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This curve looks like the top part of a paraboloid, bounded from below by another paraboloid. The basic equation
is:

y^2  (x^2 + z^2) y^2  (x^2 + z^2 + 2 y  1)^2 =

Crossed_Trough

This is a surface with four pieces that sweep up from the xz plane.

The equation is: y = x^2 z^2

Cubic_Cylinder

A drop coming out of water? This is a curve formed by using the equation:

y = 1/2 x^2 (x + 1)

as the radius of a cylinder having the xaxis as its central axis. The final form of the equation is:

y^2 + z^2 = 0.5 (x^3 + x^2)

Cubic_Saddle_1

A cubic saddle. The equation is: z = x^3  y^3

Devils_Curve

Variant of a devil's curve in 3space. This figure has a top and bottom part that are very similar to a
hyperboloid of one sheet, however the central region is pinched in the middle leaving two teardrop shaped holes. The
equation is:

x^4 + 2 x^2 z^2  0.36 x^2  y^4 + 0.25 y^2 + z^4 = 0

Folium

This is a folium rotated about the xaxis. The formula is:

2 x^2  3 x y^2  3 x z^2 + y^2 + z^2 = 0

Glob_5

Glob  sort of like basic teardrop shape. The equation is:

y^2 + z^2 = 0.5 x^5 + 0.5 x^4

Twin_Glob

Variant of a lemniscate  the two lobes are much more teardroplike.

Helix, Helix_1

Approximation to the helix z = arctan(y/x). The helix can be approximated with an algebraic equation (kept to the
range of a quartic) with the following steps:

tan(z) = y/x => sin(z)/cos(z) = y/x =>
(1) x sin(z)  y cos(z) = 0 Using the taylor expansions for sin,
cos about z = 0, sin(z) = z  z^3/3! + z^5/5!  ... cos(z) = 1  z^2/2! + z^6/6!  ... Throwing out the
high order terms, the expression (1) can be written as: x (z  z^3/6)  y (1 + z^2/2) = 0, or
(2) 1/6 x
z^3 + x z + 1/2 y z^2  y = 0 This helix (2) turns 90 degrees in the range 0 <= z <= sqrt(2)/2. By using
scale <2 2 2>, the helix defined below turns 90 degrees in the range 0 <= z <= sqrt(2) = 1.4042.

Hyperbolic_Torus

Hyperbolic Torus having major radius sqrt(40), minor radius sqrt(12). This figure is generated by sweeping a
circle along the arms of a hyperbola. The equation is:

x^4 + 2 x^2 y^2  2 x^2 z^2  104 x^2 + y^4  2 y^2 z^2 + 56 y^2 + z^4 + 104 z^2 + 784 = 0

Lemniscate

Lemniscate of Gerono. This figure looks like two teardrops with their pointed ends connected. It is formed by
rotating the Lemniscate of Gerono about the xaxis. The formula is:

x^4  x^2 + y^2 + z^2 = 0

Quartic_Loop_1

This is a figure with a bumpy sheet on one side and something that looks like a paraboloid (but with an internal
bubble). The formula is:

(x^2 + y^2 + a c x)^2  (x^2 + y^2)(c  a x)^2

99*x^4+40*x^398*x^2*y^298*x^2*z^2+99*x^2+40*x*y^2

+40*x*z^2+y^4+2*y^2*z^2y^2+z^4z^2

Monkey_Saddle

This surface has three parts that sweep up and three down. This gives a saddle that has a place for two legs and
a tail... The equation is:

z = c (x^3  3 x y^2)

The value c gives a vertical scale to the surface  the smaller the value of c, the flatter the surface will be
(near the origin).

Parabolic_Torus_40_12

Parabolic Torus having major radius sqrt(40), minor radius sqrt(12). This figure is generated by sweeping a
circle along the arms of a parabola. The equation is:

x^4 + 2 x^2 y^2  2 x^2 z  104 x^2 + y^4  2 y^2 z + 56 y^2 + z^2 + 104 z + 784 = 0

Piriform

This figure looks like a hersheys kiss. It is formed by sweeping a Piriform about the xaxis. A basic form of the
equation is:

(x^4  x^3) + y^2 + z^2 = 0.

Quartic_Paraboloid

Quartic parabola  a 4th degree polynomial (has two bumps at the bottom) that has been swept around the z axis.
The equation is:

0.1 x^4  x^2  y^2  z^2 + 0.9 = 0

Quartic_Cylinder

Quartic Cylinder  a Space Needle?

Steiner_Surface

Steiners quartic surface

Torus_40_12

Torus having major radius sqrt(40), minor radius sqrt(12).

Witch_Hat

Witch of Agnesi.

Sinsurf

Very rough approximation to the sinwave surface z = sin(2 pi x y).

In order to get an approximation good to 7 decimals at a distance of 1 from the origin would require a polynomial
of degree around 60, which would require around 200,000 coefficients. For best results, scale by something like <1
1 0.2>.


 
 




 


