#### 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