

2.7.7.2 Internal Functions
Here is a list of the internal functions in the order they appear in the "functions.inc" include file
f_algbr_cyl1(x,y,z, P0, P1, P2, P3, P4) . An algebraic cylinder is what you get if you take any 2d
curve and plot it in 3d. The 2d curve is simply extruded along the third axis, in this case the z axis. With the
SOR Switch switched on, the figureofeight curve will be rotated around the Y axis instead of being extruded along
the Z axis.
f_algbr_cyl2(x,y,z, P0, P1, P2, P3, P4) . An algebraic cylinder is what you get if you take any 2d
curve and plot it in 3d. The 2d curve is simply extruded along the third axis, in this case the z axis. With the
SOR Switch switched on, the cross section curve will be rotated around the Y axis instead of being extruded along the
Z axis.
f_algbr_cyl3(x,y,z, P0, P1, P2, P3, P4) . An algebraic cylinder is what you get if you take any 2d
curve and plot it in 3d. The 2d curve is simply extruded along the third axis, in this case the Z axis. With the
SOR Switch switched on, the cross section curve will be rotated around the Y axis instead of being extruded along the
Z axis.
f_algbr_cyl4(x,y,z, P0, P1, P2, P3, P4) . An algebraic cylinder is what you get if you take any 2d
curve and plot it in 3d. The 2d curve is simply extruded along the third axis, in this case the z axis. With the
SOR Switch switched on, the cross section curve will be rotated around the Y axis instead of being extruded along the
Z axis.
f_bicorn(x,y,z, P0, P1) . The surface is a surface of revolution.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Scale. The mathematics of this surface suggest that the shape should be different for different
values of this parameter. In practice the difference in shape is hard to spot. Setting the scale to 3 gives a surface
with a radius of about 1 unit
f_bifolia(x,y,z, P0, P1) . The bifolia surface looks something like the top part of a a paraboloid
bounded below by another paraboloid.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Scale. The surface is always the same shape. Changing this parameter has the same effect as
adding a scale modifier. Setting the scale to 1 gives a surface with a radius of about 1 unit
f_blob(x,y,z, P0, P1, P2, P3, P4) . This function generates blobs that are similar to a CSG blob with
two spherical components. This function only seems to work with negative threshold settings.

P0 : X distance between the two components

P1 : Blob strength of component 1

P2 : Inverse blob radius of component 1

P3 : Blob strength of component 2

P4 : Inverse blob radius of component 2
f_blob2(x,y,z, P0, P1, P2, P3) . The surface is similar to a CSG blob with two spherical components.

P0 : Separation. One blob component is at the origin, and the other is this distance away on the X
axis

P1 : Inverse size. Increase this to decrease the size of the surface

P2 : Blob strength

P3 : Threshold. Setting this parameter to 1 and the threshold to zero has exactly the same effect as
setting this parameter to zero and the threshold to 1
f_boy_surface(x,y,z, P0, P1) . For this surface, it helps if the field strength is set low, otherwise
the surface has a tendency to break up or disappear entirely. This has the side effect of making the rendering times
extremely long.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Scale. The surface is always the same shape. Changing this parameter has the same effect as
adding a scale modifier
f_comma(x,y,z, P0) . The 'comma' surface is very much like a commashape.
f_cross_ellipsoids(x,y,z, P0, P1, P2, P3) . The 'cross ellipsoids' surface is like the union of three
crossed ellipsoids, one oriented along each axis.

P0 : Eccentricity. When less than 1, the ellipsoids are oblate, when greater than 1 the ellipsoids
are prolate, when zero the ellipsoids are spherical (and hence the whole surface is a sphere)

P1 : Inverse size. Increase this to decrease the size of the surface

P2 : Diameter. Increase this to increase the size of the ellipsoids

P3 : Threshold. Setting this parameter to 1 and the threshold to zero has exactly the same effect as
setting this parameter to zero and the threshold to 1
f_crossed_trough(x,y,z, P0)

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_cubic_saddle(x,y,z, P0) . For this surface, it helps if the field strength is set quite low,
otherwise the surface has a tendency to break up or disappear entirely.

