pov-ray for geometry

POV-Ray for Geometry

J.C. Sifre, Institut Mathmatique de Bourgogne

July 2, 2014

Contents

1 Use of POV-Ray 11

1.1 Editor and messages window . . . . . . . . . . . . . . . . . . . . 11

1.2 First view : axes . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Camera for a conic perspective view . . . . . . . . . . . . . . . . 15

1.3.1 Location and look_at . . . . . . . . . . . . . . . . . . . . . 16

1.3.2 Adjustment of the objective : up, right, direction . . . . . . 18

1.3.3 Moving the camera : translate et rotate . . . . . . . . . . . 21

1.4 Camera for an orthographic perspective view . . . . . . . . . . . . 24

2 Geometric objects 27

2.1 Elementary solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Curve drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Tori and Hlder norm balls . . . . . . . . . . . . . . . . . . . . . 29

2.4 Constructive Solid Gemometry . . . . . . . . . . . . . . . . . . . 31

2.4.1 Intersection, union, difference . . . . . . . . . . . . . . . . 31

2.4.2 Color or texture in CGS . . . . . . . . . . . . . . . . . . . 33

2.4.3 Merge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Surfaces based on a curve . . . . . . . . . . . . . . . . . . . . . . 35

2.5.1 Revolution surfaces, cylinders, canal surfaces . . . . . . . 36

2.5.2 Linear and quadratic splines . . . . . . . . . . . . . . . . . 38

2.5.3 Elementary arc for pour cubic, bezier and natural splines . 39

2.5.4 POV-Ray cubic spline . . . . . . . . . . . . . . . . . . . . 40

2.5.5 POV-Ray bezier spline . . . . . . . . . . . . . . . . . . . . 41

2.5.6 POV-Ray cubic spline with variable steps . . . . . . . . . 43

2.5.7 POV-Ray natural spline (with variable steps) . . . . . . . 43

2.5.8 Drawing a lathe . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5.9 Drawing a prism . . . . . . . . . . . . . . . . . . . . . . . 45

2.5.10 Drawing a canal surface . . . . . . . . . . . . . . . . . . . 47

2.5.11 Using a canal surface to draw a curve . . . . . . . . . . . . 49

3

4 CONTENTS

2.5.12 Drawing parametrized spline curves . . . . . . . . . . . . 49

2.6 Bezier bicubic surfaces . . . . . . . . . . . . . . . . . . . . . . . . 51

2.6.1 Bicubic functions . . . . . . . . . . . . . . . . . . . . . . . 51

2.6.2 POV-Ray bicubic patches . . . . . . . . . . . . . . . . . . 52

2.6.3 Joining bicubic patches . . . . . . . . . . . . . . . . . . . 53

2.7 Isosurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.7.1 Volume by isosurface . . . . . . . . . . . . . . . . . . . . 55

2.7.2 Surface by isosurface . . . . . . . . . . . . . . . . . . . . 56

2.8 Algebraic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.8.1 Surface with poly and polynomial . . . . . . . . . . . . . . 58

2.8.2 Singular surface with poly . . . . . . . . . . . . . . . . . . 61

2.8.3 Complete intersection with poly . . . . . . . . . . . . . . . 63

2.9 Parametric surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Introduction

We intend to describe the use of POV-Ray (Persistence of Vision), a ray

tracing software, strictly for mathematical (and more precisely 3-dimensional

geometry) illustration on a screen , as for beamer, or for printed publication. This

implies a drastic selection of the topics, since we dont need to give an image of

the garden of our dreams or a cartoon scene of Star Wars.

Mathematical softwares like Maple and Mathematica represent parametrized

curves by a succession of little segments, and parametrized surfaces by juxtaposi-

tion and superposition of small triangles. The hidden parts of objects are precisely

determined by painters algorithm, see figure (1).

Figure 1: Hidden parts

We call this representation vector graphics, since the elementary objects are

segments and triangles.

POV-Ray is a ray tracing software. To represent an object, we have to choose

light sources from which start millions of light rays. These light rays bounce from

or go through surfaces, these surface may absorb some wave lengths. We have

to choose a position of the observer (the camera) and of the screen. The light

rays which, after having travelled in the scene, meets the screen, while going in

direction of the camera, gives a coloured point on the screen, like in the classical

5

6 CONTENTS

corpuscular theory of vision (and photography).

As it is easy to determine the intersection of a line and a surface defined by an

equation, by putting a parametrization of the line in an equation of the surface, we

guess that POV-Ray is more adapted to representation of surface defined by an

equation that that of a parametrized surface. Indeed, in this last case, the software

has essentially to determine, for each little triangle of the triangulation whether

it cuts the light ray or not. For each light ray, thousands of intersections must be

separately tested. POV-Ray can do that, but very slowly.

Moreover, representing a surface defined by an equation f .x; y; z/ D 0 is noteasy for vector graphics. The software begins by approximating the surface by

a triangulated one, essentially by separating the points of a network where the

function f is positive and is negative, after which it has to find a triangulated

surface covering approximately the no-mans land between the two (discrete)

domains. Generally, the singularities of the surface are badly represented, and

even not seen at all.

An interesting example is Hadamards double hyperboloid, of equation

..x 2/2 C z2 y2 1/..x C 2/2 C z2 y2 1/ 2 D 0;

whose principal connected component is a hyperboloid with two holes. For

a recent version of Maple, the function implicitplot3d needs a very big grid not

to show discontinuities of the surface near the two secondary components while

POV-Ray representation (with convenient light and transparency) shows very

well the shape, and the proximity of the three regular components, see figure (2).

Other more striking examples will appear later.

In this sense, ray-tracing is in some way dual of vector graphics, since it

prefers to use equations than parametrizations of objects. The reason is purely

dimensional. For POV-Ray, everything must be a surface or a volume. The lines

do not exist : generically two lines (the light ray and the line to be represented) do

not cut ! The segments should be represented by fine cylinders, the curves will be

pieces of fines cylinders : for visual continuity, these cylinders should be joined

smoothly by spherical kneecaps.

Photo-realism has a price. There is no image without light. In the environment

of the picture, any zone without a defined color will be black (as sky at night).

The camera (i.e. the observer) must be precisely defined, otherwise there is no

image.

Transparency (up to caustics) is nice in POV-Ray. But there is a trap. When

objects are transparent and reflect light like glass, any object in the scene may

reflect on them, and, for a mathematician who has to read geometric properties

CONTENTS 7

Figure 2: Hadamard hyperboloid with Maple and with POV-Ray (Hadamard-

two-holes-POV-Ray.pov)

on the objects, this may introduce unpleasant artefacts. So our theatre should not

be baroque ! Prefer Becket to Goldoni !

