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

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7
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196 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 ...
6
votes
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129 views

On the Visual Manifestation of Curves in Nature

All sorts of curves are useful in modelling and describing phenomena we observe. Trig functions, logarithms, exponentials, polynomials, hyperbolas, circles, and so forth are all very useful in this ...
6
votes
0answers
58 views

Is this solution legal?

Let $M(1,-1)$ be a point in a plane. Find its distance from a line given by $x+2y-4=0$. Later on I found a formula: $$d=\frac{\left | Ax_{0}+Bx_{0}+C \right | }{\sqrt{A^2+B^2}}$$ But I did it ...
5
votes
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230 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
votes
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107 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-...
4
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0answers
149 views

“Hard” exercises on Linear Algebra and Analytic Geometry

I started lecturing this subject called "Linear Algebra and Analytic Geometry" and in the second day of class I was approached by an undergrad student, asking for referenced that would contain "hard" ...
4
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0answers
89 views

Is there an algebraic description of the ring of analytic functions on the real projective line?

Apologies for the long question. Let $X=\mathbf P^1(\mathbf R) \subseteq \mathbf P^1(\mathbf C)$ be the real projective line. Let $\mathcal O_X$ be the sheaf of real-analytic complex-valued functions ...
4
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122 views

Tangent developable of helix.

Let $T$ be union of tangent lines to helix $C=(\cos x, \sin x,x)$. 1) I want to prove that $T - C$ is a smooth manifold and find equation for $T$. 2) I want to find how many times a line can ...
4
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68 views

Soft: Why does the existence of a singularity cause problems for deRham cohmology?

I've heard that if a variety has a singularity then the deRham theory has "problems". What exactly are these? Im guessing there is some sort of issue with the defintion of a differential form, but ...
3
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37 views

Cosine Inequality

Show that given three angles $A,B,C\ge0$ with $A+B+C=2\pi$ and any positive numbers $a,b,c$ we have $$bc\cos A + ca \cos B + ab \cos C \ge -\frac {a^2+b^2+c^2}{2}$$ This problem was given in the ...
3
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31 views

Find the radius of the circle for given conditions

A circle with center at origin passes through three points $P$, $Q$ and $R$ with the line segment $PQ$ as its diameter along $x$-axis. A line passes through $P$ intersects the chord $QR$ at point $D$. ...
3
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51 views

Construction new ellipse

Using a pencil the thread was pulled on the ellipse. Then the pencil started to rotate around the ellipse. How to prove that a new geometric figure which the pencil drew is also an ellipse (with the ...
3
votes
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61 views

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
3
votes
0answers
29 views

On the solutions of a system of inequalities avoiding Helly's theorem

Let $a_1,b_1,\cdots,a_4,b_4\in\mathbb{R},r_1,\cdots,r_4\in(0,+\infty)$. Show that, if $\not\exists (x,y)\in\mathbb{R}^2$ such that $$ \begin{cases} (x-a_1)^2+(y-b_1)^2\le r_1\\ (x-a_2)^2+(y-b_2)...
3
votes
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99 views

How to calculate a reduced volume?

Let's say we have an irregular 3D shape with volume=V ( we know V but we don't know its equation= F). Now I want to calculate another 3D shape which is exactly the same shape but one size smaller, ...
3
votes
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143 views

Calculate the flux of $\underline{v}$ across the boundary of the sector.

For $a\in(0,1)$, calculate without use of the divergence theorem the flux of $\underline{v}(x,y) = g(y/x)(-1/x,1/y)$ across the boundary of the sector $ S_a := \{(x,y)\in \Omega : 1\leqslant x^2+y^2 ...
3
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45 views

Applications of the quartic curve $x^2y^2-1=0$?

The quartic curve $x^2y^2-1=0$ is equivalent to the union of the hyperbolas $xy-1=0$ and $xy+1=0$, i.e., it's a rectangular hypobola superimposed with a copy of itself rotated by 90 degrees. Does this ...
3
votes
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63 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 $f(p)=\alpha x+\beta y$ ...
3
votes
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952 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
votes
0answers
82 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 (...
3
votes
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78 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 ...
3
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144 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
votes
0answers
44 views

Find the ratio of slope

Note : Elevation $46000$ and all dimention in $mm$ (milimeter) The pipe will be installed on a surface of module structure, that module structure has different surface. I want to know " ratio ...
2
votes
0answers
12 views

If one of the focii of ellipse is moved to infinity, then how it becomes a parabola?

Eccentricity of Parabola is 1. Eccentricity of ellipse is <1. i.e. FP / PM = 1 for parabola. Then how an ellipse will become a parabola if F1P / PM =e<1 if F1 is any one of its focus?
2
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22 views

Bending a horizontal from 0 to infinity real number line, ninety degress counter-clickwise at 1.

Can the real number line from 0 to infinity, which of course is often represented as a horizontal straight line, also be represented as being bent ninety degrees counter-clockwise at 1? I.e., if such ...
2
votes
0answers
43 views

Given any parametric curve, finding its general form?

I'll illustrate the problem I'm trying to solve with an example. Let's consider the equations $$ x = \cos (t) $$ $$ y = \sin (t) $$ We know that these are a parametric form of the unit circle. In ...
2
votes
0answers
48 views

Ring of germs of holomorphic functions at $0\in \mathbb{C}$

So I've been reading the book and they used a induction proof where they just state that for the base case the ring of germs of holomorphic functions on $\mathbb{C}$ is Noetherian. I looked at other ...
2
votes
0answers
42 views

Classify the set $\{(x,y,z):f(x,y,z)=0, \nabla f=0\}$, where $f$ is a polynomial of degree at most 3.

