-1
votes
1answer
59 views

Blow-Up over a Field

I want to prove that a function $\pi : \mathbb{C}_{*}^{n}\mapsto \mathbb{C}^{n}$ is bijective. Where $\mathbb{C}_{*}^{n}$ is the explosion of $\mathbb{C}^{n}$ and is defined as $\mathbb{C}_{*}^{n}:= ...
1
vote
1answer
134 views

Felix Klein's view on algebraic geometry

I think, as a first approach one would say that a geometry on a set $X$ is given by an inner product on $X$. Klein then links geometry to group theory by identifying a geometry on $X$ with a group of ...
0
votes
1answer
20 views

Find a projectivity to create a graph.

I have the tetrahedron {xyzt=0} in projective space with homogeneous coordinate (x,y,z,t). I need to create a graph but the tetrahedron in affine coordinate is {xyz=0} and I can't visualize the ...
1
vote
0answers
44 views

Hochschild dimension

I'm curious; if $A$ ia a commutative $k$-algebra over a field $k$ of global dimension $n$, then is its $A^e$-projective dimension $2n$ (this is also sometimes called the Hochschild cohomological ...
1
vote
2answers
27 views

Is equation for ellipse in polar coordinates correct?

Wikipedia gives the following equation for the conic sections in the polar coordinate system: $r = \frac{l}{1+e\cos\varphi}$. According to the article on conic sections, in case of an ellipse $e = ...
2
votes
2answers
56 views

Smoothness and field of fractions

If $k$ is an integral domain and $A$ is a Noetherian finitely presented $k$-algebra for which $A \otimes_k Q(k)$ is a smooth $Q(k)$ algebra, then can it be deduced that $A$ was initially smooth over ...
0
votes
0answers
32 views

Finding the point satisfying the condition

Given N interesting points on the plane. Each interesting point has integer coordinates. Also, all the interesting points form a strictly convex polygon. If we select two coordinates from these ...
2
votes
1answer
68 views

Question about Qing Liu's Algebraic Geometry book

I was just wondering what the real prerequisites are for reading Qing Liu's 'Algebraic Geometry and Arithmetic Curves', and if it is a good first book on the subject. In his preface he states that the ...
0
votes
1answer
32 views

What is the intuition behind contours and their geometric properties

What is the the intuition behind contours? Can someone explain whar are contours, their geometric properties in simple manner
0
votes
0answers
53 views

Alternative Proof why algebraic groups are smooth

Definitions: Let $k$ be a field, by an affine algebraic group we will mean an affine group scheme $G$ over $Spec(k)$ with the property that the coordinate ring of the underlying affine scheme ...
1
vote
0answers
45 views

Linear Transformation of points [duplicate]

Could you help me solve this: For projective coordinate system on the line $l$ are given points $A (2,1)$, $B (1,1)$, $C (0,1)$, $A_1 (0,1)$, $B_1 (1,5)$ and $C_1 (2,1 )$. Find a linear transformation ...
0
votes
0answers
32 views

Is it possible to solve this series of triangles with only the given information?

Consider the following: As displayed in the picture, the distance between the points is 1, so the last point's coordinates would naturally be $(c, d+4)$ Is it possible to solve for the coordinates ...
0
votes
2answers
29 views

Solving for points in a plane based on line lengths and geometry

I have the following points and lines in a plane: The problem is this: Given that we know the lengths of lines A, B and C, how can we calculate the coordinates of each point a, b and c? The ...
0
votes
1answer
37 views

Formula of signed distance from hyperplane to point

Let $H$ be a hyperplane defined by the points $p_1, p_2, ..., p_n$ and single point $x$ generally out of the hyperplane. Is there any formula to calculate the signed distance between $x$ and $H$? I ...
2
votes
0answers
73 views

Equation for the Maximum-Area Rotor in a Wankel Engine

I am attempting to construct a specialized Wankel Engine in Autodesk Inventor. For my particular project, the rotor must take up the maximum area, in order to separate the air spaces of the top and ...
1
vote
1answer
64 views

An question on effective divisor (Clifford 'S theorem)

For an effective divisor $D\ge 0$ on a curve $Y$, define $$\lvert D\rvert =\{ D' \in \mathrm{Div}(Y) \mid D'\ge 0 \;\text{ and }\; D' \sim D \}$$ where $D\sim D$ means $\exists$ a rational ...
2
votes
1answer
36 views

Question about geometry in a finite projective space

I apologize again for a dumb question! To add some context (though I think it'll largely be unnecessary): suppose $q$ is a prime, $F:= \displaystyle \mathbb{Z}/q\mathbb{Z}$ is a field. I've defined ...
0
votes
0answers
20 views

Which bound on $a,b,c,d$ is correct?

