Tagged Questions

0
votes
0answers
54 views

Multiplicity of a root

What is the multiplicity of a root $(0,0,0)$ if we have an ideal $I$ which has the next primary decomposition: $(x-y^2,z^3,y^3)$? Thanks for answers.
2
votes
0answers
40 views

Puiseux series and Resolution of Singularities

I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities. So the Newton's method gives us a Puiseux series parametrizing the branches of ...
1
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0answers
30 views

Dolbeault cohomology of $S^{2n-1} \times S^1$

Let $X=S^{2n-1} \times S^1$. I have to compute $H^{(1,0)}_{\bar{\partial}}(X)$ and $H^{(0,1)}_{\bar{\partial}}(X)$. I don't know how to do this but if we use Kunnet formula we have that ...
4
votes
2answers
56 views

Looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex

I am looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex; to make it simple, assume the following figure which is showing a sample transformation for the case when $n=2$. ...
1
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0answers
28 views

A rectangular prism has the surface area of 300 square inches. what whole number dimensions will give the prism the greatest volume. [duplicate]

it is a tough geometrical algebra problem It is tough and involves geometry and algebra. thank you
5
votes
5answers
138 views

Does $e$ have a geometric representation? [duplicate]

Just like $\pi$ is the ratio of a circle's circumference to its diameter? I know that the tangent line to the function $e^x$ has a slope of $e^x$ at that point, but is there some other geometric ...
0
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0answers
28 views

Geometry Question with irregular hexagons

Suppose you have a rectangle with sides x and y and both numbers are integers and have no factors. now draw lines inside this rectangle starting with a line at 45 degrees coming out of a corner and ...
1
vote
1answer
17 views

2D triangulation

I understood what it is from the following link: http://electronics.howstuffworks.com/gadgets/travel/gps1.htm But I want to know : In a 2D plane, if we know the (x, y) positions of three “guard” ...
1
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2answers
89 views

Finding the locus of the midpoint of chord that subtends a right angle at $(\alpha,\beta)$

There is a circle $x^2+y^2=a^2$. On any line that cuts the circle in two distinct points(it is a secant), the points of intersection with circle are taken and at those two points I draw the tangents ...
0
votes
0answers
18 views

Showing that $|e . \Theta (e)| \geq \min\{k_1,k_2\}$?

For the shape operator $\Theta :T_p(S) \to T_p(S)$ how would you show that if the principle curvatures of $S$ are non negative, $k_1, k_1 \geq 0$. Then for any tangential vector $e$ we have $|e . ...
2
votes
1answer
56 views

Number of points determining a Quadric

I know that in $\mathbb{R}^2$ that 5 points in general linear position determine a unique conic (also non-degenerate). I was wondering about the higher dimensional analogue of this. Is it true, for ...
0
votes
1answer
71 views

A curve with positive curvature is asymptotic if and only if its binormal is parallel to the unit normal of the surface

I want some one explain to me; this part is not clear to me. Q: Show that curve $c$ with positive curvature is asymptotic if and only if if its binormal $B$ is parallel to the unit normal of $S$ ...
0
votes
1answer
57 views

Show that $f(z)= \frac{-1}{z}$ maps each circle of the form $|z+ti| = (t^2-1)^{1/2}$ onto itself.

Show that $f(z)= \frac{-1}{z}$ maps each circle of the form $|z+ti| = (t^2-1)^{1/2}$ onto itself
2
votes
1answer
76 views

Deriving hyperbola equation: why can we assume vertices lie in between foci?

I'm reading through a derivation of the standard equation of a horizontal hyperbola, and while I can follow the the algebra, I'm hung up on an assumption it makes early on: that the vertices lie in ...
0
votes
1answer
102 views

question about Gaussian map

I have questions. Can anyone help me to get the idea or figure out this problem. compute the Gaussian and mean curvature for torus. notice the metric for torus is X(U,V)=((a+b cos(u))cos(v),(a+b ...
0
votes
0answers
78 views

question in differeinal geometry about gaussiam map

I have questions. Can anyone help me to get the idea or figure out this problem. Compute the asymptotic curves for the torus. notice the metric for torus X(U,V)=((a+b cos(u))cos(v),(a+b ...
7
votes
2answers
234 views

Connection between algebraic geometry and high school geometry.

if there is one thing that going to math competitions has taught me it is that I suck at high school olympiad level geometry. However I often find solace in the fact that not a lot of mathematicians ...
0
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1answer
54 views

What is “degenerate” about degenerate quadratic surfaces?

