Does not work on mobile! please play on desktop sorry
Shadows of curves in $\mathbb{CP}^2$
Click and drag the dots on the left hand side to select the coefficients of an algebraic curve. The right shows a projection of random points sampled from the algebraic curve.
Input controls
A polynomial of degree $\leq d$ is made up of monomials, which we can naturally arrange into a triangle of side length $d$.
\[\begin{gather} 1,\\ x,\quad y \\ x^2, \quad xy, \quad y^2, \\ x^3, \quad x^2 y, \quad x y^2, \quad y^3 \\ \vdots \end{gather}\]Each dot on the left triangle represents the coefficient of the associated monomial.
- Click and drag dot to set complex value of the coefficient
- Click each dot to toggle whether the coefficient is zero
- Scroll over dot to change the magnitude of coefficient
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Shift + Scroll over dot to change the phase of coefficient
- Degree: sets the degree of the polynomial
- Real coefficients: When enabled, restricts the coefficients of the polynomial to be real. In this mode, a green dot means a positive coefficient, while an orange dot means a negative coefficent.
Projection modes
Real plane
Takes real part of coordinates in affine slice.
\[[z_0:z_1:z_2] \mapsto (\text{Re}(z_1),\text{Re}(z_2))\]3D
Takes affine slice $\CC^2$, then forgets one of the four coordinates. Click and drag to rotate, Shift click and drag to pan.
\[[z_0:z_1:z_2] \mapsto (\text{Re}(z_1),\text{Re}(z_2),\text{Im}(z_1))\]Toric moment map
Computes the toric moment map for $\mathbb{CP}^2$. For more details, see toric geometry of CP2. Succinctly, identifying $\RR^2$ with $\CC$, the map is
\[[z_0:z_1:z_2] \mapsto |z_0|^2 e^{0 \pi i} + |z_1|^2 e^{\frac{2 \pi i}{3}} + |z_2|^2 e^{\frac{4 \pi i}{3}}\]Toric with Z coordinate
Computes the toric moment map, then uses the phase of $z_0$ to determine the $z$-coordinate of the point. Click and drag to rotate.
Stellar
Treat a point $[a:b:c] \in \mathbb{CP}^2$ as the quadratic polynomial $ax^2 + bx + c$, then plot the roots on the unit sphere.
Mathematical details
An algebraic equation defines a curve of complex codimension 1 inside of $\mathbb{CP}^2$. The program samples random points on this curve, drawn uniformly with respect to the induced fubini-study metric on the curve. Here is the algorithm:
- Choose a random complex line $\mathbb{CP}^1 \subset \mathbb{CP}^2$, uniform with respect to the $SU(3)$-invariant Harr measure on the space of lines in $\mathbb{CP}^2$.
- Intersect the line with the curve. This amounts to finding the roots of the degree $d$ polynomial restricted to the line, which is very fast.
- Plot the intersection points. These points are evenly distributed with respect to the Fubini-study metric due to a complex version of Crofton’s formula on $\mathbb{CP}^2$, one of a family of Kinematic formulae. I learned this method for visualizing algebraic curves from Random Points on an Algebraic Manifold by Paul Breiding and Orlando Marigliano.