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| Type | Normal form | Alternate forms |
|---|---|---|
| A0 (not singular) | x - y^2 - z^2; | x + y^2 - z^2; |
| A1 | x^2 - y^2 - z^2; | x^2 + y^2 + z^2; (isolated point) |
| A2 | x^3 - y^2 - z^2; | x^3 + y^2 - z^2; |
| A3 | x^4 - y^2 - z^2; | x^4 + y^2 - z^2; x^4 + y^2 + z^2; (isolated point) |
| A4 | x^5 - y^2 - z^2; | x^5 + y^2 - z^2; |
| D4 | x^2 y - y^3 - z^2; | x^2 y + y^3 - z^2; |
| D5 | x^2 y - y^4 - z^2; | x^2 y + y^4 - z^2; |
| D6 | x^2 y - y^5 - z^2; | x^2 y + y^5 - z^2; |
| E6 | x^3 - y^4 - z^2; | x^3 + y^4 - z^2; |
| E7 | x^3 - x y^3 - z^2; | |
| E8 | x^3 - y^5 - z^2; |
Then there are some highly degenerate surfaces where whole curves are singular.
x^2 y - z^2 = 0;
4 z^3 y^2 - 27 y^4 + 16 x z^4 -128 x^2 z^2 - 144 x y^2 z + 256 x^3 = 0;
The reason why the above singularities are important is that when you have a family of surfaces controlled by a number of parameters you will often find some surfaces which contain one of these singularities. For example consider the one parameter family of surfaces:
x^2 - y^2 - z^2 = a;
for each different value of a you get a different surface. When a < 0 you get a hyperboloid of one sheet and when a > 0 you get a hyperboloid of two sheets. When a = 0 you get a surface which contains an A1 singularity. Try using the equation
x^2 - y^2 - z^2 = a; a = 0.1;
with a taking the values 0.1, 0.05, 0, -0.05, -0.1. In a one parameter family you typically only get A1 singularities either of the type shown above or its alternate form. The alternate form is just an isolated point, which occurs in the family
x^2 + y^2 + z^2 = a;
Lots of fun can be had by taking one of the more complicated singularities such as D4 and adding lower degree terms for example try
x^2 y - y^3 - z^2 + a x^2 + b y^2 + c (x^2-y^2) + d y z = 0; a = 0.0; b = 0.0; c = 0.0; d = 0.0;
And vary the values of a, b, c, d. When a and b are non zero you get surfaces which show A2 singularities. Perhaps the most fun is had when the value of c is changed and a surfaces with three A1 points is displayed.
Another deformation to try is to take the D5 and add on multiples of y^3
x^2 y + y^4 - z^2 + a y^3 = 0; a = -0.5;
You can do the same trick with the other forms of D4 and D5 as well as any of the other higher singularities.
x^2 + y^2 + z^2 = 1;
x^4 + y^4 + z^4 = 1; (You can increase the powers to get nearer to a cube.)
x y z = 0;
4(x^2+y^2+z^2) + 16xyz = 1;
Working in complex projective 3 space there is only one cubic with 4 singular points (up to isomorphism), which is called Cayley's cubic. An equation for this is
4(x^3+y^3+z^3+w^3)-(x+y+z+w)^3 = 0;In real 3D space the other versions can look very different:
4(x^3+y^3+z^3+w^3)-(x+y+z+w)^3 = 0; w = 1;or
-5(x^2*y+x^2*z+y^2*x+y^2*z+z^2*y+z^2*x)+2*(x*y+x*z+y*z)=0;
4( t^2 x^2 - y^2)(t^2 y^2-z^2)(t^2 z^2-x^2)- (1 + 2t)(x^2 + y^2 + z^2 - 1)^2 = 0; t = 1.618034;
The second equation defines the value of t, the golden ratio. Use +/-2 for the domain bounds.
x^2 y^2 + y^2 z^2 + z^2 x^2 = 2 x y z ;
(3-v^2)(x^2+y^2+z^2-v^2)^2 -(3 v^2 - 1) p q r s = 0; p = 1 - z - x rt2; q = 1 - z + x rt2; r = 1 + z + y rt2; s = 1 + z - y rt2; v = 1.1; rt2 = sqrt(2);
You can vary the value of v to get different surfaces, try v=1, 1.1, 1.5, 1.7, 1.8, 2. Note the similarity with the Roman surface.
