intersection of two overlapping circles drawn on the surface of a sphere? I am looking for a simple understandable solution ...
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19 views

finding a locus in a $3$ dimensions

Given a Tetrahedron $OABC$ such that $O(0,0,0),A(a,0,0),B(0,b,0),C(0,0,c)$ ; $a,b,c$ are not zero. We build a plane $\pi$ that is parallel to $z$-axis and also to $AB$. Plane $\pi$ cuts plane $ABC$ ...
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109 views

Computing Euler Angles from Direction Cosines Vector

My problem simply as the following: Suppose we measured the orientation of a plane object with respect to a reference fame. (where the reference frame parallel to plane frame). The unit normal vector ...
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28 views

Equations for Intersection of Plane and circle

I am having a problem in getting the related equations for an intersection between a point on a plane and the edge of a circle in 3D space. Any suggestions? Or also a tangent plane and a point on a ...
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40 views

Volume of a regular tetrahedron vs Volume of a sphere

I have the following question: given a regular tetrahedron and a sphere that goes through the middle of all sides of the regular tetrahedron, which has a bigger volume? and what is the ratio of ...
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0answers
41 views

Determine properties of a shape defined by a function?

I have a number of functions that define 3d shapes. They take a point $(x, y, z)$ as a parameter and return a real value representing information about the shape. On the surface of the shape: $f(x) ...
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64 views

How to calculate center coordinates of two reverse arcs in 3D space

Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of points C1 and C2? ...
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107 views

Sum of euclidean norms with box constraints

minimizing the sum of euclidean norms with box constraints I am a graduate student in computer science, making a thesis on uncertainty geometry. During my thesis I came across the following ...
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38 views

Stereographic projection of the icosahedron and snub cube?

Using a steoreographic projection, the three equations associated with the icosahedron with unit circumradius, inradius, and midradius (respectively) are, $$f=z^{20} - 228z^{15} + 494z^{10} + 228z^5 ...
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38 views

shortest distance between two cones in 3-dim space

How can I find the shortest distance between two cones in 3-dim space? cone 1: apex - $(x_{0}, y_{0}, z_{0})$ angle - $\alpha_{0}$ base circle - $(cx_{0}, cy_{0}, cz_{0}, r_{0})$ cone 2: apex - ...
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0answers
35 views

Question on a statement about analytic variety irreducible at $0$.

I am trying to understand this statement, it is in "Principles of Algebraic Geometry" by Philip Griffiths and Joe Harris. In page 13, third point, they are trying to prove that an analytic variety ...
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57 views

Three reflection theorem in the context geometry on the sphere

Recently, I study geometry on the sphere in Patrick Ryan's Geometry book. A line on the unit sphere $S^2$ is defined as \begin{equation} l=\{x\in S^2: <x,z>=0\} \end{equation} for some point ...
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46 views

Create a frustum knowing only two edges

You have a 3D frustum. Consider the smallest face the "near" and it's opposite the "far" face. The near and far faces are both perfect rectangles (not squares). No face is axis-aligned. The near ...
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74 views

Locus Problem .

Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabola is $y^2(2x+a)=a(3x+a)^2$
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77 views

Linear separability in a 2×2×2 cube

Essentially, linearly separable points are just those corners that can be cut off with just one slice as marked out by a hyperplane. Can someone please explain how many cases there are for a ...
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160 views

Formula for intersection of “power” curve and parabola.

EDIT I have edited this question to make it more clear. I have spent quite some time trying to find this on Google, but haven't succeeded. I need the formula(s) to determine the intersection ...
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0answers
44 views

finding parabola equation

The centers of all circles which tangent to a circle $P(2R,0)$ with radius $R$ and to the line $x=-t$ is a Canonical parabola. Need to find the equation of that parabola and $t$ (by $R$). so: ...
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79 views

Draw the locus of points which satisfy the equation

(1) Draw the locus of points $(x,y)$ which satisfy the following equation. $bx^{3}+y^{3}+x^{2}y+bxy^{2}-4abxy-2ab^{2}x^{2}-2ay^{2}+b\left( a^{2}b^{2}+a^{2}-1\right)x+\left( a^{2}b^{2}+a^{2}-1\right) ...
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0answers
104 views

Canonical form of a curve (geometry)

I am bothering with this geometric problem more than half a day and couldn't understand it yet. Here it is: In orthonormal coordinate system K=Oxy we have a curve C: $9x^2 - 4xy + 6y^2 + 6x - 8y + ...
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51 views

How to introduce perpendicular or congruence of angles in affine space

$n$-dimensional affine point-vector space is a pair $\mathbb A^n = \langle \mathbb A, V^n \rangle$, where $\mathbb A$ is an arbitrary set, which elements are called points of affine space, $V^n$ is an ...
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0answers
47 views

Equation of a general conic from 3 points and the major axis

I have read that given 3 points on a conic and the equation ($ax+by+c=0$) of its major axis, we can write the equation of the conic ($Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$). I've seen it done by ...
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99 views

Parametrizing a section of a torus

Consider the torus obtained by rotating the circle $(x-R)^2+z^2=r^2$ about the $z$-axis, where $R>r>0$. Parametrize the part of this torus where $z>x+y$. My approach to this so far is to ...
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31 views

Is there a way to check if my Taylor Expansion is correct?

I have an exam later and I need to do Taylor expansions of functions. I have questions like: Consider the map $F:\mathbb{R}^2_x \rightarrow \mathbb{R}^2_y$, given by the equations $$y_1 = ...
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119 views

How to generalise a result regarding intersections of cones and other convex sets?

To test for a particular property of positive LTI systems using feasibility problems I've come across the following claim which, intuitively, I believe can be generalised. I think I've (rather ...
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0answers
45 views

Solving inverse square of visible scale

I'm not super-adept in mathematics, so I turn to you for help. As I read, the perceived scale of an object reduces by the inverse square as the viewed distance increases. In order to solve for this, ...
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50 views

figuring out the sum of angles from the addition of geometrical shapes within a circle

if you add a number with another you get the sum of the two.(1+1=2, 2+2=4) right? but if you take a circle and put a horizontal line through it you have 4 angles in it. put another circle with a ...
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0answers
107 views

Circle-Circle intersection coordinate system

Consider two points in the 2D Euclidean plane, the origin $0$ and $x$. One can define a co-ordinate system such that for any point $y$ in the plane, $y$ is parametrized by its distance from $0$, call ...
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96 views

A function with the same slope as $b\sqrt{\frac{x^2}{a^2}-1}$ but not imaginary in [0,a]?

For some fixed $a,b \in \mathbb{R}$, $y = b\sqrt{\frac{x^2}{a^2}-1}$ is supposed to plot the boundary of an ellipse in $\left[0,a\right]$. I came up with that function but it has the defect that it ...
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319 views

Prove a very original version of Descartes's circle theorem

Prove: I define the radius of three mutually externally tangent to be $d,e,f$ respectively. The circle with radius $x$ is internally tangent to all three circles. Then $$ddeeff+ddeexx+ddffxx+eeffxx ...