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_cushion(x,y,z, P0)

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_devils_curve(x,y,z, P0)

P0 : Field Strength (Needs a negative field strength or a negated function)
f_devils_curve_2d(x,y,z, P0, P1, P2, P3, P4, P5) . The f_devils_curve_2d curve can be
extruded along the z axis, or using the SOR parameters it can be made into a surface of revolution. The X and Y
factors control the size of the central feature.
f_dupin_cyclid(x,y,z, P0, P1, P2, P3, P4, P5)

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Major radius of torus

P2 : Minor radius of torus

P3 : X displacement of torus

P4 : Y displacement of torus

P5 : Radius of inversion
f_ellipsoid(x,y,z, P0, P1, P2) . f_ellipsoid generates spheres and ellipsoids. Needs
"threshold 1". Setting these scaling parameters to 1/n gives exactly the same effect as performing a
scale operation to increase the scaling by n in the corresponding direction.

P0 : X scale (inverse)

P1 : Y scale (inverse)

P2 : Z scale (inverse)
f_enneper(x,y,z, P0)

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_flange_cover(x,y,z, P0, P1, P2, P3)

P0 : Spikiness. Set this to very low values to increase the spikes. Set it to 1 and you get a sphere

P1 : Inverse size. Increase this to decrease the size of the surface. (The other parameters also
drastically affect the size, but this parameter has no other effects)

P2 : Flange. Increase this to increase the flanges that appear between the spikes. Set it to 1 for
no flanges

P3 : Threshold. Setting this parameter to 1 and the threshold to zero has exactly the same effect as
setting this parameter to zero and the threshold to 1
f_folium_surface(x,y,z, P0, P1, P2) . A 'folium surface' looks something like a paraboloid glued to a
plane.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Neck width factor  the larger you set this, the narrower the neck where the paraboloid meets
the plane

P2 : Divergence  the higher you set this value, the wider the paraboloid gets
f_folium_surface_2d(x,y,z, P0, P1, P2, P3, P4, P5) . The f_folium_surface_2d curve can be
rotated around the X axis to generate the same 3d surface as the f_folium_surface , or it can be extruded
in the Z direction (by switching the SOR switch off)

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Neck width factor  same as the 3d surface if you are revolving it around the Y axis

P2 : Divergence  same as the 3d surface if you are revolving it around the Y axis

P3 : SOR Switch

P4 : SOR Offset

P5 : SOR Angle
f_glob(x,y,z, P0) . One part of this surface would actually go off to infinity if it were not
restricted by the contained_by shape.

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_heart(x,y,z, P0)

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_helical_torus(x,y,z, P0, P1, P2, P3, P4, P5, P6, P7, P8, P9) . With some sets of parameters, it looks
like a torus with a helical winding around it. The winding optionally has grooves around the outside.

P0 : Major radius

P1 : Number of winding loops

P2 : Twistiness of winding. When zero, each winding loop is separate. When set to one, each loop
twists into the next one. When set to two, each loop twists into the one after next

P3 : Fatness of winding?

P4 : Threshold. Setting this parameter to 1 and the threshold to zero has s similar effect as
setting this parameter to zero and the threshold to 1

P5 : Negative minor radius? Reducing this parameter increases the minor radius of the central torus.
Increasing it can make the torus disappear and be replaced by a vertical column. The value at which the surface
switches from one form to the other depends on several other parameters

P6 : Another fatness of winding control?

P7 : Groove period. Increase this for more grooves

P8 : Groove amplitude. Increase this for deeper grooves

P9 : Groove phase. Set this to zero for symmetrical grooves
f_helix1(x,y,z, P0, P1, P2, P3, P4, P5, P6)

P0 : Number of helixes  e.g. 2 for a double helix

P1 : Period  is related to the number of turns per unit length

P2 : Minor radius (major radius > minor radius)

P3 : Major radius

P4 : Shape parameter. If this is greater than 1 then the tube becomes fatter in the y direction

P5 : Cross section type

P6 : Cross section rotation angle (degrees)
f_helix2(x,y,z, P0, P1, P2, P3, P4, P5, P6) . Needs a negated function

P0 : Not used

P1 : Period  is related to the number of turns per unit length

P2 : Minor radius (minor radius > major radius)

P3 : Major radius

P4 : Not used

P5 : Cross section type

P6 : Cross section rotation angle (degrees)
f_hex_x(x,y,z, P0) . This creates a grid of hexagonal cylinders stretching along the zaxis. The
fatness is controlled by the threshold value. When this value equals 0.8660254 or cos(30) the sides will touch,
because this is the distance between centers. Negating the function will inverse the surface and create a honeycomb
structure. This function is also useful as pigment function.