Suppose that $f(x,y)$ is a nonzero polynomial of degree at most 2. Observe the following set: $$S=\{(x,y) : f(x,y)=0 ,\; \partial_{x}f(x,y)=0,\; \partial_{y}f(x,y)=0 \}.$$ Note that this set is the ...
2
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24 views

Equation curves

Introduce equation curves to the canonical form, finding an appropriate rectangular coordinate system. a) $5x^2+12xy-22x-12y-19=0$ b) $9x^2+24xy+16y^2-230x+110y-475=0$ Could somebody do one task. I ...
2
votes
0answers
49 views

Prove that planes $AOA',BOB'$ and $COC'$ pass through the line $\frac{x}{l_1+l_2+l_3}=\frac{y}{m_1+m_2+m_3}=\frac{z}{n_1+n_2+n_3}$

$O$ is the origin and lines $OA,OB$ and $OC$ have direction cosines $l_1,m_1,n_1;l_2,m_2,n_2;l_3,m_3,n_3$ respectively.If lines $OA',OB'$ and $OC'$ bisect angles $BOC,COA$ and $AOB$,respectively,prove ...
2
votes
0answers
28 views

about transverse complete intersection

There are several questions about transverse complete intersection arising from L. Guth's paper: http://www.ams.org/journals/jams/0000-000-00/S0894-0347-2015-00827-X/home.html We say a polynomial $P$...
2
votes
0answers
25 views

Locus of point satisfying a condition

Consider a fixed point $O$ and $n$ fixed straight lines. Through $O$ a variable line is drawn intersecting the fixed lines in $P_1,P_2,\ldots,P_n$. On this variable line, a point $P$ is taken such ...
2
votes
0answers
67 views

What theorems or frameworks explain why the roots of well-behaved functions $h : \mathbb{R} \leftarrow \mathbb{R}^2$ seem to be made up of “pieces”?

First, some terminology: given functions $g,f:Y \leftarrow X$, the equalizer of $g$ and $f$ is defined to be the set of all solutions $x \in X$ to the equation $g(x)=f(x)$ in $Y$. Okay. The following ...
2
votes
0answers
58 views

Eccentricity of $9x^2 + 4y^2 - 24y + 144 = 0$

For a National Board Exam Review: Compute the eccentricity of a given curve $9x^2 + 4y^2 - 24y + 144 = 0$ Answer is $0.75$ I try: $$9x^2 + 4y^2 - 24y + 144 = 0$$ $$9x^2 + 4(y^2 - 6y + 9) = -...
2
votes
0answers
57 views

Find the equations of the lines tangent to the circle $x^2+y^2=r^2$ that pass through the point $(a,0)$?

Find the equations of the lines tangent to the circle $x^2+y^2=r^2$ that pass through the point $(a,0)$. My book explains that the equation of this line is $y=m(x-a)$ and then we make the ...
2
votes
0answers
57 views

Is the following a conic section

All vectors are in $\mathbb{R}^3$ and only $\mathbf{r} = \left[ x; y; z \right]$ is unknown. My question is does the following system define a conic section in the $x-y$ plane and, if so, how can I ...
2
votes
0answers
39 views

Faster Alternative than Calculating Euclidian Distance to determine which Coordinate has Max Distance from a fixed coordinate (eg (0,0))

I am developing a program that needs me to determine which coordinate in a $2$-D figure has maximum distance from a fixed coordinate. Let me demonstrate: $3$ points: $(1,3), (2,2), (5,0) $; Fixed ...
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0answers
31 views

Finding the transformation matrix of a projective transformation in RP^2

So I want to understand how to find the matrix that represents the projective transformation that sends 4 given points to 4 given images, I know that 4 points are necessary to determine it but I can't ...
2
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0answers
122 views

Analytic-geometrical properties of dodecahedron

Consider the following projection of a dodecahedron: An equilateral triangle can be projected to make points $A, B, C, D, E, F$ intersect with it's edges. What would be the mathematical proof (if ...
2
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0answers
57 views

About parametric equation of a line in $3$-space

$a.$ Given coordinates $(x, y, z )$ with origin $(0,0,0)$, parameterize the line through the points $(4,5,6)$ and $(1,2,3).$ $b.$ Take components of your answer to Part $(a)$ to give three ...
2
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0answers
123 views

how to find angle between two added up vectors in cartesian space

I would like to find the angle between two vectors (theta) -> v1 From i to i+1 v1=(xi1-xi , yi1-y1) and v2 from i+1 to i+2 v2=(xi2-xi1, yi2-yi1), which are shown as in the figure (but v1 and v2 can be ...
2
votes
0answers
81 views

A variational strategy for finding a family of curves?

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ ...
2
votes
0answers
34 views

finite morphism (algebraic) vs finite morphism (analytic)

Let $X$ and $Y$ be two algebraic varieties (reduced schemes of finite type) over $\mathbb{C}$. Let $f : X \to Y$ be a morphism of schemes. Let $X^{an}$, $Y^{an}$ and $f^{an}$ the corresponding ...
2
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87 views

Locus of centre of circle in Lambert theorem

A beautiful theorem, when three tangents to a parabola form a triangle,the focus of the parabola lies on the circumcircle of the triangle. But what is the locus of the centre of the circumcircle of ...
2
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37 views

Is there a Focal Point/Area/Line of a Parabola for not perpendicular Lines

I'm not sure if this is mathematical enough for this forum, since it's my first post, but please don't be too harsh! So my question is: If the incoming lines of a Parabola come in perpendicular to ...
2
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0answers
70 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
votes
0answers
273 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 ...
2
votes
0answers
278 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
votes
0answers
82 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 $${J_{sphere}=\rho^2\sin\phi}$$ ...
2
votes
0answers
55 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 ...