We will even have sometimes to forget some properties of objects. For

instance, light implies shadows, which are not easy to make as smooth as in real

life. Sometimes the shadows make some details hard to read (if we do not want to

multiply the light sources), and POV-Ray allows to forget the shadows (but keep

intensity of light). Augmented reality will sometimes be synonym of heating with

it.

Finally, a great quality of POV-Ray is that we can control everything, since

everything is coded. The language is a layer over C++. The object-oriented

properties of C++ are widely used by the authors of POV-Ray, to collect common

properties of objects of very different origin and construction. But object-oriented

programming may be essentially ignored by the common user, I mean the author

of these lines, and any mathematician. A C++ programmer will understand

immediately some choices of POV-Ray, but it is not necessary to know C++,

since POV-Rays authors have partly bridled C++ for users comfort (essentially

to avoid type declarations, since the strictly geometric and optical context allows

POV-RAy to guess many types).

For some rather complex objects, it may be useful to prepare POV-Rays work

in another application, for instance Mathematica of Maple, when some computer

algebra is needed. We shall give some important examples.

Of course, all the POV-Ray sources of all the pictures of this document are

8 CONTENTS

given together with it, and we recommend warmly their reading.

The principal source is the manual included as Help in the software, but

it incomplete and sometimes ambiguous (even in POV-Ray 3.7). Some other

sources may be found on Internet. A good collection of examples of scenes for

advanced user is Oyonale/POV-Ray objects and scripts. At a basic level (but

with pedagogical purpose), there is also the well known site forLohmuller. Some

others (in french) are outdated. The movie Dimensions is made with POV-Ray,

and free on Internet. You may find many mathematical illustrations in the site of

Konrad Polthier.

The absolute reference is the complete source, provided here in the file

povray-3.7-stable.zip. The details of the *.h and *.CPP files which contain

information on geometry and treatment of light (written in C++) are in the sub-

directory source/backend. For instance, the parameters for determining camera

and screen are in source/backend/camera.h. Hard to read, but it is possible to

find there an answer to a local question.

In this preliminary document, we describe POV-Rays principal techniques to

draw basic objets, surfaces and even curves.

In the lectures, we shall first review and comment them.

(The following material is not in these preliminary notes.)

Then, we shall review the most useful directives of C++ bridled by POV-

Ray : it gives tools for programming repetitive actions.

After that, we show how to use Mathematica andMaple for some important

calculations to prepare POVRays rendering in cases where a formal calculus

software is necessary.

Finally, we present themost interesting contribution of POV-Ray to geometry :

the use of transparency to show hidden parts of surfaces or intersection (or even

contact) of surfaces.

Here is a little gallery of some geometric objects enhanced by POV-Ray

transparency, see figures (3) and (4) which were made for [La-Si], [La-Si2],

[La-Si-Dru-Gar-Pa].

1) The segments delimited on the rulings of an hyperboloid by a secant sphere.

2) A DAlembert cyclide, circled by two families of circles, any circle of the

first family being cospherical to any cercle of the second.

3) A parabolic Dupin cyclide tangent along a curve to a cylinder.

4) Three contacts of two generic Dupin cyclides imply a common contact

curve.

CONTENTS 9

Figure 3: Hyperboloid generatrices cut by a sphere and DAlembert cyclide

(Generatrices-cut-by-Sphere.pov and DAlembert-Cyclide.pov)

Figure 4: Contacts between Dupin cyclides (Parabolic-Tangent-Cyclides.pov

and Tritangent-Dupin-Cyclides.pov)

10 CONTENTS

Chapter 1

Use of POV-Ray

1.1 Editor and messages window

The version of POV-Ray in use is 3.7 (stable, and essentially definitive).

You open POV-Ray editor by clicking on POV-Ray icon, or on a file of type

*.pov in Windows explorer. In the editor, to may work simultaneously on several

images, each in an editor window. To close one of them, you type Ctrl W or

select (in the upper bar)Window/Close window. If your editor is already opened,

and you want to open a new POV file by clicking on it in Windows explorer, you

may find this message (see figure (1.1) :

Figure 1.1: Editor window already opened

The problem is solved by clicking as on the picture.

Editor fonts are very small, and you enlarge them by travelling through the

menus

Options/Editor Window/Editor Preferences (Color, Keystrokes, etc.)

but for a shorter access, type Alt Enter (together). It takes you to a window where

you can choose font size (figure (1.2). Dont forget to quit by OK.

11

12 CHAPTER 1. USE OF POV-RAY

Figure 1.2: Choice of font size

Once the code written, you launch image making by clicking on the Run

button on the upper bar. You can interrupt at any time a too low execution by

clicking on Pause.

While working, POVay writes a work report in a Message Window at the

bottom of the screen. (figure 1.3) : which disappears when the image is drawn.

Figure 1.3: Message window

his window may be opened (and closed) by Alt M even after drawing the image.

It is important for debugging. We can ask, for instance, the display of the value of

a variable A during execution at a specified moment of this execution. It suffices

to write

#warning str(A, L, P)

in the code. Then, at that moment of execution, in the Message Window, will

appear the value of A, with P decimals, written in characters of size at least L.

1.1. EDITOR AND MESSAGES WINDOW 13

Then the value ofAwill be given, preceded by the warning : Parse warning :

. You can write also some text to identify the variable :

#warning concat("A = ", str(A, L, P))

Then you will see

#Parse warning : A = 1.35

if the value of A is 1:35 at that moment.

You can also enlarge the characters in the Message window by clicking on

Options/Message Window/Text font, in the same way as above.

Sometimes, by mouse lapsus (!), you may loose text-colouring (necessary

for coding confort). You find it by going to Options/Editor Window/Color

Scheme/custom, which opens a window (figure 1.4), where you choose POV-

Ray language.

Figure 1.4: Choice of text colouring

POV-Ray is case sensitive.

Some words are directives of the language (in fact C++) and are preceded

by # , like #declare . The parameters of directives (functionss, procedures,

macros) are generally put between braces.

Comments on a line are preceded by // . Comments on several lines should

be preceded by /* and followed by */ .


You include a file by

#include "[file name]"

for example the very frequent #include color.inc

The numbers are integers or of type float. The vectors of coordinates a, b, c

are denoted by . If the name given to the vector is V , its coordinates are

V:x, V:y, V:z. In particular, x, y, z are reserved words for POV-Ray.