P0 : No effect (but the syntax requires at least one parameter)
f_hex_y(x,y,z, P0) . This is function forms a lattice of infinite boxes stretching along the zaxis.
The fatness is controlled by the threshold value. These boxes are rotated 60 degrees around centers, which are
0.8660254 or cos(30) away from each other. This function is also useful as pigment function.

P0 : No effect (but the syntax requires at least one parameter)
f_hetero_mf(x,y,z, P0, P1, P2, P3, P4, P5) . f_hetero_mf (x,0,z) makes multifractal height
fields and patterns of '1/f' noise 'Multifractal' refers to their characteristic of having a fractal dimension
which varies with altitude. Built from summing noise of a number of frequencies, the hetero_mf parameters determine
how many, and which frequencies are to be summed. An advantage to using these instead of a height_field {} from an
image (a number of height field programs output multifractal types of images) is that the hetero_mf function domain
extends arbitrarily far in the x and z directions so huge landscapes can be made without losing resolution or having
to tile a height field. Other functions of interest are f_ridged_mf and f_ridge .

P0 : H is the negative of the exponent of the basis noise frequencies used in building these
functions (each frequency f's amplitude is weighted by the factor f  H ). In landscapes, and many natural forms, the
amplitude of high frequency contributions are usually less than the lower frequencies. When H is 1, the
fractalization is relatively smooth ("1/f noise"). As H nears 0, the high frequencies contribute
equally with low frequencies as in "white noise".

P1 : Lacunarity' is the multiplier used to get from one 'octave' to the next. This parameter affects
the size of the frequency gaps in the pattern. Make this greater than 1.0

P2 : Octaves is the number of different frequencies added to the fractal. Each 'Octave' frequency is
the previous one multiplied by 'Lacunarity', so that using a large number of octaves can get into very high
frequencies very quickly.

P3 : Offset is the 'base altitude' (sea level) used for the heterogeneous scaling

P4 : T scales the 'heterogeneity' of the fractal. T=0 gives 'straight 1/f' (no heterogeneous
scaling). T=1 suppresses higher frequencies at lower altitudes

P5 : Generator type used to generate the noise3d. 0, 1, 2 and 3 are legal values.
f_hunt_surface(x,y,z, P0)

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_hyperbolic_torus(x,y,z, P0, P1, P2)

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Major radius: separation between the centers of the tubes at the closest point

P2 : Minor radius: thickness of the tubes at the closest point
f_isect_ellipsoids(x,y,z, P0, P1, P2, P3) . The 'isect ellipsoids' surface is like the intersection of
three crossed ellipsoids, one oriented along each axis.

P0 : Eccentricity. When less than 1, the ellipsoids are oblate, when greater than 1 the ellipsoids
are prolate, when zero the ellipsoids are spherical (and hence the whole surface is a sphere)

P1 : Inverse size. Increase this to decrease the size of the surface

P2 : Diameter. Increase this to increase the size of the ellipsoids

P3 : Threshold. Setting this parameter to 1 and the threshold to zero has exactly the same effect as
setting this parameter to zero and the threshold to 1
f_kampyle_of_eudoxus(x,y,z, P0, P1, P2) . The 'kampyle of eudoxus' is like two infinite planes with a
dimple at the center.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Dimple: When zero, the two dimples punch right through and meet at the center. Nonzero values
give less dimpling

P2 : Closeness: Higher values make the two planes become closer
f_kampyle_of_eudoxus_2d(x,y,z, P0, P1, P2, P3, P4, P5) The 2d curve that generates the above surface
can be extruded in the Z direction or rotated about various axes by using the SOR parameters.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Dimple: When zero, the two dimples punch right through and meet at the center. Nonzero values
give less dimpling

P2 : Closeness: Higher values make the two planes become closer

P3 : SOR Switch

P4 : SOR Offset

P5 : SOR Angle
f_klein_bottle(x,y,z, P0)

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_kummer_surface_v1(x,y,z, P0) . The Kummer surface consists of a collection of radiating rods.

