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#### 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

where Order is an integer number from 2 to 7 inclusively that specifies the order of the equation. A1, A2, ... An are float values for the coefficients of the equation. There are m 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. Remember `cubic` is actually a 3rd order polynomial and `quartic` is 4th order.

| 2nd | 3rd | 4th | 5th | 6th | 7th |  |  | 5th | 6th | 7th |  |  | 6th | 7th
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 | x5z |  | A43 | y2z2 | xy2z2 | x2y2z2 |  | A83 | z | xz
A4 | x | x2 | x3 | x4 | x5 | x5 |  | 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

      }

Poly, cubic and quartics are just like quadrics in that you don't have to understand what one is 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 doesn't work, try rotating or translating the shape by some small amount.

There are really so many different polynomial shapes, we can't even begin to list or describe them all. If you are interested and mathematically inclined, an excellent reference book for curves and surfaces where you'll find more polynomial shape formulas is:

"The CRC Handbook of Mathematical Curves and Surfaces"

David von Seggern

CRC Press, 1990

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