Interactive surface pictures

Interactive drawings of algebraic surfaces

About the pictures

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 shown here all are ruled surfaces, i.e. each surface contains an infinite family of straight lines.

Quadric cone

Hyperbolic paraboloid

Hyperboloid of one sheet

A cone asymptotic to a hyperboloid

The hyperboloid and a tangent plane

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 rational curve of degree 4.

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

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.

The Geometry Center and JGV

The Geometry Center existed at the University of Minnesota prior to 1998; it was funded by the National Science Foundation. Although it is now closed, its webpages still exist and are hosted by the University of Illinois. Here is the current URL:

http://geom.math.uiuc.edu/

The Java GeomView software was developed at the Geometry Center. You don't need this software for viewing my pages. If, however, you'd like to obtain it for your own webpage, you can download it from the following URL:

http://geom.math.uiuc.edu/java/JGV/
About Java
- When you view one of these figures, your computer downloads the Java applet (a small application) from the U of MN math department web server, along with the surface coordinates and related data from my web page. When you rotate, translate, or scale the figure by dragging it with your computer's mouse, the computations are done on your computer in real time. Thus, you don't have to wait for the server to re-draw the figure.
Polyhedral approximation vs. ray tracing.
- A triangulation of a surface is a literally correct form of polyhedral approximation. Generally, we are more likely to use approximations of a surface by collections of other polygonal figures. Quadrilaterals are very common; for example that is what you'd have with a parametric sketch created in Matlab, Maple, or Mathematica. And we're often not looking at actual planar polygons. But this turns out not to matter!! Indeed, the edges of our polygon approximate curve segments on the surface. These are projected onto the plane of the computer screen, finally giving an actual polygon. This is finally filled in with whatever color is given in my webpage data.
- If we use a grid with relatively few vertices to plot our surface, we may get a picture that's not as smooth as what can be done by tools such as surf. But, as mentioned above, we are able to take advantage of the cross-platform capabilities of the Java language, thereby obtaining figures that can be rotated much more readily.