Let $ABCD$ be a unit square. Four points E,F,G, and H are chosen on the sides $AB,BC,CD,$ and $DA$ respectively. Let the length of the quadrilateral be $a,b,c,d$. Then which of these is always true? ...
1
vote
1answer
46 views

multiplicity of intersection of projective curves

This example is taken from Silverman and Tate's Rational Points on Elliptic Curves. Suppose we have the line $C_1: x+y=2$ and the circle $C_2: x^2 + y^2 = 2$. We see that these intersect at the ...
1
vote
0answers
32 views

Lateral area of a cone with oblique base (elliptical base)

Can someone help me find the lateral area of a straight circular cone with oblique base ( inclined elliptical base) as seen at the following link. ...
0
votes
1answer
114 views

Algebra, Geometry and Algebraic Geometry

I want to know, what is the difference between Algebra, Geometry and Algebraic Geometry ? Your reply is highly appreciated.
1
vote
1answer
36 views

Order of $A/2A$ for $A$ an Abelian variety

Let $A$ be an Abelian variety over $\mathbb R$ of dimension $g$. Then the size of $A(\mathbb R)/2A(\mathbb R)$ is $(\# A(\mathbb R)[2])/2^g$. I'm wondering how one might go about proving such a ...
1
vote
1answer
49 views

What is the nature of this surface?

What is the nature of the surface whose equation is (it depends on $m$) $$x^2+2y^2+(m+1)z^2+2xy-2yz-2x+2y-4z+m^2+4=0$$
1
vote
0answers
37 views

Determining how accurate an ellipse fit is

So I have an image of bacteria particles which are often shaped very irregularly with many grooves. Im trying to fit ellipses onto these particles so I can get a better, more smooth analysis of the ...
2
votes
1answer
83 views

Chern class of line bundle and vector bundle

Let $L$ is a Line bundle and $E$ a vector bundle of rank $r$ then how can we prove that $$c_1(L\otimes E)=rc_1(L)+c_1(E)$$ where here $c_1$ means first chern class
0
votes
2answers
34 views

How do we define touching lines?

If two curves are touching at one point and intersect one another, how do we define it? If two lines are touching at a point then $L\cap K=\{q\}$ for two lines L and K and q is the touching point. ...
1
vote
0answers
44 views

Algorithm for finding nearest distance from a point to a curved surface in space

I need to write an algorithm which can find the nearest distance from a point in space to a 3D curved surface which is straight in vertical direction but its projection is an arc of a circle (Similar ...
3
votes
1answer
168 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
1
vote
1answer
51 views

Computing the intersections of arbitrary number of planes

Imagine three sets of planes (A, B, and C). The planes in each set are parallel and evenly spaced and so can be defined by a plane equation and a scalar representing the distance between planes. If ...
1
vote
0answers
93 views

product of sums

This is a question which has puzzled me for a while. I would be very thankful if somebody can help me with it. Assume you have $S$ rectangles appearing in front of your screen one by one. Each ...
17
votes
3answers
295 views

What is the largest circle that fits in $\sin(x)?$

Imagine dropping a circle into the trough of $\sin(x)$. Would it reach the bottom or get wedged between two points on the curve? Depends on the size of the circle. So, what is the radius of the ...
2
votes
3answers
89 views

Representing 2D line as two variables, all cases

Is it possible to represent a line in the general case using only two variables (namely floats or doubles)? By this I mean ...
5
votes
1answer
66 views

Problem about parallel curve- differential geometry

Let $\alpha (s)$ , $s\in [0,L]$, be a smooth positively oriented regular Jodan curve which is arc-length parametrized. The curve $\beta(s)=\alpha (s) +\lambda n(s)$, where $\lambda$ is a positive ...
1
vote
1answer
46 views

Crossing Circles

On a plane, you are only allowed to draw circles. After drawing 1 circle, can you ALWAYS draw another so that the new circle crosses all existing circles at 2 points? Why?
1
vote
2answers
65 views

Does congruence guarantee length conversion?

