Infinite Solid Primitives

6.5.3.1 Plane

The plane primitive is a simple way to define an infinite flat surface. The plane is not a thin boundary or can be compared to a sheet of paper. A plane is a solid object of infinite size that divides POV-space in two parts, inside and outside the plane. The plane is specified as follows:

PLANE:
    plane
    {
        <Normal>, Distance
        [OBJECT_MODIFIERS...]
    }

The <Normal> vector defines the surface normal of the plane. A surface normal is a vector which points up from the surface at a 90 degree angle. This is followed by a float value that gives the distance along the normal that the plane is from the origin (that is only true if the normal vector has unit length; see below). For example:

  plane { <0, 1, 0>, 4 }

This is a plane where straight up is defined in the positive y-direction. The plane is 4 units in that direction away from the origin. Because most planes are defined with surface normals in the direction of an axis you will often see planes defined using the x, y or z built-in vector identifiers. The example above could be specified as:

  plane { y, 4 }

The plane extends infinitely in the x- and z-directions. It effectively divides the world into two pieces. By definition the normal vector points to the outside of the plane while any points away from the vector are defined as inside. This inside/outside distinction is important when using planes in CSG and clipped_by. It is also important when using fog or atmospheric media. If you place a camera on the "inside" half of the world, then the fog or media will not appear. Such issues arise in any solid object but it is more common with planes. Users typically know when they've accidentally placed a camera inside a sphere or box but "inside a plane" is an unusual concept. In general you can reverse the inside/outside properties of an object by adding the object modifier inverse. See "Inverse" and "Empty and Solid Objects" for details.

A plane is called a polynomial shape because it is defined by a first order polynomial equation. Given a plane:

  plane { <A, B, C>, D }

it can be represented by the equationA*x + B*y + C*z - D*sqrt(A^2 + B^2 + C^2) = 0.

Therefore our example plane{y,4} is actually the polynomial equation y=4. You can think of this as a set of all x, y, z points where all have y values equal to 4, regardless of the x or z values.

This equation is a first order polynomial because each term contains only single powers of x, y or z. A second order equation has terms like x^2, y^2, z^2, xy, xz and yz. Another name for a 2nd order equation is a quadric equation. Third order polys are called cubics. A 4th order equation is a quartic. Such shapes are described in the sections below.

6.5.3.2 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 7 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. 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 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 don't 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 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. We suggest you find a good reference or text book if you want to investigate the subject further.

6.5.3.3 Quadric

The quadric object can produce shapes like paraboloids (dish shapes) and hyperboloids (saddle or hourglass shapes). It can also produce ellipsoids, spheres, cones, and cylinders but you should use the sphere, cone, and cylinder objects built into POV-Ray because they are faster than the quadric version.

Note: do not confuse "quaDRic" with "quaRTic". A quadric is a 2nd order polynomial while a quartic is 4th order.

Quadrics render much faster and are less error-prone but produce less complex objects. The syntax is:

QUADRIC:
    quadric
    {
        <A,B,C>,<D,E,F>,<G,H,I>,J
        [OBJECT_MODIFIERS...]
    }

Although the syntax actually will parse 3 vector expressions followed by a float, we traditionally have written the syntax as above where A through J are float expressions. These 10 float that define a surface of x, y, z points which satisfy the equation A x² + B y² + C z² + D xy + E xz + F yz + G x + H y + I z + J = 0

Different values of A, B, C, ... J will give different shapes. If you take any three dimensional point and use its x, y and z coordinates in the above equation the answer will be 0 if the point is on the surface of the object. The answer will be negative if the point is inside the object and positive if the point is outside the object. Here are some examples:

X² + Y² + Z² - 1 = 0	Sphere
X² + Y² - 1 = 0	Infinite cylinder along the Z axis
X² + Y² - Z² = 0	Infinite cone along the Z axis

The easiest way to use these shapes is to include the standard file shapes.inc into your program. It contains several pre-defined quadrics and you can transform these pre-defined shapes (using translate, rotate and scale) into the ones you want. For a complete list, see the file shapes.inc.

6.5.3 Infinite Solid Primitives

6.5.3.1 Plane

6.5.3.2 Poly, Cubic and Quartic

6.5.3.3 Quadric