2.4.3.2 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 15 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.

Cubic and quartic polynomial terms

	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	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
  }

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.