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_kummer_surface_v2(x,y,z, P0, P1, P2, P3) . Version 2 of the kummer surface only looks like radiating
rods when the parameters are set to particular negative values. For positive values it tends to look rather like a
superellipsoid.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Rod width (negative): Setting this parameter to larger negative values increases the diameter
of the rods

P2 : Divergence (negative): Setting this number to 1 causes the rods to become approximately
cylindrical. Larger negative values cause the rods to become fatter further from the origin. Smaller negative numbers
cause the rods to become narrower away from the origin, and have a finite length

P3 : Influences the length of half of the rods. Changing the sign affects the other half of the
rods. 0 has no effect
f_lemniscate_of_gerono(x,y,z, P0) . The "Lemniscate of Gerono" surface is an hourglass shape.
Two teardrops with their ends connected.

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_lemniscate_of_gerono_2d(x,y,z, P0, P1, P2, P3, P4, P5) . The 2d version of the Lemniscate can be
extruded in the Z direction, or used as a surface of revolution to generate the equivalent of the 3d version, or
revolved in different ways.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Size: increasing this makes the 2d curve larger and less rounded

P2 : Width: increasing this makes the 2d curve fatter

P3 : SOR Switch

P4 : SOR Offset

P5 : SOR Angle
f_mesh1(x,y,z, P0, P1, P2, P3, P4) The overall thickness of the threads is controlled by the
isosurface threshold, not by a parameter. If you render a mesh1 with zero threshold, the threads have zero thickness
and are therefore invisible. Parameters P2 and P4 control the shape of the thread relative to this threshold
parameter.

P0 : Distance between neighboring threads in the x direction

P1 : Distance between neighboring threads in the z direction

P2 : Relative thickness in the x and z directions

P3 : Amplitude of the weaving effect. Set to zero for a flat grid

P4 : Relative thickness in the y direction
f_mitre(x,y,z, P0) . The 'Mitre' surface looks a bit like an ellipsoid which has been nipped at each
end with a pair of sharp nosed pliers.

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_nodal_cubic(x,y,z, P0) . The 'Nodal Cubic' is something like what you would get if you were to
extrude the Stophid2D curve along the X axis and then lean it over.

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_noise3d(x,y,z)
f_noise_generator(x,y,z, P0)

P0 : Noise generator number
f_odd(x,y,z, P0)

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_ovals_of_cassini(x,y,z, P0, P1, P2, P3) . The Ovals of Cassini are a generalization of the torus
shape.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Major radius  like the major radius of a torus

P2 : Filling. Set this to zero, and you get a torus. Set this to a higher value and the hole in the
middle starts to heal up. Set it even higher and you get an ellipsoid with a dimple

P3 : Thickness. The higher you set this value, the plumper is the result
f_paraboloid(x,y,z, P0) . This paraboloid is the surface of revolution that you get if you rotate a
parabola about the Y axis.

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_parabolic_torus(x,y,z, P0, P1, P2)

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Major radius

P2 : Minor radius
f_ph(x,y,z) = atan2( sqrt( x*x + z*z ), y ) When used alone, the "PH" function gives a
surface that consists of all points that are at a particular latitude, i.e. a cone. If you use a threshold of zero
(the default) this gives a cone of width zero, which is invisible. Also look at f_th and f_r
f_pillow(x,y,z, P0)
f_piriform(x,y,z, P0) . The piriform surface looks rather like half a lemniscate.
f_piriform_2d(x,y,z, P0, P1, P2, P3, P4, P5, P6) . The 2d version of the "Piriform" can be
extruded in the Z direction, or used as a surface of revolution to generate the equivalent of the 3d version.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Size factor 1: increasing this makes the curve larger

P2 : Size factor 2: making this less negative makes the curve larger but also thinner

P3 : Fatness: increasing this makes the curve fatter

P4 : SOR Switch

P5 : SOR Offset

P6 : SOR Angle
f_poly4(x,y,z, P0, P1, P2, P3, P4) . This f_poly4 can be used to generate the surface of
revolution of any polynomial up to degree 4. To put it another way: If we call the parameters A, B, C, D, E; then
this function generates the surface of revolution formed by revolving "x = A + By + Cy2 + Dy3 + Ey4" around
the Y axis.