Suppose that a linear transformation $M:R^2 \rightarrow R^2$ maps a triangle $ABC$ to a congruent triangle $A'B'C'$ ($\{A, B, O\}, \{B, C, O\},\{C, A, O\}$ are not colinear, and $A,B,C\neq O$) Is it ...
3
votes
1answer
50 views

Example: Irreducible component - affine varieties

Again, I know how to prove the statement. But, I cannot find any example. Please help me for finding an example. Thank you:)
2
votes
1answer
69 views

Affine variety example.

I know how to show the statement. But I cannot find an example (the part I underlined by a yellow pen) please help me for finding an example. Note: For example, can I consider the following ...
1
vote
2answers
73 views

How to Represent a 3D Line under Polar Coordinates

In one of my applications, I need to represent a line under 3D polar coordinates system. In 2D, we can define a line by a distance to the origin and then a angle indicating the direction of the line ...
0
votes
1answer
127 views

Proof: The coordinates of the witch of Agnesi curve

I need to prove that the coordinates ofthe witch of Agnesi curve is: $$x=2a\cot \theta$$ and $$y=2a\sin ^2 \theta$$ Any idea how to prove it? And I don't understand how we got $a$... (because the ...
0
votes
2answers
82 views

How to extend a line when its slope is defined

I want to extend my line by few centimetres on both sides without changing the slope. Is there any way to do this
1
vote
1answer
35 views

How do I convert OS coordinates (X and Y) to longitude and latitude coordinates?

How do I convert OS coordinates (X and Y) - Eastings and Northings to longitude and latitude coordinates? For example X and Y below X (Eastings): 347904 Y (Northings): 287484
1
vote
0answers
18 views

How do I figure out the grid references necessary to create a circular 1 mile radius around one location?

How do I figure out the grid references necessary to create a circular 1 mile radius around one location? For instance: Grid Reference: SO 47904 87484 X (Eastings): 347904 Y (Northings): 287484 ...
1
vote
1answer
75 views

How do I figure out the longitude and latitude coordinates necessary to create a circular 1 mile radius around one location?

How do I figure out the longitude and latitude coordinates necessary to create a circular 1 mile radius around one location? I don't mind how many coordinates that takes. For instance: Latitude = ...
0
votes
1answer
68 views

Parametrization of a Curve

I am working through my script of Algebraic Geometry and have some questions to Parametrization: What is the exact definition of Parametrization? We wrote in our script: $k \to C$ with $t\mapsto ...
6
votes
1answer
146 views

Exercise 1.11 of Eisenbud

I'm doing the exercises from Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, and I don't understand part of one of them, ex. 1.11 a): Exercise 1.11 a: Over $\mathbb{C}$, ...
0
votes
1answer
43 views

Closest point out of collision

I am tackling IMO very hard problem from geometry "How to find closest point out of collision". To explain it better I've drawn an image explaining it. Basically: given a point on 2D plane( yellow ...
1
vote
2answers
42 views

Is there a family of lines having two (or more) distinct envelopes?

The question: It's on the title. It just needs some clarification: $i)$ The problem takes place in the real plane. $ii)$ The "envelope" has its definition of the $E_2$ in this Wikipedia page: ...
0
votes
1answer
55 views

Automorphisms of Affine plane

I am working on the automorphisms of the affine plane and projective plane. Can we extend an automorphism of an affine to projective plane? As we know that we extend affine plane to projective ...
2
votes
1answer
53 views

Distance from curve to plane

What is the function which defines the distance from the plane to the curve as a function of the angle. Currently shows the distance for a circle inclined 15 degrees to the plane. 10.729 degrees @ 45 ...
0
votes
0answers
65 views

equivariant map $\mathbb{CP}^1 \to \mathbb{CP}^2:$ where does it send the boundary

Start from a map $\mathbb{CP}^1 \ni (u:v) \mapsto (x:y:z) \in \mathbb{CP}^2$ given by $$ x = au^2, \quad y = av^2, \quad z = uv,$$ where $a \in \mathbb{R}$ is a parameter. The image of the map is the ...