## Interactive drawings of algebraic surfaces

• Please  don't  do this at home.
• Actually I feel very strongly that it's more beneficial to actively view
the surface pictures by rotating them with the mouse, rather than just
looking at the view that I've chosen to present.
• To view the interactive picture of any particular surface, please click on
either the thumbnail sketch or the corresponding text.
• Further info about the pictures.
A quadric surface is defined by a polynomial equation of degree 2.

The quadric surfaces in the following row all are ruled surfaces,
i.e. each surface contains an infinite family of straight lines.
 Hyperbolic paraboloid

 Hyperboloid of one sheet

 A cone asymptotic to a hyperboloid

 The hyperboloid and a tangent plane

The quadric surfaces in the following row are not ruled surfaces.
 Ellipsoid

 Hyperboloid of two sheets

 An elliptic paraboloid (To appear soon)

Tangent surfaces of space curves.

The tangent surface of a space curve with no singular points is the union of all of the tangent lines of the curve.
• The tangent surface of the twisted cubic.
 The image of a rectangular coordinate patch

 A view with equal length tangent line segments
 The tangent surface of another curve.
• The tangent surface of a curve is singular at every point of the curve itself.

Images under generic projection.

Given a smooth surface in P4 (projective 4-space), generic projection to P³ refers to the process of centrally projecting it it from a generic point of P4. In the case of a smooth surface in P5, we project from a generic line in P5. {Or we can iterate the process of projecting from a point.}
 Cubic ruled surface

 The Steiner surface
• The image of a smooth surface, under generic projection to P³, is a surface with a 1-dimensional singular locus. Most of the singular points are ordinary double points (where two smooth sheets of the surface cross transversally). There are finitely many pinch points and finitely many triple points.
Projective duality For a smooth surface X in P³, the dual variety parametrizes the set of tangent planes of X. For a non-smooth surface, we work with the closure of the set of tangent planes at smooth points.
 The Cayley Surface
• The Cayley surface is dual to the Steiner surface.

Another ruled surface Here is a ruled surface of degree 4.
 A quartic ruled surface
• The lines on this surface are secant lines of a twisted cubic curve.

More about the pictures