P0 : Constant

P1 : Y coefficient

P2 : Y2 coefficient

P3 : Y3 coefficient

P4 : Y4 coefficient
f_polytubes(x,y,z, P0, P1, P2, P3, P4, P5) . The 'Polytubes' surface consists of a number of tubes.
Each tube follows a 2d curve which is specified by a polynomial of degree 4 or less. If we look at the parameters,
then this function generates "P0" tubes which all follow the equation " x = P1 + P2y + P3y2 + P4y3 +
P5y4 " arranged around the Y axis. This function needs a positive threshold (fatness of the tubes).

P0 : Number of tubes

P1 : Constant

P2 : Y coefficient

P3 : Y2 coefficient

P4 : Y3 coefficient

P5 : Y4 coefficient
f_quantum(x,y,z, P0) . It resembles the shape of the electron density cloud for one of the d orbitals.

P0 : Not used, but required
f_quartic_paraboloid(x,y,z, P0) . The 'Quartic Paraboloid' is similar to a paraboloid, but has a
squarer shape.

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_quartic_saddle(x,y,z, P0) . The 'Quartic saddle' is similar to a saddle, but has a squarer shape.
f_quartic_cylinder(x,y,z, P0, P1, P2) . The 'Quartic cylinder' looks a bit like a cylinder that is
swallowed an egg.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Diameter of the "egg"

P2 : Controls the width of the tube and the vertical scale of the "egg"
f_r(x,y,z) = sqrt( x*x + y*y + z*z ) When used alone, the "R" function gives a surface
that consists of all the points that are a specific distance (threshold value) from the origin, i.e. a sphere. Also
look at f_ph and f_th
f_ridge(x,y,z, P0, P1, P2, P3, P4, P5) . This function is mainly intended for modifying other surfaces
as you might use a height field or to use as pigment function. Other functions of interest are f_hetero_mf
and f_ridged_mf .

P0 : Lambda

P1 : Octaves

P2 : Omega

P3 : Offset

P4 : Ridge

P5 : Generator type used to generate the noise3d. 0, 1, 2 and 3 are legal values.
f_ridged_mf(x,y,z, P0, P1, P2, P3, P4, P5) . The "Ridged Multifractal" surface can be used to
create multifractal height fields and patterns. 'Multifractal' refers to their characteristic of having a fractal
dimension which varies with altitude. They are built from summing noise of a number of frequencies. The f_ridged_mf
parameters determine how many, and which frequencies are to be summed, and how the different frequencies are weighted
in the sum. An advantage to using these instead of a height_field{} from an image is that the
ridged_mf function domain extends arbitrarily far in the x and z directions so huge landscapes can be made without
losing resolution or having to tile a height field. Other functions of interest are f_hetero_mf and f_ridge .

P0 : H is the negative of the exponent of the basis noise frequencies used in building these
functions (each frequency f's amplitude is weighted by the factor fE H ). When H is 1, the fractalization is
relatively smooth. As H nears 0, the high frequencies contribute equally with low frequencies

P1 : Lacunarity is the multiplier used to get from one "octave" to the next in the
"fractalization". This parameter affects the size of the frequency gaps in the pattern. (Use values
greater than 1.0)

P2 : Octaves is the number of different frequencies added to the fractal. Each octave frequency is
the previous one multiplied by "Lacunarity". So, using a large number of octaves can get into very high
frequencies very quickly

P3 : Offset gives a fractal whose fractal dimension changes from altitude to altitude. The high
frequencies at low altitudes are more damped than at higher altitudes, so that lower altitudes are smoother than
higher areas

P4 : Gain weights the successive contributions to the accumulated fractal result to make creases
stick up as ridges

P5 : Generator type used to generate the noise3d. 0, 1, 2 and 3 are legal values.
f_rounded_box(x,y,z, P0, P1, P2, P3) . The Rounded Box is defined in a cube from <1, 1, 1> to
<1, 1, 1>. By changing the " Scale" parameters, the size can be adjusted, without affecting the Radius
of curvature.

