Algebraic Surfaces

Algebraic surfaces are surfaces whose points (x,y,z) satisfy a polynomial of three variables p(x,y,z)=0. For example, a sphere is an algebraic surface since it can be described as

(x - a)² + (y - b)² + (z - c)² = r²

The degree of an algebraic surface is the highest degree of all of its terms. Thus, spheres are degree two algebraic surfaces. It is very difficult to display algebraic surfaces correctly because some of them may extend to infinity and some of them are extremely complicated.

POVRAY can raytrace any algebraic surface of degree less than or equal to 7. In general, this range is sufficient for most applications because algebraic surfaces of degree higher than 7 are rarely used and are very complex. POVRAY supports two forms. One for quadric, cubic and quartic surfaces which are commonly used. The other is a general form.

Quadric Surfaces

To specify a quadric surface, use quadric {}:

quadric {
     < A, B, C >,
     < D, E, F >,
     < G, H, I >,
     J
     // pigment and finish
}

where these ten values are coefficients of a quadric polynomial as follows:

Ax² + By² + Cz² + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0

For example, to trace the following quadric surface

10x² + y² + 2z² + 3xy + 4xz + yz + 5x + y + 10z = 0

use the following

quadric {
     < 10,  1,  2 >,
     <  3,  4,  1 >,
     <  5,  1, 10 >,
     0
     // pigment, finish and transformations
}

Note that in general geometric transformations are required to bring the surface into your camera's coverage. Also note that the above equation is different from what has been discussed in class. The result a flat ellipsoid as shown below:

Click here to download the scene file.

Cubic Surfaces

POVRAY uses the following cubic equation for tracing a cubic surface:

A₁x³ + A₂x²y + A₃x²z + A₄x² + A₅xy² + A₆xyz + A₇xy + A₈xz² + A₉xz + A₁₀x + A₁₁y³ + A₁₂y²z + A₁₃y² + A₁₄yz² + A₁₅yz + A₁₆y + A₁₇z³ + A₁₈z² + A₁₉z + A₂₀ = 0

There are 20 coefficients organized as follows:

cubic {
     < A1, A2, A3, ...., A20 >
     sturm      // optional
     // pigment, finish and transformations
}

All 20 coefficients are in a single vector. Note that a new keyword sturm is introduced. This keyword is used only when the degree of an algebraic surface is greater than 2. The purpose of using this keyword is for more accurate ray and object intersection. As a result, it is more time consuming than without using it. Charles Sturm was a well-known German mathematician of the last century who discovered a method for determining the ranges of the roots of a single-variable polynomial.

To trace the following cubic surface

x³ - 0.11111xz² + y² = 0

use the following:

cubic {
//     x3  x2y  x2z  x2  xy2
     < 1,  0,   0,   0,   0,
//     xyz xy   xz2       xz   x
       0,  0,  -0.11111,  0,   0,
//     y3  y2z  y2  yz2  yz
       0,  0,   1,  0,   0,
//     y   z3   z2  z    Const
       0,  0,   0,  0,   0 >
     sturm
     clipped_by {
          box { < -2, -2, -2 >, < 2, 2, 2 > }
     }
}

Because cubic surfaces extend to infinity, frequently a clipped_by {} is used to cut off unwanted part. It may require several tries to get the result right. Here is the result:

Click here to download this scene file.

Quartic Surfaces

Quartic surfaces require 35 coefficients as shown below:

A₁x⁴ + A₂x³y + A₃x³z + A₄x³ + A₅x²y² + A₆x²yz + A₇x²y + A₈x²z² + A₉x²z + A₁₀x² + A₁₁xy³ + A₁₂xy²z + A₁₃xy² + A₁₄xyz² + A₁₅xyz + A₁₆xy + A₁₇xz³ + A₁₈xz² + A₁₉xz + A₂₀x + A₂₁y⁴ + A₂₂y³z + A₂₃y³ + A₂₄y²z² + A₂₅y²z + A₂₆y² + A₂₇yz³ + A₂₈yz² + A₂₉yz + A₃₀y + A₃₁z⁴ + A₃₂z³ + A₃₃z² + A₃₄z + A₃₅ = 0

These coefficients are organized as follows:

quartic {
     < A1, A2, ...., A35 >
     sturm      // optional
     // pigment, finish and transformations
}

To trace the following quartic surface

x⁴ + x²y² - 3x²y + 2.5x²z² + 1.5y²z² - 3yz² + 1.5z⁴ = 0

use the following

quartic {
//     x^4   x^3y   x^3z   x^3   x^2y^2
     < 1,    0,     0,     0,    1,
//     x^2yz x^2y   x^2z^2 x^2z  x^2
       0,    -3,    2.5,     0,    0,
//     xy^3  xy^2z  xy^2   xyz^2 xyz
       0,    0,     0,     0,    0,
//     xy    xz^3   xz^2   xz    x
       0,    0,     0,     0,    0,
//     y^4   y^3z   y^3    y^2z^2 y^2z
       0,    0,     0,     1.5,   0,
//     y^2   yz^3   yz^2   yz    y
       0,    0,     -3,    0,   0
//     z^4   z^3    z^2    z     Const
       1.5,    0,     0,     0,    0 >
     sturm
     // pigment, finish and transformations
}

The result is the following well-known cross-cup surface

Click here to download the scene file.

General Algebraic Surfaces

For general algebraic surfaces, use the following syntax:

poly {
     n,                         // degree of the surface
     < A1, A2, ..... >          // coefficients
     sturm                      // optional
     // pigment, finish and transformations
}

where the degree n can be an integer in the range 2 and 7. The correspondence between coefficients and their positions in the vector is quite tedious and will not be listed here. If you need it, please click here to download a PostScript format table.