1.2 First view : axes

As said in the introduction we need :

a camera, called camera , one or several light source light_source , a background where this light is reflected, otherwise the background will

be black (which is possible and sometime interesting on a screen, but has to be

avoided in print).

The simplest possible background for a mathematical picture which you want

to convert in an EPS file (for TEXand publication) is :

background {White}

You can also define a sky_sphere , and perhaps you want to place your

object in relation to a sort of floor, with the code plane . It will be easily seen

latter.

A first surprise is the choice of coordinates in POV-Ray. Let < a; b; c > be a

vector. For the coordinates in this order,

the first, the abscissa, a is horizontal from left to right ; the second b is vertical from bottom to top ; the third c is horizontal from < 0; 0; 0 > to horizon.Thus, the usual y-coordinate in mathematics is the z-coordinate in POV-Ray,

and the usal z-coordinate in mathematics is the y coordinate in POV-Ray.

The reason of this choice in POV-Ray is logical : the natural plane of

reference is frontal. However, the plane of the image produced by POV-Ray will

be on a screen orthogonal to direction of vision, and this direction may be any

vector in R3.

1.3. CAMERA FOR A CONIC PERSPECTIVE VIEW 15

An important point : the letters x, y, z are strictly reserved word of POV-Ray.

They denote respectively the vectors :

x D< 1; 0; 0 >; y D< 0; 1; 0 >; z D< 0; 0; 1 >

and nothing else. Then 3 x, is < 3; 0; 0 >.The basic axes of POV-Ray are represented on figure (1.5), produced by the

joined file POV-unit-vectors.pov). The tiles of the floor have unit size, to give

Figure 1.5: POV-Ray unit vectors (POV-Unit-Vectors.pov)

a precise idea of size of objects. We keep this habit in the sequel, to help vision

of examples, which does not mean that mathematical scenes should have a check

floor !

1.3 Camera for a conic perspective view

In this section, we describe conic perspective, that is with vanishing point

(the Renaissance perspective of Brunelleschi). Latter, we shall consider briefly

orthographic perspective, which is an orthogonal projection. They ave not the

same use, and for photo-realism, the first is more natural.

We have to place two linked up elements, which must form in mind a solid :

the camera and the screen of intersection of the light rays arriving to the camera,

and which contains the final image. Both are determined in POV-Ray under the

same name camera . The principal parameter for its determination ar :


location, up, right, direction, look_at, rotate, translate

For instance, we may write :

camera{location up right

direction look_at

rotate translate}

For our purpose (geometry) it would be better to introduce them in the

preceding order, we shall see why.

1.3.1 Location and look_at

To put the center of the objective of the camera at the point ! D< a; b; c >,we write

location

The axis of the camera will join the point ! to the point D< a0; b0; c0 > (inabsolute coordinates), keeping horizon parallel to the original one, when we write

look_at

To understand what happens, let us associate to the camera a moving frame

of origin ! and of basis vectors X , Y , Z. In any move, the direction of the

look (or objective) is in the direction of Z, the vertical of the machine itself is

Y , and the line joining the two hands of the cameraman is X . Then, specifying

look_at means for the cameraman the following move : from the

original position where X D x, Y D y, Z D z (but ! has already been changedby location ), and without changing !, he first turns around the Y D y axis soas to find the point in the new vertical plane X D 0 ; after which he turns theobjective in this vertical plane (keeping ! fixed) to put the point at the middle

of the image.

If the angle between the horizontal plane and the new Z is the objective

latitude , and the angle between the initial axis x and the new horizontal X is

the objective longitude ', the move of the camera is in fact a rotation of around

the initial x axis followed by a rotation of ' around the initial y axis (which is


less natural for a human cameraman..). We must keep in mind that for POV-Ray,

the angles are counted with the left hand rule, that is the sense opposed to the

trigonometric one. It means that if you use your left thumb as axis, when you close

the other fingers, they turn in the positive direction. It gives, in mathematical

notations with vertical vectors and square matrices (but POV-Ray coordinates) :

X Y Z

D0@ cos' 0 sin '0 1 0 sin' 0 cos'

1A0@1 0 00 cos sin 0 sin cos

1A

The sign of sin is due to the fact that the first coordinate in the latitude rotation

is z and not y, and the left hand convention. This calculus has been given to

prevent some errors due to mathematical habits. It gives :8Y D< sin ' sin ; cos ; cos' sin >Z D< sin ' cos ; sin ; cos' cos > :

(1.1)

This vector should be u=kuk, where u D< a0 a; b0 b; c0 c >. Then ' isdetemined by the first and third coordinates of a0 a, and is determined by therespective second coordinates of b0 b and c0 c :(

D Arcsin.Z2/' D Arctg.Z1=Z3/:

(1.2)

The norm of u is not used here.

An example for camera adjustments will be the following simple objects. On

the floor (with one unit check) put at one unit down :

a yellow unit cube put on the floor its upper face being at the y zero level) ; put on four brown legs (of height one unit) a unit blue cube ; in front of then, a green cone with unit height, see figure (1.6).With the basic code

camera{location look_at

}

we see figure (1.7) on the left. The right picture shows where is the camera

for this result. The following code

camera{location look_at

}


Figure 1.6: Two cubes and a cone (Cubes-cone.pov)

Figure 1.7: Cubes and cone viewed from the camera, and together with it

(Cubes-cone1.pov and Camera-Cubes-Cone1.pov)

gives figure (1.8) where the camera looks at the foot of the right leg of the blue

cube.

1.3.2 Adjustment of the objective : up, right, direction

Now, we have to determine the position of the rectangular screen which picks

up the light rays in their last travel in direction of the camera. The light rays in

direction of the observer but not cutting this rectangle are not considered.

This rectangle (the screen) will be centered on the Z axis at a distance which

we (for this document) denote by d of the point !. With respect to the moving

frame .!; X; Y; Z/, the X dimension which we denote byW and the Y dimension


Figure 1.8: When the cameraman looks at a foot of a leg (Cubes-cone2.pov and

Camera-Cubes-Cone2.pov)

which we denote by H . For POV-Ray, this will be specified by the code

up H*y

right W*x

direction d*z

The lettersH , L d of our mathematical description should be replaced by the

numerical values chosen by the user. You must be advised that the total height of

the screen is H , its total width is L. Then, in the moving frame, the extremities

of the screen have (POV-Ray) coordinates :

< W=2;H=2; d >; < W=2;H=2; d >;< W=2;H=2; d >; < W=2;H=2; d > :

In fact, X , Y , Z may be replaced by other vectors, but it will not interest us

for our purpose. We pass to the examples.

