Documentation Talk:Reference Section 4.2

Next is the updated section about poly & polynomial, from 2 to 35 --Le Forgeron 18:22, 21 December 2010 (UTC)

Poly, Cubic and Quartic

Higher order polynomial surfaces may be defined by the use of a poly shape. The syntax is

POLY:
  poly {
    Order, <A1, A2, A3,... An>
    [POLY_MODIFIERS...]
    }
POLY_MODIFIERS:
  sturm | OBJECT_MODIFIER

Poly default values:

sturm : off

where Order is an integer number from 2 to 35 inclusively that specifies the order of the equation. A1, A2, ... An are float values for the coefficients of the equation. There are n such terms where n = ((Order+1)*(Order+2)*(Order+3))/6.

The cubic object is an alternate way to specify 3rd order polys. Its syntax is:

CUBIC:
  cubic {
    <A1, A2, A3,... A20>
    [POLY_MODIFIERS...]
    }

Also 4th order equations may be specified with the quartic object. Its syntax is:

QUARTIC:
  quartic {
    <A1, A2, A3,... A35>
    [POLY_MODIFIERS...]
    }

The following table shows which polynomial terms correspond to which x,y,z factors for the orders 2 to 7. Remember cubic is actually a 3rd order polynomial and quartic is 4th order.

	2^nd	3^rd	4^th	5^th	6^th	7^th		5^th	6^th	7^th		6^th	7^th
A₁	x²	x³	x⁴	x⁵	x⁶	x⁷	A₄₁	y³	xy³	x²y³	A₈₁	z³	xz³
A₂	xy	x²y	x³y	x⁴y	x⁵y	x⁶y	A₄₂	y²z³	xy²z³	x²y²z³	A₈₂	z²	xz²
A₃	xz	x²z	x³z	x⁴z	x⁵z	x⁶z	A₄₃	y²z²	xy²z²	x²y²z²	A₈₃	z	xz
A₄	x	x²	x³	x⁴	x⁵	x⁶	A₄₄	y²z	xy²z	x²y²z	A₈₄	1	x
A₅	y²	xy²	x²y²	x³y²	x⁴y²	x⁵y²	A₄₅	y²	xy²	x²y²	A₈₅		y⁷
A₆	yz	xyz	x²yz	x³yz	x⁴yz	x⁵yz	A₄₆	yz⁴	xyz⁴	x²yz⁴	A₈₆		y⁶z
A₇	y	xy	x²y	x³y	x⁴y	x⁵y	A₄₇	yz³	xyz³	x²yz³	A₈₇		y⁶
A₈	z²	xz²	x²z²	x³z²	x⁴z²	x⁵z²	A₄₈	yz²	xyz²	x²yz²	A₈₈		y⁵z²
A₉	z	xz	x²z	x³z	x⁴z	x⁵z	A₄₉	yz	xyz	x²yz	A₈₉		y⁵z
A₁₀	1	x	x²	x³	x⁴	x⁵	A₅₀	y	xy	x²y	A₉₀		y⁵
A₁₁		y³	xy³	x²y³	x³y³	x⁴y³	A₅₁	z⁵	xz⁵	x²z⁵	A₉₁		y⁴z³
A₁₂		y²z	xy²z	x²y²z	x³y²z	x⁴y²z	A₅₂	z⁴	xz⁴	x²z⁴	A₉₂		y⁴z²
A₁₃		y²	xy²	x²y²	x³y²	x⁴y²	A₅₃	z³	xz³	x²z³	A₉₃		y⁴z
A₁₄		yz²	xyz²	x²yz²	x³yz²	x⁴yz²	A₅₄	z²	xz²	x²z²	A₉₄		y⁴
A₁₅		yz	xyz	x²yz	x³yz	x⁴yz	A₅₅	z	xz	x²z	A₉₅		y³z⁴
A₁₆		y	xy	x²y	x³y	x⁴y	A₅₆	1	x	x²	A₉₆		y³z³
A₁₇		z³	xz³	x²z³	x³z³	x⁴z³	A₅₇		y⁶	xy⁶	A₉₇		y³z²
A₁₈		z²	xz²	x²z²	x³z²	x⁴z²	A₅₈		y⁵z	xy⁵z	A₉₈		y³z
A₁₉		z	xz	x²z	x³z	x⁴z	A₅₉		y⁵	xy⁵	A₉₉		y³
A₂₀		1	x	x²	x³	x⁴	A₆₀		y⁴z²	xy⁴z²	A₁₀₀		y²z⁵
A₂₁			y⁴	xy⁴	x²y⁴	x³y⁴	A₆₁		y⁴z	xy⁴z	A₁₀₁		y²z⁴
A₂₂			y³z	xy³z	x²y³z	x³y³z	A₆₂		y⁴	xy⁴	A₁₀₂		y²z³
A₂₃			y³	xy³	x²y³	x³y³	A₆₃		y³z³	xy³z³	A₁₀₃		y²z²
A₂₄			y²z²	xy²z²	x²y²z²	x³y²z²	A₆₄		y³z²	xy³z²	A₁₀₄		y²z
A₂₅			y²z	xy²z	x²y²z	x³y²z	A₆₅		y³z	xy³z	A₁₀₅		y²
A₂₆			y²	xy²	x²y²	x³y²	A₆₆		y³	xy³	A₁₀₆		yz⁶
A₂₇			yz³	xyz³	x²yz³	x³yz³	A₆₇		y²z⁴	xy²z⁴	A₁₀₇		yz⁵
A₂₈			yz²	xyz²	x²yz²	x³yz²	A₆₈		y²z³	xy²z³	A₁₀₈		yz⁴
A₂₉			yz	xyz	x²yz	x³yz	A₆₉		y²z²	xy²z²	A₁₀₉		yz³
A₃₀			y	xy	x²y	x³y	A₇₀		y²z	xy²z	A₁₁₀		yz²
A₃₁			z⁴	xz⁴	x²z⁴	x³z⁴	A₇₁		y²	xy²	A₁₁₁		yz
A₃₂			z³	xz³	x²z³	x³z³	A₇₂		yz⁵	xyz⁵	A₁₁₂		y
A₃₃			z²	xz²	x²z²	x³z²	A₇₃		yz⁴	xyz⁴	A₁₁₃		z⁷
A₃₄			z	xz	x²z	x³z	A₇₄		yz³	xyz³	A₁₁₄		z⁶
A₃₅			1	x	x²	x³	A₇₅		yz²	xyz²	A₁₁₅		z⁵
A₃₆				y⁵	xy⁵	x²y⁵	A₇₆		yz	xyz	A₁₁₆		z⁴
A₃₇				y⁴z	xy⁴z	x²y⁴z	A₇₇		y	xy	A₁₁₇		z³
A₃₈				y⁴	xy⁴	x²y⁴	A₇₈		z⁶	xz⁶	A₁₁₈		z²
A₃₉				y³z²	xy³z²	x²y³z²	A₇₉		z⁵	xz⁵	A₁₁₉		z
A₄₀				y³z	xy³z	x²y³z	A₈₀		z⁴	xz⁴	A₁₂₀		1

