## Mathematics 535: Introduction to Algebraic Geometry, Fall 2008

# Movie of the week - Week 1

The moving image above, which was at the top of the mathematics 535 home page during
the first week shows the twisted cubic as the intersection of 3
quadric surfaces. The cubic rotates about the z axis, which is common
to the green and purple quadrics.
The transparent red, green and purple quadrics (equations
y=x^2,z=yx,xz=y^2) have common intersection the twisted cubic, which is
clearly seen if you look into the open end of the green quadric
as it rotates, as well as through the transparent back side.
Note that pairwise quadrics contain a common line as
well. The green and red quadric contain a common line at infinity,
so it does not appear in this affine picture of the real locus clipped
to lie in a sphere. The horizontal and vertical lines in the
intersection of the purple quadric with the red and green quadrics respectively
are clearly visible. As the quadrics rotate, the transparent
colors combine to produce browns and grays in addition to the original
quadric colors. This picture was made with the help of a nice surface
drawing program called surf
which allows careful study of surfaces by drawing portions in affine 3
space and using transparency to examine self intersections and hidden
portions of the surface.