The code

camera { location look_at

up 1*y right 4/3*x direction 1*z }

gives figure (1.9). The code


Figure 1.9: d D 1 (Cubes-cone3.pov and Camera-Cubes-Cone3.pov)



gives figure (1.10).

Figure 1.10: d D 2 (Cubes-cone4.pov and Camera-Cubes-Cone4.pov)

Even if the code contains the reserved word x, y, z in low case, they make

reference to the moving frame vectors X , Y , Z as can be seen in the following

example. The code



gives figure (1.11). As we see a little part of the blue cube and of the horizon,


Figure 1.11: d D 2 and look at < 1;1; 0 > (Cubes-cone5.pov and Camera-Cubes-Cone5.pov)

we notice that the scaling of the screen is not related to the distance of ! to but

only the number d defined above. The distance from the camera to the screen is

exactly 2, which is smaller than half the distancep17 from ! to .

1.3.3 Moving the camera : translate et rotate

Suppose now that you want to turn your camera around its view direction with

an angle, say 45 degrees, not for fun but for some geometric reason (for instance

in an animation). POV-Rays coding of rotations does not give immediately the

answer. Of course, it offers a procedure : rotate . But its working is particular.

If, in the declaration

camera { location ... look_at ...

up ... right ... direction ... }

you put a geometric transformation such as rotate or

translate , it will be executed after the code

location, look_at, up, right, direction

wherever you write the transformation in the camera{...} code. In fact it

transforms the bloc camera-screen just defined as a solid, and the transformations

are executed with respect to the initial frame .O; x; y; z/. In particular, the location

of the camera will not be preserved. Moreover, we must know how the rotation is

carried out.

For POV-Ray, an object (like the camera-screen) is rotated by a code like


object{... (construction of the geometric object)

rotate

}

where the object is :

rotated around the x-axis with an angle alpha (left hand convention) ;

rotated around the y-axis with an angle beta (left hand convention) ;

rotated around the z-axis with an angle gamma (left hand convention).

To describe concretely what happens, suppose the angles are between 0 and

90 degrees, and you stand up, looking in the z-direction (POV-Rays horizon),

with your object in front of you. The first rotation is natural. You lean the object

with an angle . After that, keeping it at hand, you turn yourself around the y-axis

with an angle (in the right direction, left hand rule). But the last movement is

physically very unnatural : you should turn around the z-axis (from which you

have turned away), with a determined angle . You guess that no cameraman

has in mind these angular parameters when he executes the instructions of his

director, even in a complicated S.F. movie !

So, if you need to turn the camera naturally around its view axis (from a

position where horizon is horizontal) it is better to write the description of the

pair camera-screen in another way. Here is a solution, which we shall explain.


#declare ScreenDirection = 2;

#declare ScreenHeight = 1;

#declare ScreenWidth = 4/3;

#declare CameraLocation = ;

#declare CameraLookAt = ;

#declare U = CameraLookAt-CameraLocation;

#declare NU = sqrt(pow(U.x,2)+pow(U.y,2)+pow(U.z,2));

#declare Un = U/NU;

#declare rhau = 45;

#declare theta = - asin(Un.y)*180/pi;

#declare phi = atan(Un.x/Un.z)*180/pi;

// Camera and screen

camera {location look_at z

up H*y right W*x

direction ScreenDirection*z

rotate rhau*z

rotate theta*x

rotate phi*y

translate CameraLocation

}

Let ! be the desired location of the camera, the look_at point, H the

screen height,W its width, ScreenDirection the camera-screen distance, and

the angle (in the left hand convention) of rotation you want for your camera.

We first calculate the vision vector u D !. We let the parameterslocation and look_at at their initial value (see the code above). We choose

the up , right , and direction parameters. The normed vision vector is u=kuk.This defines a latitude (oriented with left hand convention, i.e. head down),

and a longitude ' (from left to right). Then we execute in this order a rotation

of (translated in degrees) around the z axis (which leans the camera on the

left), followed by a rotation of around the x axis (we lean our camera) and a

rotation of ' around the y-axis (from left to right), where the angles and ' are

determined from u=kuk by the formulas (1.2). All the angles must be in degrees.Finally, we translate the camera to the desired point !.


Thus, we dont use the parameters location and look_at , but it is due to

POV-Rays conventions for rotations for which the authors have chosen anything

else than Euler angles... Starting from cameras position of figure (1.11), and

Figure 1.12: d D 2, look at< 1;1; 0 > and turn of 45 degrees (Cubes-cone6.povand Camera-Cubes-Cone6.pov)

turning of an angle of 45 degrees around the vision axis (with left hand convention)

gives figure (1.12).

1.4 Camera for an orthographic perspective view

This is approximately what you obtain with telephotography (and big zoom-

ing). The image has no depth of field. Parallel lines in space stay parallel on the

screen. This is Monges descriptive geometry projection. Its interest is to retrieve

real lengths from measurements on the screen. In particular, we shall use it for

plane figures.

The parameters of the camera code are the same as conic perspective, with a

new parameter orthographic :

camera{orthographic

location ... etc.

}

The parameters up and right have the samemeaning as in conic perspective,

with a particularity : the screen has actually these dimensions H and W , but the

parameter d of direction has no more importance, and should be deleted.

1.4. CAMERA FOR AN ORTHOGRAPHIC PERSPECTIVE VIEW 25

In fact, only the objects in the half-plane in front of the camera are considered.

If the camera is inside one of them, you cannot be sure whether POV-Ray

represents a part of it or not, so avoid this situation, and use orthographic

perspective only to represent objects in front of you : when you look at a leopard

through telephoto lens, your camera is not in his mouth.

Chapter 2

Geometric objects

2.1 Elementary solids

The four elementary solids are (see figure (2.1))

Figure 2.1: Elementary solids (Elementary-solids.pov and Elementary-

solids2.pov)

The revolution cylinder

object{cylinder{A B r pigment{color ...}}}

where A and B are vectors representing the centers of circular faces, and

r the radius. It should be assigned a color. The simplest way to do it is

pigment{color ...} . This cylinder is full, it is a volume, and not a surface. For

a hollow cylinder (a cylindrical surface), add open just after the radius. Commas

are accepted but not necessary between the parameters.

27

28 CHAPTER 2. GEOMETRIC OBJECTS

The boxobject{box{A B pigment{color ...}}}

which s a rectangle parallelipiped with great diagonal joining the points A and

B . It may not be opened, and is a convex volume. No comma necessary.

The sphereobject{sphere{A r pigment{color ...}}}

of center A and radius r . No comma necessary between A and r .