Polynomial shapes can be used to describe a large class of shapes including the torus, the lemniscate, etc. For example, to declare a quartic surface requires that each of the coefficients (A1 ... A35) be placed in order into a single long vector of 35 terms. As an example let's define a torus the hard way. A Torus can be represented by the equation: x⁴ + y⁴ + z⁴ + 2 x² y² + 2 x² z² + 2 y² z² - 2 (r_02 + r_12) x² + 2 (r_02 - r_12) y² - 2 (r_02 + r_12) z² + (r_02 - r_12)² = 0

Where r_0 is the major radius of the torus, the distance from the hole of the donut to the middle of the ring of the donut, and r_1 is the minor radius of the torus, the distance from the middle of the ring of the donut to the outer surface. The following object declaration is for a torus having major radius 6.3 minor radius 3.5 (Making the maximum width just under 20).

// Torus having major radius sqrt(40), minor radius sqrt(12)
quartic {
  < 1,   0,   0,   0,   2,   0,   0,   2,   0,
  -104,   0,   0,   0,   0,   0,   0,   0,   0,
  0,   0,   1,   0,   0,   2,   0,  56,   0,
  0,   0,   0,   1,   0, -104,  0, 784 >
  sturm
  }

If you are afraid of messing up with the numerous coefficients, an alternative syntax is available as polynomial which does not care about the order of the coefficients (as long as you do not define one more than once, otherwise only the value of the last definition is kept) and default with all coefficients to 0 (so you can hopefully also save some typing if you have a lot of 0):

POLYNOMIAL:
  polynomial {
    Order, [COEFFICIENTS...]
    [POLY_MODIFIERS...]
    }
COEFFICIENTS:
  xyz(<x_power>,<y_power>,<z_power>):<value>[,]
POLY_MODIFIERS:
  sturm | OBJECT_MODIFIER

Same torus as above, with the polynomial syntax:

// Torus having major radius sqrt(40), minor radius sqrt(12)
polynomial { 4,
  xyz(4,0,0):1,   
  xyz(2,2,0):2,  
  xyz(2,0,2):2,
  xyz(2,0,0:-104,  
  xyz(0,4,0):1,
  xyz(0,2,2):2,
  xyz(0,2,0):56,
  xyz(0,0,4):1,
  xyz(0,0,2):-104, 
  xyz(0,0,0):784
  sturm
  }

Poly, cubic and quartics are just like quadrics in that you do not have to understand one to use one. The file shapesq.inc has plenty of pre-defined quartics for you to play with.

Polys use highly complex computations and will not always render perfectly. If the surface is not smooth, has dropouts, or extra random pixels, try using the optional keyword sturm in the definition. This will cause a slower but more accurate calculation method to be used. Usually, but not always, this will solve the problem. If sturm does not work, try rotating or translating the shape by some small amount.

There are really so many different polynomial shapes, we cannot even begin to list or describe them all. We suggest you find a good reference or text book if you want to investigate the subject further.

