Gallery of animated algebraic surfaces

Warning. This gallery is the very beginning of a project, with only a small fraction of what is planned. The moving pictures are animated gif files, with around 200K bytes each. The quality is not very high (20 pictures of 200x200 pixels for each sequence), in a compromise to keep them within reasonable file sizes.

Comments and suggestions are welcome. The development of this gallery is momentarily suspended.

Quadratic surfaces

• Rotating paraboloid, like a traditional radar antenna. Equation: x^2+z^2=y
• Rotating cone. Equation: x^2-y^2=z^2
• Hyperboloid of one sheet. Equation: x^2-y^2+z^2=1
• Hyperboloid of two sheets. Equation: x^2-y^2+z^2=-1
• Deformation of hyperboloid, which deforms from one sheet into two sheets, with a cone at the middle of the deformation (when the two sheets touch each other). This illustrates a general phenomenon for the deformation of the simplest surface singularities: rational double point of type A₁. Equation: x^2+y^2-z^2=s, for s going from -1 to 1.

Cubic surfaces

• First of all, a simple cubic sheet. Equation: x^2*y+y^2*z+z^2*x=0.1
• A sheet with several holes, under rotation. Equation: x^3+y^3+z^3=x+y+z
• The same sheet with a zoom, to let you peek into its inside.
• Surface with 4 singular points. It is well known that a cubic surface can have at most 4 singular points (if it does not contain singular curves). Equation: x^2+y^2+z^2+2*x*y*z=1
• Deformation around the above surface, with rotation at the same time. Equation: x^2+y^2+z^2+2*s*x*y*z=1, for s goint from 0 to 2.
• The same deformation as above, but without rotation.
• The end of the deformation under rotation. Equation: x^2+y^2+z^2+4*x*y*z=1

Quartic surfaces

• A deformation of quartics. We arrive at singular surfaces at an intermediate stage as well as at the end. Equation: (x^2-1)^2+(y^2-1)^2+(z^2-1)^2=s, for s going from 0.5 to 2.
• Riemann surface in 2-1 cover over the plane, with two ramification points. Vertically placed, under rotation. Equation: z^2*x^2+(z^2+1)*y^2=5*(z^4+z^2)
• A Kummer surface with 16 singular points. It is known that a quartic surface can contain at most 16 isolated singular points. Equation: x^4+y^4+z^4-(0.5*x+1)^2*(x^2+y^2+z^2)-(x^2*y^2+x^2*z^2+y^2*z^2) +(0.5*x+1)^4=0

Surfaces of higher degrees

• Bath's sextic with 65 singular points. We can see 50 of them on the picture. Equation: 4*(2.618*x^2-y^2)*(2.618*y^2-z^2)*(2.618*z^2-x^2) -4.236*(x^2+y^2+z^2-1)^2=0
• Klein's bottle with only one side, a realisation according to Ian Stewart. Equation: (x^2+y^2+z^2+2*y-1)*((x^2+y^2+z^2-2*y-1)^2-8*z^2) +16*x*z*(x^2+y^2+z^2-2*y-1)=0
• If you still find it hard to imaging what happens inside this strange bottle for the above surface, here is it again, but with two caps cut off. So that you can see what happens inside. Same equation as above.

These animated sequences are produced by WIMS, in an application which uses Povray as background engine.

© 1998, by XIAO, Gang. Commercial exploitation of the pictures is prohibited without explicit written authorization.

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