The truncated coneobject{cone{A r B r pigment{color ...}}}

wher A and B are centers of circular faces of respective radii r and r 0. It is

full but may be hollow with open after the second radius. The two radii are

necessary, even if one of then is null.

They may be translated by translate , rotated by rotate , expanded in

the three directions by scale , which admits a three dimensional parameter

< a; b; c > for expanding of ratio a in x direction, and so on. In figure (2.1)

we have expanded the sphere to obtain an ellipsoid. Then all ellipsoid are basic

shapes in POV-Ray.

2.2 Curve drawing

We have said that POV-Ray does not draw curves. He draws only surfaces and

volumes (in fact only surfaces). A simple strategy to construct a curve consists in

drawing a great number of closed little spheres all along the curve. Figure (2.2)

on the left shows what happens. We choose a very little number of spheres to

show the graphic disadvantage.

Another strategy could be to draw a sequence of cylinders. But curvature

implies an unpleasant discontinuity, see figure (2.2) on the right. With some

patience, a better solution consists in drawing a sequence of cylinders, two

successive cylinders having a circular face center in common, together with the

spheres centered at that points realizing a kind of kneecap to fill the discontinuity.

Now it seems smooth (even if it is not), see figure (2.3).

2.3. TORI AND HLDER NORM BALLS 29

Figure 2.2: Basic curve drawing, with spheres or cylinders (Parabola.pov and

Parabola1.pov)

Figure 2.3: Nice curve drawing, with spheres and cylinders (Parabola2.pov)

2.3 Tori and Hlder norm balls

We present in figure (2.4) two uses of transparency to show the singular tori,

the left one being more abstract. In both cases, the shadows are suppressed. In the

inside view, the tiles are grey and white, which shows that the light source is also

inside the torus.

Figure 2.4: Two view of the three types of tori (Blue-Tori.pov and Glass-

Tori.pov)

A torus of vertical (i.e. y) axis centered at < 0; 0; 0 > is constructed with


Figure 2.5: Inside a singular torus (Singular-torus-inside.pov)

object{torus{R r pigment{color ...}}

where R is the radius of the fixed circle, and r the radius of the turning circles.

When R r , the torus is singular. and transparency helps to look inside, seefigures (2.4) and (2.5). Another useful object is called by POV-Rays authors

Figure 2.6: Different kinds of superellipsoids (Superellipsoid.pov and Superel-

lipsoid2.pov)

the super-ellipsoid. It is in fact the set of points < a; b; c > (with POV-Ray

conventions) such that

f .a; b; c/ Djaj2=e C jbj2=e

e=nC jcj2=n 1 D 0;

2.4. CONSTRUCTIVE SOLID GEMOMETRY 31

where e and n are arbitrary positive values. POV-Rays code is :

\begin{verbatim}

object{superellipsoid{< e, n> pigment{color ...}}}

For D 2=e and D 2=n, this is the set of points such that

k.a; b/k; c

D

1, where k k denotes Hlder norm of parameter when 1 and 1. As asuper-ellipsoid is defined for all e and n > 0, it may not be convex. In figure (2.6),

the parameters e and n take the values 3, 2, and 1, with e constant for each color.

On the left, the same with the values 3=2, 1, and 1=2, so that they are convex.

These surfaces are defined centered at < 0; 0; 0 > and in a precise position.

They may be transformed by any affine transformation.

2.4 Constructive Solid Gemometry

Commonly named CSG, Constructive Solid Gemometry gives techniques to

code compound geometric objects obtained from simpler ones with set theoretic

operations : union, intersection, difference.

2.4.1 Intersection, union, difference

The code

intersection{box{

pigment{Red}

}

sphere{ 1

pigment{Blue}

}

no_shadow

}

gives figure (2.7). Colors may be surprising, and we shall describe them later

in this section.

In the above code, we had only to write the word intersection followed

by the direct descriptions of the desired object. Ie we ad assigned to names

such as A, B , ... in a #declare directive (for example to use them for several


Figure 2.7: Intersection of a red box and a blue sphere (CSG0.pov)

actions), then the intersection action needs the word object when we want

their intersection, like in the following code.

#declare A = box{

pigment{Red}

}

#declare B = sphere{ 1

pigment{Blue}

}

intersection{object{A} object{B}

no_shadow

}

A little constraint due to the Object-Oriented nature of the underlying C++

(no more comment).

The operation union is absolutely equivalent to writing the descriptions of

the objects one after the other or the assigned variables A, B , etc. under the form :

object{A}

object{B}

So, why putting them together in a union{...} code ? To work on all of

them simultaneously. Indeed, we may have to translate, rotate, scale, the union

of objects, impose them a common no_shadow , etc. In that case, putting all of

them in a union is necessary.

Figure (2.8) shows the union or two overlapping cubes A and B , and images

of A\B , A nB , B nA by translations to see all of them on the same picture. Thecode is

2.4. CONSTRUCTIVE SOLID GEMOMETRY 33

Figure 2.8: A n B , A \ B , B n A, A [ B . (CSG1.pov)

union{

object{intersection{object{A} object{B}}}

object{difference{object{A} object{B}}

translate}

object{difference{object{B} object{A}}

translate}

object{ union{object{A} object{B}}

translate}

no_shadow

}

2.4.2 Color or texture in CGS

Even if we have not yet studied texture, we may just say that a volume has

a texture attached to its boundary, i.e. the surface which bounds it (even when

transparency allows refraction indices and so on). In case of opaque objects, this

texture is strictly attached to the boundary, since we cannot see anything inside.

This is the case we consider here.

Suppose that the boundaries @A and @B have no surface element in common

i.e. have at most curves in common. Then, for opaque objects, color or texture

attribution of the compound object is simple : if some partial surface remains after

the set theoretic operation, it keeps its initial texture/color. For example, in figure


(2.8), the concave part of the boundary of B nA comes from A, and A is red, thenit must be red. The same is true for the frontal face of A \ B .

When A and B have a part of face in common, POV-Ray doesnt know which

texture he has to assign to it, and ... does anything which has no geometric

meaning, and probably anything else than what you would expect, see figure (2.9)

for a wooden and a glass cube. In that case, you had rather combine object without

Figure 2.9: Fake texturewhenA andB have a part of face in common. (CSG2.pov)

color or texture and attribute afterwards a texture to the set theoretic result.

