Duals of surfaces

Visulising duals of surfaces (preprint)

Algebraic varities in P3

The technique used in the preprint can also be used to help visulise algebraic varities defined in $\mathbb{P}^3$. The classification of cubics define the surfaces in three dimensional cubic space. For example a cubic with 4 double pints is defined by homogenous polynomial $x y z + x y w + x z w + y z w=0$. As this is a homogeneous polynomial if $(x,y,z,w)$ is a solution then so is $(a x, a y,a z,a w)$. This surface has four double point at $(0,0,0,1)$, $(1,0,0,0)$, $(0,1,0,0)$, and $(0,0,1,0)$ which easily shown as all derivative vanish if any three of the variables are zero. Part of the surface can be generated by fixing a plane at infinity, say $w=1$, and take the projection $(x,y,z,w)\to(x/w,y/w,z/w)$. In effect this calculates the intersection of the surface with the plane $w=1$. This results in the 3D equation $x y z + x y + x z + y z=0$ which has a single double point the other nodes lie at infinity.

One way all four nodes can be sean is to choose a different plane at infinity, for instance $x+y+z+w=0$, choosing a rotation $R:(x,y,z,w)\to(X,Y,Z,W)$ which maps this plane onto $W=\text{constant}$, and calculate the surface $f(X,Y,Z)=0$, in this case the equation $4+8 X Y Z-4*X^2-4*Y^2-4*Z^2$. This is the aproach taken on the cubic page.

Here we use a more general method. Four planes are taken $x=1$, $y=1$, $z=1$ and $w=1$ and the intesection of the surface with each planes calculated. So four 3D surfaces are calculated $y z + y w + z w + y z w=0$, $x z + x w + x z w + z w=0$, $x y + x y w + x w + y w=0$ and $x y z + x y + x z + y z=0$. For each surface we only need to calculate the surface with each variable in the range $[-1,1]$, there four planes are suficient to cover the whole of $\mathbb{P}^3$. It we conside the case of $\mathbb{P}^2$, the set of lines through the origin in $\mathbb{R}^3$, then every line will intersect one of three faces of the unit cube, $x=1$, $y=1$, or $z=1$. There is no need to find the intersection with the other three faces $x=-1$, $y=-1$, $z=-1$.

Once the four surfaces have been calculated they can be combined together to give a surface in $\mathbb{R}^4$. This surface can be rotated in 4D and projected to into 3D. First three rotations in 4D are applied $$ R_{xw}=\begin{pmatrix}\cos(\theta)&0&0&-\sin(\theta)\\0&1&0&0\\0&0&1&0\\\sin(\theta)&0&0&\cos(\theta)\end{pmatrix}, R_{yw}=\begin{pmatrix}1&0&0&0\\0&\cos(\phi)&0&-\sin(\phi)\\0&0&1&0\\0&\sin(\phi)&0&\cos(\phi)\end{pmatrix}, R_{zw}=\begin{pmatrix}1&0&0&0\\ 0&1&0&0 \\ 0&0&\cos(\psi)&-\sin(\psi)\\0&0&\sin(\psi)&\cos(\psi)\end{pmatrix}. $$ Then the points are projected to the lower half of the unit 3-sphere, $$(x,y,z,w)\to \frac{-\operatorname{sign} w}{\sqrt{x^2+y^2+z^2+w^2}}(x,y,z,w)$$ and finally a stereographic projection is applied mapping point on the 3-sphere to interior of the 3 sphere $$(x,y,z,w)\to\frac{1}{1-w}(x,y,z)$$.

The following figure shows the result of this algorithm, the colours represent which plane was used for the intersection. Note how the four surfaces knit together to form a surface which is smooth apart from at a the four nodes. The entire surface fits within the unit sphere and all four node always visable. The two projections only differ in the rotation in 4D.

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Written by Richard Morris - rich@singsurf.org - Home Page

Last modified: April 4, 1997