P0 : Radius of curvature. Zero gives square corners, 0.1 gives corners that match "sphere {0,
0.1}"

P1 : Scale x

P2 : Scale y

P3 : Scale z
f_sphere(x,y,z, P0)
f_spikes(x,y,z, P0, P1, P2, P3, P4)

P0 : Spikiness. Set this to very low values to increase the spikes. Set it to 1 and you get a sphere

P1 : Hollowness. Increasing this causes the sides to bend in more

P2 : Size. Increasing this increases the size of the object

P3 : Roundness. This parameter has a subtle effect on the roundness of the spikes

P4 : Fatness. Increasing this makes the spikes fatter
f_spikes_2d(x,y,z, P0, P1, P2, P3) =2D function : f = f( x, z )  y

P0 : Height of central spike

P1 : Frequency of spikes in the X direction

P2 : Frequency of spikes in the Z direction

P3 : Rate at which the spikes reduce as you move away from the center
f_spiral(x,y,z, P0, P1, P2, P3, P4, P5)

P0 : Distance between windings

P1 : Thickness

P2 : Outer diameter of the spiral. The surface behaves as if it is contained_by a sphere of this
diameter

P3 : Not used

P4 : Not used

P5 : Cross section type
f_steiners_roman(x,y,z, P0) . The "Steiners Roman" is composed of four identical triangular
pads which together make up a sort of rounded tetrahedron. There are creases along the X, Y and Z axes where the pads
meet.

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_strophoid(x,y,z, P0, P1, P2, P3) . The "Strophoid" is like an infinite plane with a bulb
sticking out of it.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Size of bulb. Larger values give larger bulbs. Negative values give a bulb on the other side of
the plane

P2 : Sharpness. When zero, the bulb is like a sphere that just touches the plane. When positive,
there is a crossover point. When negative the bulb simply bulges out of the plane like a pimple

P3 : Flatness. Higher values make the top end of the bulb fatter
f_strophoid_2d(x,y,z, P0, P1, P2, P3, P4, P5, P6) . The 2d strophoid curve can be extruded in the Z
direction or rotated about various axes by using the SOR parameters.

P0 : Field Strength

P1 : Size of bulb. Larger values give larger bulbs. Negative values give a bulb on the other side of
the plane

P2 : Sharpness. When zero, the bulb is like a sphere that just touches the plane. When positive,
there is a crossover point. When negative the bulb simply bulges out of the plane like a pimple

P3 : Fatness. Higher values make the top end of the bulb fatter

P4 : SOR Switch

P5 : SOR Offset

P6 : SOR Angle
f_superellipsoid(x,y,z, P0, P1) . Needs a negative field strength or a negated function.

P0 : eastwest exponentx

P1 : northsouth exponent
f_th(x,y,z) = atan2( x, z )
f_th() is a function that is only useful when combined
with other surfaces. It produces a value which is equal to the "theta" angle, in radians, at any point.
The theta angle is like the longitude coordinate on the Earth. It stays the same as you move north or south, but
varies from east to west. Also look at f_ph and f_r
f_torus(x,y,z, P0, P1)

P0 : Major radius

P1 : Minor radius
f_torus2(x,y,z, P0, P1, P2) . This is different from the f_torus function which just has the major and
minor radii as parameters.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Major radius

P2 : Minor radius
f_torus_gumdrop(x,y,z, P0) . The "Torus Gumdrop" surface is something like a torus with a
couple of gumdrops hanging off the end.

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_umbrella(x,y,z, P0)

P0 : Field Strength (Needs a negative field strength or
a negated function)
f_witch_of_agnesi(x,y,z, P0, P1, P2, P3, P4, P5) . The "Witch of Agnesi" surface looks
something like a witches hat.

P0 : Field Strength (Needs a negative field strength or
a negated function)

P1 : Controls the width of the spike. The height of the spike is always about 1 unit
f_witch_of_agnesi_2d(x,y,z, P0, P1, P2, P3, P4, P5) . The 2d version of the "Witch of Agnesi"
curve can be extruded in the Z direction or rotated about various axes by use of the SOR parameters.