In Wikipedia the table of quadratic surfaces is divided into 2 parts, the second being "degenerate quadrics". Why is this distinction made? and what does the word degenerate means in this case?
3
votes
1answer
182 views

SVD and how to get two points on a 3D line from the representation of the line by means of two intersecting planes?

I have a 3D line represented by the intersection of these two planes $a_1x+b_1y+c_1z+d_1=0$ $a_2x+b_2y+c_2z+d_2=0$ I need to compute two 3D points $P_1=(x_1,y_1,z_1)$ and $P_2=(x_2,y_2,z_2)$ ...
0
votes
0answers
47 views

Torus equation in terms of tangent

So if I have an equation for a torus in $F(a,b) = (X, Y, Z)$ where $X = (R + r\cos a)\cos b$ and $0 < r < R$, how would I go about rewriting this equation for $X$ in terms of $\tan(a/2)$ and ...
0
votes
1answer
59 views

Equation with 2 variables tricky problem

So let's say I have some random equation $6yx^3 - 3yx + 5 = 0$, but could be anything. How would I go about finding a value for $y$, that makes it so that this equation only holds true if $x$ is ...
0
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1answer
48 views

A question on where a chord $AB$ inside an ellipse with a point $P$ on it, attains its maximum for the ratio $\frac{AP}{PB}$

My question is this. Given an ellipse $E$ with equation $\frac{x^2}{9}$+$\frac{y^2}{4}=1$ and $P(\frac{3}{\sqrt{5}},\frac{2}{\sqrt{5}})$ is a point inside $E$. $AB$ is a chord of $E$ through $P$ and ...
3
votes
1answer
85 views

Intersection of two $n$-dimensional quadratic inequalities?

I have two quadratic inequalities of the form $$ a_1x^TAx + b_1^Tx + c_1 \le 0\\ a_2x^TAx + b_2^Tx + c_2 \le 0 $$ where $A\in\mathbb{R}^{n\times n}$ is positive semidefinite, $x\in\mathbb{R}^n$, ...
0
votes
0answers
79 views

A problem in differential geometry

How can we get $ \large w= \frac{1}{k(a)} + \frac{1}{k(a+pi)}$ by using those $4$ facts I got? Let $y (a)$ be a simple closed planar curve with curvature $k > 0$ parametrized by $a$, where $a$ is ...
3
votes
1answer
128 views

Calculating the distance between a camera and a target using camera output

I have a 640x480 camera that recognizes a rectangle that is 1ftx2ft. Is it possible to calculate the distance between the camera and the rectangle? Edit: The horizontal angle of view is 54°.
1
vote
0answers
89 views

Great Circle Center of Circle + radius

I need to draw a great circle arc between two latitude and longitude points. For sake of example we will use the coordinates for LAX and JFK. JFK is 40.64°N / 73.78°W LAX is 33.94°N / 118.41°W My ...
1
vote
2answers
243 views

Is the area of intersection of convex polygons always convex?

I am interested specifically in the intersection of triangles but I think this is true of all convex polygons am I correct? Also is the largest possible inscribed triangle of a convex polygon always ...
3
votes
0answers
45 views

Intersection of a hyperplane and another object

Does there exist a 4-dimensional object whose intersection with a hyperplane always produces a torus, regardless of the orientation of the hyperplane? I am assuming that the hyperplane passes through ...
4
votes
1answer
83 views

Intersecting circles: finding the centre of one using the other?

Hi guys I'm working out a problem which needs me to solve these circle equations but it's been a while since I've had to do this stuff. I need to find the y coordinate of the red circle radius r/4 ...
10
votes
3answers
355 views

How does intuition fail for higher dimensions?

From this answer: Now, Algebraic Geometry is one of the oldest, deepest, broadest and most active subjects in Mathematics with connections to almost all other branches in either a very direct ...
0
votes
1answer
53 views

Lines which intersect the postive half axis of x

We have to find out which lines intersect the postive half axis of x. According to this formula we can determine if the angle between two points(A[x1,y1] and B[x2,y2]) of the line (angle A0B where 0 ...
0
votes
1answer
96 views

Spherical coordinates to cartesian coordinates.

