2
votes
1answer
26 views

Counting the dimension of a component of $\mathsf{hilb}^{2t+1}_{3}$

Consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, parametrizing varieties of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Consider the component $ ...
3
votes
0answers
54 views

Quadric question

I'm trying to prove that given 3 disjoint lines in $\mathbb{P}^{3}$ there exists a non-singular quadric containing them. The exercise is from the following link: ...
1
vote
0answers
32 views

any conic in $\mathbb{A}^2$

Exercise 3.1 in Hartshorne's Algebraic Geometry: Show that any conic in $\mathbb{A}^2$ is isomorphic to $\mathbb{A}^1$ or $\mathbb{A}^1-\{0\}$. when the conic given by $x^{2}+y^{2}-1$, what the ...
0
votes
1answer
64 views

Ellipse Tangents in 3D

I know that we can find the tangent of the ellipse in 2D by taking the derivative of the equation defining the ellipse. But I'm little bit confused about finding the ellipse tangent in 3D. Where the ...
0
votes
1answer
61 views

Intersection of conics

By conic we understand a conic on the projective plane $\mathbb{P}_2=\mathbb{P}(V)$, where $V$ is $3$-dimensional. I'd like to ask how to find the number of points in the intersection of two given ...
3
votes
0answers
94 views

General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
2
votes
0answers
126 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 ...
0
votes
0answers
95 views

rotating parabolas in 3D to get part of a circle

Say you have a unit-circle with its center at (0,0), and you "cut out" the upper-right quadrant. You rotate this segment around the Y-axis and the orthographic projection is the upper-right segment of ...
1
vote
2answers
119 views

pick point on parabola so 2 conditions are true

You have a parabola $$y=ax^2+bx+c $$ We know $a,b$ & $c$. On this parabola you have to pick a point A where the following conditions are true: 1) If you draw a tangent line in this ...
5
votes
1answer
219 views

Determine if a conic is degenerate with the determinant.

There is a natural bijection between conics (written as homogeneous quadratic) and 3x3 matrices: $$C=aX^2+2bXY+cY^2+2dXZ+2eYZ+fZ^2\Leftrightarrow \left(\begin{array}{ccc} a&b&d\\ ...
3
votes
2answers
43 views

What is the rationale for the factor of $4$ in the Conics parabola equation?

The Conics form of a parabola equation is $4p(y-k)=(x-h)^2$ where $(h,k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus. (Which is also the same distance from the ...
2
votes
1answer
94 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 ...
3
votes
1answer
529 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
107 views

In an equation that looks like the standard form of an ellipse, what must the constant on the RHS equal for exactly one solution?

I am working on a homework question: What must be the value(s) of $c$ for the following equation to have exactly 1 solution? The equation is of the standard form of the equation for an ellipse, ...
1
vote
1answer
146 views

Apollonius ellipse equation $y^2=x \left( p-\frac{p}{2a}x \right) $ to standard form.

I am looking for a way to understand the last steps found at this site: http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=196&bodyId=203 The page finishes with showing in ...
3
votes
1answer
116 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 ...
3
votes
1answer
477 views

Canonical form of conic section

I have $x^2+2xy-2y^2+x-4y=0$ and I have to find its canonical form, but I'm a little confused.. I'd like to understand very well what I have to do.. Can you help me, please? Thanks!
2
votes
1answer
400 views

Parametrization of a conic and rational solutions

How can we parametrize the conic $C$: $x^2+y^2 = 5$, by considering a variable line through $(2,1)$ and hence all rational solutions of $x^2 + y^2 = 5$? I'm thinking let $x = \sqrt{5}\cos t$, and $y ...
3
votes
1answer
138 views

The probability of $Ax^2+Bxy+Cy^2 = 1$ defining an ellipse.

In Keith Kendig's paper, Stalking the Wild Ellipse (published in the American Mathematical Monthly, November 1995), he says that if $A, B, C$ are chosen at random, the probability that the Cartesian ...
4
votes
1answer
211 views

Conics in $\mathbb{A}^2$

I'm trying to solve Exercise 3.1 in Hartshorne's Algebraic Geometry: Show that any conic in $\mathbb{A}^2$ is isomorphic to $\mathbb{A}^1$ or $\mathbb{A}^1-\{0\}$. I know from a previous exercise ...
1
vote
4answers
227 views

Difficult equations to rewrite as ellipses

I have this equality that defines an elliptic boundary. I am trying to rewrite it in the form of the equation of an ellipse, but I am having trouble doing that. How would I go about rewriting this ...
-9
votes
1answer
685 views

A Hunt for a Mathematical Machine That Gives Points

The central question is : Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us ? Explanation: ...
0
votes
1answer
255 views

Irreducible conic implies that the underlying matrix is invertible

I guess that it is true that a conic (2nd degree homogeneous equation in complex variables) is irreducible (i.e can't be factorized over polynomials) if and only if the underlying matrix of ...
2
votes
2answers
228 views

Trying to piece together an integral addition theorem

If we have a curve $C:\{ P(x,y) = 0 \}$ and define $\omega=\frac{\mathrm{d}x}{y}$ then is $$\int_0^A \omega + \int_0^B \omega = \int_0^{A \oplus B} \omega$$ (with $\oplus$ being addition on a group ...