Isosurface Object

Details about many of the things that can be done with the isosurface object are discussed in the isosurface tutorial section. Below you will only find the syntax basics:

isosurface {
  function { FUNCTION_ITEMS }
  [contained_by { SPHERE | BOX }]
  [threshold FLOAT_VALUE]
  [accuracy FLOAT_VALUE]
  [max_gradient FLOAT_VALUE]
  [evaluate P0, P1, P2]
  [open]
  [max_trace INTEGER] | [all_intersections]
  [OBJECT_MODIFIERS...]
  }

Isosurface default values:

contained_by : box{-1,1}
threshold    : 0.0
accuracy     : 0.001
max_gradient : 1.1

function { ... } This must be specified and be the first item of the isosurface statement. Here you place all the mathematical functions that will describe the surface.

contained_by { ... } The contained_by object limits the area where POV-Ray samples for the surface of the function. This container can either be a sphere or a box, both of which use the standard POV-Ray syntax. If not specified a box {<-1,-1,-1>, <1,1,1>} will be used as default.

contained_by { sphere { CENTER, RADIUS } }
contained_by { box { CORNER1, CORNER2 } }

threshold This specifies how much strength, or substance to give the isosurface. The surface appears where the function value equals the threshold value. The default threshold is 0.

function = threshold

accuracy The isosurface finding method is a recursive subdivision method. This subdivision goes on until the length of the interval where POV-Ray finds a surface point is less than the specified accuracy. The default value is 0.001.
Smaller values produces more accurate surfaces, but it takes longer to render.

max_gradient POV-Ray can find the first intersecting point between a ray and the isosurface of any continuous function if the maximum gradient of the function is known. Therefore you can specify a max_gradient for the function. The default value is 1.1. When the max_gradient used to find the intersecting point is too high, the render slows down considerably. When it is too low, artefacts or holes may appear on the isosurface. When it is way too low, the surface does not show at all. While rendering the isosurface POV-Ray records the found gradient values and prints a warning if these values are higher or much lower than the specified max_gradient:

Warning: The maximum gradient found was 5.257, but max_gradient of
the isosurface was set to 5.000. The isosurface may contain holes!
Adjust max_gradient to get a proper rendering of the isosurface.

Warning: The maximum gradient found was 5.257, but max_gradient of
the isosurface was set to 7.000. Adjust max_gradient to
get a faster rendering of the isosurface.

For best performance you should specify a value close to the real maximum gradient.

evaluate POV-Ray can also dynamically adapt the used max_gradient. To activate this technique you have to specify the evaluate keyword followed by three parameters:

P0: the minimum max_gradient in the estimation process,
P1: an over-estimating factor. This means that the max_gradient is multiplied by the P1 parameter.
P2: an attenuation parameter (1 or less)

In this case POV-Ray starts with the max_gradient value P0 and dynamically changes it during the render using P1 and P2. In the evaluation process, the P1 and P2 parameters are used in quadratic functions. This means that over-estimation increases more rapidly with higher values and attenuation more rapidly with lower values. Also with dynamic max_gradient, there can be artefacts and holes.

If you are unsure what values to use, start a render without evaluate to get a value for max_gradient. Now you can use it with evaluate like this:

P0 : found max_gradient * min_factor
min_factor being a float between 0 and 1 to reduce the max_gradient to a minimum max_gradient. The ideal value for P0 would be the average of the found max_gradients, but we do not have access to that information.
A good starting point is 0.6 for the min_factor
P1 : sqrt(found max_gradient/(found max_gradient * min_factor))
min_factor being the same as used in P0 this will give an over-estimation factor of more than 1, based on your minimum max_gradient and the found max_gradient.
P2 : 1 or less
0.7 is a good starting point.

When there are artifacts / holes in the isosurface, increase the min_factor and / or P2 a bit. Example: when the first run gives a found max_gradient of 356, start with

#declare Min_factor= 0.6;
isosurface {
  ...
  evaluate 356*Min_factor,  sqrt(356/(356*Min_factor)),  0.7
  //evaluate 213.6, 1.29, 0.7
  ...
  }

This method is only an approximation of what happens internally, but it gives faster rendering speeds with the majority of isosurfaces.

open When the isosurface is not fully contained within the contained_by object, there will be a cross section. Where this happens, you will see the surface of the container. With the open keyword, these cross section surfaces are removed. The inside of the isosurface becomes visible.

Note: Using open slows down the render speed, and it is not recommended to use it with CSG operations.

max_trace Isosurfaces can be used in CSG shapes since they are solid finite objects - if not finite by themselves, they are through the cross section with the container.
By default POV-Ray searches only for the first surface which the ray intersects. But when using an isosurface in CSG operations, the other surfaces must also be found. Therefore, the keyword max_trace must be added to the isosurface statement. It must be followed by an integer value. To check for all surfaces, use the keyword all_intersections instead.
With all_intersections POV-Ray keeps looking until all surfaces are found. With a max_trace it only checks until that number is reached.

Documentation Talk:Reference Section 4.2

Poly, Cubic and Quartic

Isosurface Object

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