All Questions
Tagged with algebraic-curves plane-geometry
9 questions
7
votes
2
answers
607
views
Will (general points + small number of arbitrary points) impose independent condtions on plane curves?
It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known that a small number ($\le d+1$) points always impose ...
- 1,509
5
votes
2
answers
666
views
Recognize this plane curve?
An aspect of my work led to a plane curve with implicit equation
$$
x^2+y^2 = 3 (y/2)^{2/3} + 1
$$
Actually, I started with the parametrization below and derived from it the
equation above:
\begin{...
- 151k
5
votes
0
answers
333
views
Which equation of a Butterfly?
Let $A, B$ be two points and $L$ be a line on the Euclidean Plane. Take two points $J, G$ on the line $L$ such that $JG=constant$. Let $AJ$ meet $BG$ at $P$, $AG$ meet $BJ$ at $Q$, then the locus of ...
- 477
4
votes
1
answer
495
views
Cubic curve closest to the given set of points
Assume we are given the set $S$ of $n$ points on the real plane and want to draw a parametrized cubic curve (actually a segment of Bézier spline) with fixed startpoint in such a way, that the ...
- 41
4
votes
0
answers
248
views
Minimal $b_2$ in Sarkisov's construction
In the paper On the structure of conic bundles. Math. USSR, Izv.,
120:355–390, 1982, Theorem 5.10, Sarkisov constructed the first example of non-rational, rationally connected $3$-fold $X$ with $H^{3}...
- 6,995
4
votes
0
answers
111
views
A question about complex plane algebraic curves
I would like to ask a question about plane projective curves. Let $C\subset{\mathbb P}_2={\mathbb P}(V)$ be a plane curve of degree $n\geq 3$. Then we have a non splitted exact sequence
$$0\...
- 41
2
votes
1
answer
220
views
Maximal sets of algebraic curves, closed under rotation, dilation, and translation, that pairwise intersect at most twice
Consider a set of nontrivial algebraic curves on the plane groovy if that set is closed under rotation, dilation, and translation, and has the property that no two members of the set intersect more ...
- 785
1
vote
0
answers
106
views
Length of isoline $x(1-x)y(1-y)=c$
For the integral appearing in this answer, it may be beneficial to derive the length $L(c)$ of the isoline:
$$x(1-x)y(1-y) = c,$$
where $x,y$ are ranging in $[0,1]$, and constant $c\in [0,\frac1{16}]$....
- 34.3k
0
votes
1
answer
231
views
Divide angles by coefficients relate to Fibonacci sequence
In the left Figure, consider a right triangle $OPA$ with $\angle {AOP} = 90^\circ$. Let $\ell$ be the reflection of $PO$ in $PA$ and $\ell$ meets $OA$ at $A_1$. Let $O_1$ be the center of the circle $(...
- 1,776