(r1^2 - dy^2 - (dx + r0)^2)(r1^2 - dy^2 - (dx - r0)^2)* (x^4+y^4+z^4)+ 2((r1^2 - dy^2 - (dx + r0)^2 )* (r1^2 - dy^2 - (dx - r0)^2)(x^2 y^2+x^2 z^2+y^2 z^2))+ 2 ri^2 ((-dy^2-dx^2+r1^2+r0^2)(2 x dx+2 y dy-ri^2)-4 dy r0^2 y)* (x^2+y^2+z^2)+ 4 ri^4(dx x+dy y)(-ri^2+dy y+dx x)+ 4 ri^4 r0^2 y^2+ri^8 = 0; r0=2.1; r1=2; dx=2; dy=0; ri=2;
Use +-5 for domain limits. Where
r0 = Major radius of generating Torus. r1 = Minor radius of generating Torus. dx,dy = Torus displacement. ri = Inversion radius.
Other examples to try:
r0=1.9; r1=2; dx=2; dy=0; ri=2; r0=2; r1=2; dx=2; dy=0; ri=2; r0=2.1; r1=2; dx=2; dy=0; ri=2; r0=4; r1=2; dx=2; dy=0; ri=2; r0=4.5; r1=2; dx=2; dy=0; ri=2; r0=1.1; r1=2; dx=5.5; dy=0; ri=3.0;
Look at this one with edges on and elements off.
64 (1-z)^3 z^3- 48 (1-z)^2 z^2 (3 x^2+3 y^2+2 z^2)+ 12 (1-z) z (27 (x^2+y^2)^2-24 z^2 (x^2+y^2)+ 36 sqrt(2) y z (y^2-3 x^2)+4 z^4)+ (9 x^2+9 y^2-2 z^2) (-81 (x^2+y^2)^2-72 z^2 (x^2+y^2)+ 108 sqrt(2) x z (x^2-3 y^2)+4 z^4) = 0;
x^4 + y^4 + z^4 - 5 (x^2 + y^2 + z^2 ) + 11.8 = 0;
Use +/- 3 for the domain bounds. It looks quite nice if 12.5 is used as the constant.
(x^2+y^2+z^2-a*k^2)^2-b*((z-k)^2-2*x^2)*((z+k)^2-2*y^2) = 0; k=5; a=0.95; b=0.8;
Use +/- 5 for the domain bounds.
4 (x^2 + y^2 + z^2 - 13)(x^2 + y^2 + z^2 - 13)(x^2 + y^2 + z^2 - 13) + 27 (3 x^2 + y^2 -4 z^2 - 12)^2 = 0;
Use +/- 5 for domain bounds.
a1*p1+(a2*z+a3)*p3+a4*z^3+a5*z^2+a6*z+a7 = 0; p1 = 2*x^3-6*x*y^2; p3 = x^2+y^2; a3 = -3*a1; a5 = a2^2/(3.0*a1); a6 = -a2; a7 = a1; a1 = -1.0; a2 = -1.0; a4 = 1.0;
Use +/-2 for the domain bounds.
a1*p1+(a2*z+a3)*p3+a4*z^3+a5*z^2+a6*z+a7 = 0; p1 = 2*x^3-6*x*y^2; p3 = x^2+y^2; a1 = -0.8; a2 = 0.0; a3 = 0.0; a4 = 1.0; a5 = 0.8; a6 = 0.0; a7 = 0.0;
25*z^3+16*z*y^2+60*x^2*y+50*z^2 = 0; p1 = 2*x^3-6*x*y^2; p3 = x^2+y^2; a1 = 3.0; a2 = 0.8;
Use +-4 for domain bounds.
x^2+y^2+z^3+3.2*(x^3-3*x*y^2) = 0;
16*p*q*r-s^3 = 0; p = 1-z-w2*x; q = 1-z+w2*x; r = 1+z+w2*y; s = 1+z-w2*y; w2 = 1.414214;
3 z + 9 z^2*b + b^4 + 15/7*b*re + a*(x^2 + y^2 + 9 z^2)^5 = 0; b = x^2 + y^2; re = ((x,y)*(x,y)*(x,y)*(x,y)*(x,y)*(x,y)).(1,0); a = 0.0;
Use +/-3 for the domain). Try a=0, 0.3, 0.64, 1.0.
z + ((x,y)*(x,y)*(x,y)*(x,y)*(x,y)).(1,0);
Use +/-2 for the domain bounds. This example illustrates some more complex features of the syntax, namely the use of vectors and the fact that 2D vectors can be used as complex numbers. Hence (x,y)*(x,y) calculates the square of the complex number (x,y) and (x,y).(1,0) calculates the dot product of two vectors. Here it calculates the real part of (x,y)^5.
Many of the equations here have been pinched from other place on the web, have a look to see some other wonderful surface.
and Richard Morris' own
Web page, applet and Algebraic Surface program by Richard Morris, copyright 1990-2003.
Maths home page Personal home page
Email pfaf@webmaster.org or rjm@amsta.leeds.ac.uk.
