Jholsenback: 1 revision: Reference Migration Initial Load

2012-03-15T19:08:47Z

1 revision: Reference Migration Initial Load

Jholsenback: 1 revision: Initial Load (TF)

2012-03-11T22:31:23Z

1 revision: Initial Load (TF)

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[[Category:Include Files]]
<dl>
<dt><code>Bicorn</code>
<dd>This curve looks like the top part of a paraboloid, bounded from below by another paraboloid.
The basic equation is: 
<dd>y^2 - (x^2 + z^2) y^2 - (x^2 + z^2 + 2 y - 1)^2 = 0

<dt><code>Crossed_Trough</code>
<dd>This is a surface with four pieces that sweep up from the x-z plane.
<dd>The equation is: y = x^2 z^2

<dt><code>Cubic_Cylinder</code>
<dd>A drop coming out of water? This is a curve formed by using the equation:
<dd>y = 1/2 x^2 (x + 1)
<dd>as the radius of a cylinder having the x-axis as its central axis. The final form of the equation is:
<dd>y^2 + z^2 = 0.5 (x^3 + x^2)

<dt><code>Cubic_Saddle_1</code>
<dd>A cubic saddle. The equation is: z = x^3 - y^3

<dt><code>Devils_Curve</code>
<dd>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:
<dd>x^4 + 2 x^2 z^2 - 0.36 x^2 - y^4 + 0.25 y^2 + z^4 = 0

<dt><code>Folium</code>
<dd>This is a folium rotated about the x-axis. The formula is:
<dd>2 x^2 - 3 x y^2 - 3 x z^2 + y^2 + z^2 = 0

<dt><code>Glob_5</code>
<dd>Glob - sort of like basic teardrop shape. The equation is:
<dd>y^2 + z^2 = 0.5 x^5 + 0.5 x^4

<dt><code>Twin_Glob</code>
<dd>Variant of a lemniscate - the two lobes are much more teardrop-like.

<dt><code>Helix, Helix_1</code>
<dd>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:
<dd>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.

<dt><code>Hyperbolic_Torus</code>
<dd>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:
<dd>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

<dt><code>Lemniscate</code>
<dd>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:
<dd>x^4 - x^2 + y^2 + z^2 = 0

<dt><code>Quartic_Loop_1</code>
<dd>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:
<dd>(x^2 + y^2 + a c x)^2 - (x^2 + y^2)(c - a x)^2
<dd>-99*x^4+40*x^3-98*x^2*y^2-98*x^2*z^2+99*x^2+40*x*y^2
<dd>+40*x*z^2+y^4+2*y^2*z^2-y^2+z^4-z^2

<dt><code>Monkey_Saddle</code>
<dd>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:
<dd><code>z = c (x^3 - 3 x y^2)</code>
<dd>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).

<dt><code>Parabolic_Torus_40_12</code>
<dd>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:
<dd>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

<dt><code>Piriform</code>
<dd>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:
<dd>(x^4 - x^3) + y^2 + z^2 = 0.

<dt><code>Quartic_Paraboloid</code>
<dd>Quartic parabola - a 4th degree polynomial (has two bumps at the bottom) that has been swept around the z axis. The equation is:
<dd>0.1 x^4 - x^2 - y^2 - z^2 + 0.9 = 0

<dt><code>Quartic_Cylinder</code>
<dd>Quartic Cylinder - a Space Needle?

<dt><code>Steiner_Surface</code>
<dd>Steiners quartic surface

<dt><code>Torus_40_12</code>
<dd>Torus having major radius sqrt(40), minor radius sqrt(12).

<dt><code>Witch_Hat</code>
<dd>Witch of Agnesi.

<dt><code>Sinsurf</code>
<dd>Very rough approximation to the sin-wave surface z = sin(2 pi x y).
<dd>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>.
</dl>

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