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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 = 0
- This is a surface with four pieces that sweep up from the x-z plane.
- The equation is: y = x^2 z^2
- 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 x-axis as its central axis. The final form of the equation is:
- y^2 + z^2 = 0.5 (x^3 + x^2)
- A cubic saddle. The equation is: z = x^3 - y^3
- Variant of a devil's curve in 3-space. 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
- This is a folium rotated about the x-axis. The formula is:
- 2 x^2 - 3 x y^2 - 3 x z^2 + y^2 + z^2 = 0
- Glob - sort of like basic teardrop shape. The equation is:
- y^2 + z^2 = 0.5 x^5 + 0.5 x^4
- Variant of a lemniscate - the two lobes are much more teardrop-like.
- 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 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 of Gerono. This figure looks like two teardrops with their pointed ends connected. It is formed by rotating the Lemniscate of Gerono about the x-axis. The formula is:
- x^4 - x^2 + y^2 + z^2 = 0
- 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
- 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 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
- This figure looks like a hersheys kiss. It is formed by sweeping a Piriform about the x-axis. A basic form of the equation is:
- (x^4 - x^3) + y^2 + z^2 = 0.
- 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 - a Space Needle?
- Steiners quartic surface
- Torus having major radius sqrt(40), minor radius sqrt(12).
- Witch of Agnesi.
- Very rough approximation to the sin-wave 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>.