Without entering the details, we give what we think (by pure experimentation)

to be the rule for textures with eventual transparency. The face arising from points

AAAAA BBBBB AAA BBB

Figure 2.10: Repartition of textures in a difference, intersection, union.

common to the two interiors of the objects are transformed into a two layer face,

with attribution of texture (or color) like in figure (2.10). It is clear in figure

(2.11). For example in intersection (figure (2.11), the small cube), te top left faces

have texture consisting of two layers. The exterior is blue, as lass like the right

2.5. SURFACES BASED ON A CURVE 35

Figure 2.11: Texture attribution in CSG (CSG3.pov)

big cube B , under which a second layer in wood like the left big cube A, and this

explains the blue wooden faces. When the exterior layer of a common face

is wood, the interior glass layer is invisible. Other tests has been carried out with

two glass objets of different colors, which seem to confirm this description.

2.4.3 Merge

As is said in POV-Ray documentation, the operator union keeps the inter-

section in the object (at the price of a complicated rule for inside texture). A much

simpler union is merge which forgets the inside parts.

Since in general using merge means trying to construct an homogeneous

object with the two pieces, we shall not play with textures in that case. Indeed,

when the two objects have different textures, the logic of POV-Rays automatic

texturing of a = merge i even less clear. The user is invited to merge only non

textured objects, an graft a texture on the result afterwards.

2.5 Surfaces based on a curve

We describe surfaces constructed in a somewhat mechanic way from a curve :

revolution surfaces, cylindrical surfaces, and canal surfaces. The types of curves

on which they are based is of different possible spline types, which in general

means : interpolated from a finite number of points. POV-Ray offers several

types of interpolation, which we study precisely. Some of these curves may be


drawn (as they are actually parametrized), some others not since they are only a

formalization of a directing curve for the surface. Here are the details.

2.5.1 Revolution surfaces, cylinders, canal surfaces

A revolution surface is for POV-Ray a lathe (like a polishing lathe, in french

un tour), with y-axis, and based on a meridian curve drawn in the xOy (vertical)

plane. The code is

lathe{spline_type | [parameters for the spline] [options],

[color, texture, etc.] [sturm]}

where spline_type is the type of spline curve (in the present case, the

meridian curve), to be chosen among the following

linear_spline, quadratic_spline, cubic_spline, natural_spline

The other entries are options, color, and the important reserved word sturm

explained later. For example, the code

lathe{quadratic_spline 5

texture{T_Wood14

finish{specular 0.35 roughness 0.05 ambient 0.3}

}

sturm

no_shadow

}

gives figure (2.12), in initial position and after rotation-translation. An

orthographic projection (figure (2.13)) shows clearly the three parabola arcs of

the directing curve, as we have chosen the quadratic_spline spline type.

There is another way to draw a revolution surface, the code is sor (like

Surface Of Revolution. It is described precisely in section 3.4.5.1.15 in the Help

of POV-Ray version 3:7. Differences with lathe are given there.

A cylindric surface is for POV-Ray a prism with y direction, based on a

horizontal directing curve in the xOz plane. The code is


Figure 2.12: A simplified (hollow) chess pawn (Lathe.pov and Lathe0.pov)

Figure 2.13: Orthographic projection of the simplified pawn (Lathe1.pov)

prism{spline_type | [parameters for the spline] [options],


with the same kind of parameters, now for the horizontal directing curve.

A canal surface (envelope of a family of spheres) is for POV-Ray a

sphere_sweep with a soul curve, the set of centers of a one parameter family

of spheres, and we have to specify the law of variation of radii of the spheres

enveloped by our surface. The code is

sphere_sweep{spline_type | [parameters for the spline] [options],



In the three cases, the 3-dimensional object is based on a curve (or polygon)

which must take one of the following forms :

linear_pline : a polygonal line ; quadratic_spline : a juxtaposition of pieces of parabola, not even

C1 but useful for simple curved surfaces of revolution (parther useless for

mathematicians) :

bezier_pline , cubic_pline and natural_spline : a C1 juxtapo-sition of third degree parametrized curves, but with different join techniques

corresponding to different needs.

For a lathe or a prims , the base point parameters of the spline are 2-

dimensional (respectively in thexoY and thexOz planes), and for a sphere_sweep

they are 3-dimensional.

We shall see later how to use these splines for these surfaces, but we have

now to give a detailed description of all these splines, since POV-Rays use is

not always standard.

2.5.2 Linear and quadratic splines

A linear_spline is simply a polygon, specified by a sequence of points,

with two of three coordinates. Two when it is used for a lathe or prism, three for

a sphere-sweep (since the soul curve may not be plane).

We are given n points p1, . . . , pn. The code

linear_spline n p_1 ... p_n

defines, for each i from 2 to n, the unique segment joining pi1 to pi .

The polygon is their union. Notice the number of points just after the word

linear_spline .

For a quadratic_spline , we are given a sequence of nC 1 points p0, p1,. . . , pn. A curve will be drawn, passing at the successive points p1, p2, . . . , pn

(but not p0, used only as a control point for the first parabola arc). The code

quadratic_spline n+1 p_0 p_1 ... p_n

defines, for each i D 2; : : : ; n, the image of the arc i restriction to 0; 1 ofthe unique parametrized arc i : 1; 1! E D R2 or R3 such that :

i is a polynomial of degree 2 ; i.0/ D pi1, i.1/ D pi ;


0i.0/ Dpi pi2

2.

This define entirely i (see figure (2.14)) :

8t 2 0; 1; i.t / D pi1 C tpi pi2

2C t2

pi2 C pi2

pi1:

The curve is the union of these arcs. The restriction of i to 1; 0 is never

pi

pi1

pi2

Figure 2.14: POV-Rays quadratic spline

represented ! The arc i depends only on the three points pi2, pi1, pi . It implies

that the curve is not even C1 in general. but it may give nice lathes (chess

pawns, chair legs ...).

Much more interesting are the following splines.

2.5.3 Elementary arc for pour cubic, bezier and natural splines

In the three cases, the spline is a juxtaposition of third degree parametrized

arcs constructed in the same following way (which is in fact generally called a

Bezier curve).

Let us begin by a small algebraic reminder.

Given four real numbersp, q, v,w, there exists a unique functionf polynomial

of degree at most 3 such that :

f .0/ D p; f .1/ D q; f 0.0/ D v; f 0.1/ D w: (2.1)

These conditions are linear, so it suffices to determine f for :

.p; q; v; w/ D .1; 0; 0; 0/, which defines f0, .p; q; v; w/ D .0; 1; 0; 0/, which defines f1, .p; q; v; w/ D .0; 0; 1; 0/, which defines g0, .p; q; v; w/ D .0; 0; 0; 1/, which defines g1.


