1
vote
0answers
73 views

Parameterizing a spherical cap in cylindrical coordinates

In calculating the mean curvature for a surface of the form $(f(z)\cos(\theta), f(z)\sin(\theta), z)$ I need to use some checks to ensure I haven't made any mistakes along the way. The one I would ...
0
votes
1answer
84 views

which algebraic curves admit (nice?) geometric characterizations?

Caveat lector: this a super soft question. In an effort to help my students, across many different courses, better appreciate and put to work the distinction between geometric objects & ...
1
vote
2answers
840 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 ...
3
votes
1answer
122 views

Every conic in $\Bbb{P}^2$ equivalent to $XZ - Y^2$ - what is meant by hint here?

I am looking at Miles Reid's UAG book. There he claims that every projective conic is projectively equivalent to $XZ = Y^2$. He asks to show that $Q$ a non-degenerate quadratic form is such that ...
0
votes
3answers
171 views

Equation of a line in homogenous coordinates given 2 points in affine coordinates

So if I have 2 points $A$ and $B$ such that $F(A) = (1; a, a^3)$, and $F(B) = (1; b, b^3)$. how do I find the equation of this line in homogeneous coordinates? So I know how to get a line the ...
-1
votes
1answer
198 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 ...
0
votes
1answer
54 views

Prove that $QM = PN$ [Coordinate Geometry]

If the points $(P,Q)$, $(M,N)$ and $(P-M,Q-N)$ are collinear. Then show that $QM=PN$.   Basic Points and formulae: Distance between two points = $\sqrt{(x_2-x_1)^2+(y_2-y_2)^2} $ Where, ...
3
votes
0answers
116 views

Local Coordinate Systems under Integral Extension

Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be an integral extension of regular local rings of dimension $d$ (of course, $\varphi$ is a local homomorphism). Furthermore, assume that $A$ contains ...