I want to find out the distance between the centers of $2$ circles. Say, circle $1$ $(\theta,\phi)$ circle $2$ $(\theta,\phi)$ The radius of this circle is found using $d\tan(\theta)$ where $d$ is ...
1
vote
4answers
112 views

Defining/constructing an ellipse

Years ago I was confronted with a (self imposed) problem, which unexpectedly resurfaced just recently... I don't know whether it makes sense to explain the background or not, so I'll be brief. If I ...
3
votes
0answers
87 views

The geometric intuition behind the fact that $y-x^3=0$ in $\mathbb{P}^2(\mathbb{R})$ has a singularity at infinity

I apologize if this question is sophomoric as my knowledge of projective geometry is rather elementary. But I'm curious if there exists a good intuitive geometric explanation for why the curve ...
2
votes
0answers
62 views

Blowup of a line and a point

I need to construct a morphism $f:X\to \mathbb{A}^3$ which is surjective, $X$ needs to be irreducible as does each fiber, and the dimension of $f^{-1}(0,0,0)$ must be $2$ while the dimension of the ...
0
votes
1answer
96 views

Finding Angle Between Lines represented by Homogenous Equations

I am trying to find angle between two lines represented by a homogeneous equation The equation is : $ 7x^2 + 4xy + 4y^2 = 0 $ When i use the standard formula $ \theta = \arctan \frac {2 \sqrt {h^2 ...
1
vote
0answers
20 views

Degrees of parabolic subgroups

Suppose a finite reflection group $G$ has the degrees $d_1,\ldots,d_n$. Let $G^*$ be a parabolic subgroup of $G$. What are the degrees of $G^*$. Since $|G^*|$ divides $|G|$ it is clear that the ...
1
vote
2answers
166 views

Integration of Sinusoid Over Complete Period

How can be $$\frac{1}{0.5}\int_0^1 t\sin{(2\pi t)}\ dt = \frac{-1}{\pi}$$ and inside this interval sin signal is defined, i.e. both its $+ive$ part and $-ive$ part of the wave is present. ...
0
votes
1answer
47 views

Given two vectors, how do you get the equation of a plane that partitions the space into two parts exactly midway between the vectors?

I'm reading a book on information retrieval and in it there is an example where they have two sets of vectors. They compute the centroid vectors for both the sets and then give the equation of a ...
1
vote
1answer
145 views

Explaining projective space to master students

I am teaching an introductory course in algebraic geometry for masters and it turns out that many of them are not at all familiar with the notion of projective space. So it is necessary to spend ...
0
votes
0answers
26 views

Building invariant $S_n$ structures from two invariant $Z_n$ structures

Take two mathematical structures with a $Z_n$ symmetry (cyclic symmetry). Which are the different ways, in "gluing" these structures, to obtain a mathematical structure with a $S_n$ symmetry ...
3
votes
1answer
71 views

Computing the trajectory of an orbiting body so that it collides with another orbiting body

I am creating a 2D game in which two space ships, orbiting around a planet under the influence of gravity, fire projectiles at each other, which are also under the influence of gravity. I'm creating ...
0
votes
0answers
20 views

Can we always fit a hypercube snugly within an angular cone?

Let $A$ be an arbitrary square matrix with side length $n$. I have a property $p(x)$, and I would like to find some $x$ satisfying it within the angular cone $Ax \ge 0$. Unfortunately, due to the ...
10
votes
3answers
370 views

Walking on a torus

Everyone knows that when you walk on a sphere along a straight line, you eventually get back to the point you started from. I'm wondering about the same question for the torus. Obviously there are ...
2
votes
3answers
103 views

Which is the probability to a random line to be parallel to a specific other line?

In my perception, using the common sense, is less common, or less probable, to a random line be parallel that not to be, because to be parallel a line needs obey a restrictive rule. But anyone can, ...
0
votes
1answer
394 views

Definition of Geodesic - Distance between two points on the same latitude

I do have a problem understanding the concept of geodesics. As I understand it a geodesic is the shortest distance between two points on any manifolds. Let's consider a spherical earth as depicted in ...
1
vote
1answer
71 views

Given lattice G; parameters of torus R^2/G?

This should be a simple, known result, but I can't seem to find it. Given a lattice $\Gamma = m\mathbb{Z} \times n\mathbb{Z}$, $\mathbb{R}^2/\Gamma$ is topologically a torus. For suitable $m$ and ...
0
votes
0answers
438 views

Converting standard equation for a paraboloid to a parametric one

I have the equation for a hyperbolic paraboloid in $x$, $y$, and $z$: $$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$$ I also have the parametric equations for the same parabaloid: $$x = a u ...
6
votes
1answer
273 views

How do mathematicians think about high dimensional geometry?

Many ideas and algorithms come from imagining points on 2d and 3d spaces. Be it in function analysis, machine learning, pattern matching and many more. How do mathematicians think about higher ...
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vote
2answers
60 views

Giving variables in a coordinate ring different weights

I am reading a book and I am curious about a certain notion. Consider $R = k[x_1,x_2,x_3,x_4,t]$ and let $G = \{\underbrace{x_1 x_3-x_2^2 + t x_3^2}_{f_1}, \underbrace{x_1 x_4-x_2 x_3 +t ...

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