Then (see figure (2.15))

f0.t / D .2t C 1/.t 1/2; f1.t / D t2.2t 3/ (2.2)g0.t / D t .t 1/2; g1.t / D t2.t 1/: (2.3)

The general f is f D pf0 C qf1 C vg0 C wg1.

Figure 2.15: Functions f0, f1, g0 and g1

This may be naturally extended to vector functions 0; 1! E D R2 or R3.

2.5.4 POV-Ray cubic spline

We are given nC 2 points p0, p1, . . . , pn, pnC1. The points p0 and pnC1 areonly control points. The code

cubic_spline n+2 p_0 p_1 ... p_n p_{n+1}

defines, for each i from 2 to n, the unique arc i : 0; 1! E D R2 or R3 ofdegree 3 such that :

i.0/ D pi1; i.1/ D pi (2.4)

0i.0/ D

pi pi22

; 0i.1/ DpiC1 pi1

2: (2.5)


This arc i is, for t 2 0; 1 :

i.t / D t332

pi1 pi

C piC1 pi2

C t2

pi2

5

2pi1 C 2pi

piC1

2

C t pi pi2

2C pi1:

This draws n 1 elementary arcs from p1 to pn.Since 0i .0/ D 0i1.1/, this arc is, by construction, C1, wich is (graphically)

much better than a quadratic_spline . It needs two extra-control points p0 and

pnC1, but no information more, and is rather simple. Nevertheless, we must have

in mind the slope of the line .pi1piC1/ to determine the direction of the tangent

at pi , which is uneasy when we have in mind a shape and look for the way of

drawing it. Another disadvantage is that this interpolation supposes implicitly

equality between the time intervals of parameters, which may have unpleasant

geometric consequences.

2.5.5 POV-Ray bezier spline

We are given a sequence of points denoted by : p1, r1, s1, q1, p2, r2, s2, q2,

. . . , pi , ri , si , qi , . . . , pn, rn, sn, qn. The code

bezier_spline 4n p_1 r_1 s_1 q_1 ... p_n r_n s_n q_n

defines, for each i from 1 to n, the unique arc i of degree 3 : 0; 1! E D R2or R3) such that :

i.0/ D pi ; i.1/ D qi (2.6)

0i .0/ D 3.ri pi/; 0i.1/ D 3.qi si/: (2.7)

The tangent at pi contains ri ant the tangent at qi contains si . Exactly what

you obtain with the pen tool of Adobe Illustrator (and any other graphic tool).

The coefficient 3 in the derivatives could be arbitrary. Figure (2.16) suggests a

justification of this consensual choice of 3. If

p1 D< 0; 0 >; r1 D< 0; 1 >; s1 D< 1; 1 >; q1 D< 1; 0 >;

the arc 1 is 1.t / < 3t2 2t3; 3t .1 t / >, see figure (2.16). The code


p1 q1

r1 s1

Figure 2.16: Basic Bezier curve (left Illustrator, right Prism-Bezier.pov)

prism{bezier_spline 0, 0.01, 8,

pigment{Blue}

sturm}

gives figure (2.16) on the right, which proves that POV-Ray and Illustrator (and

everybody) agree. The n little arcs are constructed independently and this implies

p1

r1 s1

q1 D p2

r2 s2q2

Figure 2.17: Two consecutive Bezier arcs for C1 linking (Illustrator)

a repetition for continuity : even if qi D piC1 both must be specified in the code.A necessary and sufficient condition to obtain a C1 arc is : riC1 piC1 D qi si(recall that qi D piC1), see figure (2.17).


2.5.6 POV-Ray cubic spline with variable steps

We are given reals t0 < t1 < < tn < tnC1, and nC 2 points p0, p1, . . . , pn,pnC1. The code

cubic_spline 2n+2 t_0 p_0 t_1 p_1 ... t_{n+1} p_{n+1}

defines the unique curve : t0; tnC1 ! E whose restriction to ti1; ti foreach i from 2 to n, is the unique arc i of degree 3 : ti1; ti ! E D R2 or R3such that :

i.0/ D pi ; i.1/ D qi (2.8)

0.ti/ D1

2

pi pi1ti ti1

C piC1 pitiC1 ti

(2.9)

0.ti1/ D the analoguous for the index i 1: (2.10)

This defines finally an arc of domain t1; tn, and an interpolating curve.

2.5.7 POV-Ray natural spline (with variable steps)

We are given reals t0 < t1 < < tn < tnC1, and nC 2 points p0, p1, . . . , pn,pnC1. The code

natural_spline 2n+2 t_0 p_0 t_1 p_1 ... t_{n+1} p_{n+1}

defines the unique curve : t0; tnC1 ! E whose restriction to ti1; ti hasdegree 3, .ti/ D pi , even for i D 0, nC 1, of class C2 on the whole domain,and with zero acceleration at t0 and at tnC1. This defines also an arc of domain

t0; tnC1, and an interpolating curve. This is the classical spline.

These two last splines define an arc from an interval of R to E, and allow

a draw of the arc itself, unlike the preceding splines for which the code defines

a global object used to construct formally a directing curve for a special surface

(and for which we want to draw the surface and not the spline).

2.5.8 Drawing a lathe

Each of the above types of splines may serve to draw a lathe. The curve is

not necessarily closed, but you should avoid to use a curve which cuts the y-axis.

The code


lathe{quadratic_spline 6

pigment{color Blue}

no_shadow

rotate

}

gives figure (2.18), a lathe with quadratic_spline meridian curve has ben

turned of 90 degrees and put on a floor at y D 4 to show control points (smallgreen balls, also turned) and some tangents (small red segments, in fact cylinders).

Notice that the tangent down on the right is indeed parallel to the segmen joining

the (images b rotation of) points p0 and p2, the first being a control point, the

second on the meridian curve.

Figure 2.18: Lathe for quadratic spline meridian, without and with Sturm option

(Lathe2-no-sturm.pov and Lathe2.pov)

On the right, the option sturm avoids some problems of intersection of light

rays and the surface which appeared on the left. The reader is invited to compare

the base points and the control points (little green balls on figure) with the code.

For a cubic_spline meridian cuve, the code

lathe{cubic_spline 7

pigment{Blue}

no_shadow

rotate sturm}

gives figure (2.19, left). Now, there are two extra control points (isolated green


balls on figure), but now the lathe is C1. Dont forget to write the total number of

points (even control points). For a bezier_spline meridian cuve, the code

Figure 2.19: Lathe for cubic spline and bezier spline meridian (Lathe3.pov and

Lathe4.pov)

lathe{bezier_spline 8

texture{T_Bottle_Glass}

no_shadow

rotate sturm}

gives figure (2.19, right). The surface is in glass, to show control and base

points. Notice that each elementary curve is defined by four points.

