Triangle and Smooth Triangle

where <CORNERn> is a vector defining the x, y, z coordinates of each corner of the triangle.

Because triangles are perfectly flat surfaces it would require extremely large numbers of very small triangles to approximate a smooth, curved surface. However much of our perception of smooth surfaces is dependent upon the way light and shading is done. By artificially modifying the surface normals we can simulate as smooth surface and hide the sharp-edged seams between individual triangles.

The **smooth triangle** primitive is used for just such purposes. The smooth triangles use a formula called Phong normal interpolation to calculate the surface normal for any point on the triangle based on normal vectors which you define for the three corners. This makes the triangle appear to be a smooth curved surface. A smooth triangle is defined by

where the corners are defined as in regular triangles and < NORMALn> is a vector describing the direction of the surface normal at each corner.

These normal vectors are prohibitively difficult to compute by hand. Therefore smooth triangles are almost always generated by utility programs. To achieve smooth results, any triangles which share a common vertex should have the same normal vector at that vertex. Generally the smoothed normal should be the average of all the actual normals of the triangles which share that point.

Infinite Solid Primitives

Plane

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:

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:

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 only important when using planes in CSG and **clipped_by**.

it can be represented by the equation

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.

Poly, Cubic and Quartic

where ORDER is an integer number from 2 to 7 inclusively that specifies the order of the equation. T1, T2, ... Tm are float values for the coefficients of the equation. There are m such terms where

An alternate way to specify 3rd order polys is:

Also 4th order equations may be specified with:

Here's a more mathematical description of quartics for those who are interested. Quartic surfaces are 4th order surfaces and can be used to describe a large class of shapes including the torus, the lemniscate, etc. The general equation for a quartic equation in three variables is (hold onto your hat):

a00 x^4 + a01 x^3 y + a02 x^3 z+ a03 x^3 + a04 x^2 y^2+ a05 x^2 y z+ a06 x^2 y + a07 x^2 z^2+a08 x^2 z+a09 x^2+ a10 x y^3+a11 x y^2 z+ a12 x y^2+a13 x y z^2+a14 x y z+ a15 x y + a16 x z^3 + a17 x z^2 + a18 x z + a19 x+ a20 y^4 + a21 y^3 z + a22 y^3+ a23 y^2 z^2 +a24 y^2 z+ a25 y^2 + a26 y z^3 + a27 y z^2 + a28 y z + a29 y+ a30 z^4 + a31 z^3 + a32 z^2 + a33 z + a34 = 0

To declare a quartic surface requires that each of the coefficients (a0 ... a34) 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:

x^4 + y^4 + z^4 + 2 x^2 y^2 + 2 x^2 z^2 + 2 y^2 z^2 - 2 (r_0^2 + r_1^2) x^2 + 2 (r_0^2 - r_1^2) y^2 - 2 (r_0^2 + r_1^2) z^2 + (r_0^2 - r_1^2)^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).

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. The syntax for using a pre-defined quartic is:

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. See the sub-directory **math** in the scene files directory for examples of polys in scenes.

"The CRC Handbook of Mathematical Curves and Surfaces" David von Seggern CRC Press, 1990

Quadric

A quadric is defined in POV-Ray by

where A through J are float expressions that define a surface of x, y, z points which satisfy the equation

A x^2 + B y^2 + C z^2 + 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:

X^2 + Y^2 + Z^2 - 1 = 0 Sphere X^2 + Y^2 - 1 = 0 Infinite cylinder along the Z axis X^2 + Y^2 - Z^2 = 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. You can invoke them by using the syntax:

The pre-defined quadrics are centered about the origin < 0,0,0> and have a radius of 1. Don't confuse radius with width. The radius is half the diameter or width making the standard quadrics 2 units wide.

Some of the pre-defined quadrics are,

Ellipsoid Cylinder_X, Cylinder_Y, Cylinder_Z QCone_X, QCone_Y, QCone_Z Paraboloid_X, Paraboloid_Y, Paraboloid_Z

For a complete list, see the file **shapes.inc**.

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