In any case, a meridian curve needs not be closed for a lathe , as can be seen

on figure (2.12) on the right.

2.5.9 Drawing a prism

Unlike in a lathe , the directing curve of a prism has to be closed. That is :

the last base point has to be identical to the first (base point, and not control point).

This allows, for a same code prism to represent several cylindrical surfaces,

since when POV-Ray meets an already written base point, he closes the spline,

and, if there are new points, starts a new spline for a new surface (called a new

sub-prism). For example, for two linear sub-prisms, the code has the form


\begin{verbatim}

prims{linear_spline a b n+m+2 p_0 p_1 ... p_{n-1} p_0

q_0 q_1 ... q_{m-1} q_0 [options],

[texture, etc.] [sturm]}

Notice that the total number of points has to be written. The numbers a and b

are the y-coordinate of the extreme levels of the prism.

Figure 2.20: Prisms (Prism1.pov)

In the following example (figure (2.20))

prism{linear_spline 0 0.5 18

pigment{Blue}}

the base curve is made of four polygons. POV-Ray chooses as base surface

the boolean addition of the domains bounded by them.

The same remarks apply to other spline types. For example, if the base of the

prism is the boolean sum of two domains bounded by a cubic_spline , the code

is

prims{cubic_spline a b n+2+m+2 p_0 p_1 ... p_{n-1} p_0 p_{n+1}

q_0 q_1 ... q_{m-1} q_0 q_{m+1}

[options], etc.


The control points are p0, pnC1, q0, qnC1. The closure points for the two

curves are p0 and q0.

If the base of the prism is the boolean sum of two domains bounded by a

succession of elementary bezier_spline curves, the code is

prims{cubic_spline a b 4*(n+m) p_1 r_1 s_1 q_1

p_2 r_2 s_2 q_2

...

p_n r_n s_n q_n

p_1 r_1 s_1 q_1

p_2 r_2 s_2 q_2

...

p_m r_m s_m q_m

[options], etc.

where q1 D p2, . . . , qn1 D pn and the same with primes to ensure continuityof the two Bezier curves, and qn D p1 to close the first Bezier curve, and q0m D p01to close the second. For example, figure (2.16) on the right results from code

prism{cubic_spline 0, 0.01, 6,

pigment{Blue}

sturm}

(followed by 90 degrees rotation).

2.5.10 Drawing a canal surface

For a sphere_sweep , we choose control and base points for the directing

curve, and control and base values for the variable radius of the mobile sphere.

In the code, each (base or control) point must be followed by the corresponding

(base or control) radius for the one parameter family of spheres.

For a linear_spline , the code is

sphere_sweep{cubic_spline n p_1 R_1 ... p_n R_n

[options, color, etc.]}


Figure 2.21: A canal surface with linear spline, outside and inside (Canal-Linear-

Spline.pov and Canal-Linear-Spline-Inside.pov)

Figure (2.21) is coded by

sphere_sweep{linear_spline 3 1 1 1}

where we have omitted texture coding for the blue glass (to see later). In

this example, sphere radii are constant. In genera, for a linear_spline canal

surface, sphere radii vary linearly on each segment, which means that the surface

has conic parts joined by spherical sectors.

For a cubic_spline sphere-sweep, the code is

sphere_sweep{cubic_spline n+2 p_0 R_0 p_1 R_1 ... p_{n+1} R_{n+1}

[options, color, etc.]}

We show now an example of sphere_sweep with a cubic_spline directing

curve, figure (2.22). The code is

sphere_sweep{cubic_spline 5

1 1 1/2 1 }

There is no Bezier version of sphere_sweep surfaces.


Figure 2.22: A canal surface with cubic spline (Canal-Cubic-Spline.pov)

2.5.11 Using a canal surface to draw a curve

A natural question is : why not representing curves in R3 by a solid

sphere_sweep ? It would avoid to code a sausage chain. In fact, it is

easy, since the sequence of points in a linear_spline (or any other spline) may

be programmed. The parabola arc of figure (2.2) may be coded as

#declare nbsegs=100;

#macro ff(tt)

tt*tt

#end

sphere_sweep{linear_spline 1+nbsegs

#declare i=0;

#while (i


Figure 2.23: A curve represented as a thin canal surface (Canal-Curve.pov)

In both cases, the sequence of base and control parameters and points define a

global arc . This arc become a function t 7! .t/when we use spline . Indeed,spline is a function whose parameter is t and which returns .t/. It may be

used for example to draw the arc as a chain of small spheres.

Choose a personal arbitrary name, for example MySpline and write the code

#declare MySpline =

spline{cubic_spline n

t_0 p_0 t_1 p_1 ... t_n p_n t_{n+1} p_{n+1}

}

where, between the brackets of spline , you write the description of the

cubic_spline like in subsection 2.5.6 or subsection 2.5.7. Then, for each value

of t , the code MySpline(t) returns .t/, with which you can do what you want.

You may now draw the chain of spheres which represent the curve :

#declare tt = -3;

#while (tt

2.6. BEZIER BICUBIC SURFACES 51

2.6 Bezier bicubic surfaces

Using bicubic_patch reduces drastically memory needs in representing

surfaces. A patch is defined by four base points and twelve control points. We

see here what they are precisely, how to code them in POV-Ray, and how to put

several patches together to obtain a well controled C1-surface.

2.6.1 Bicubic functions

We know, and we resume with slightly different notations, that given four

points p0, p1, p2, p3 in E D Rk, the curve parametrized on 0; 1 by :

t 7! f .t/ D3X

iD0

3

i

!t i.1 t /3ipi D

3XiD0

Bi.t /pi

is the unique curve of degree 3 such that f .0/ D p0, f .1/ D p3 ; f 0.0/ D 3.p1 p0/, f 0.1/ D 3.p3 p2/.We shall examine a two variables analog .u; v/ 7! f .u; v/ of this result. From

now on, we denote by fu, fv, fuv, etc. the partial derivatives of f .

A bicubic polynomial or bicubic function is a mapping f : 0; 1 0; 1 !E D Rk whose each component is a polynomial of degree 3 with respect toeach variable.

The four Bernstein polynomials Bi : t 7!3

i

t i.1 t /3i , i D 0; : : :3 form a

basis of the vector space of degree 3 polynomials. Then a bicubic functionwrites uniquely

f .u; v/ D3X

iD0

3XjD0

uivjpi;j ; pi;j 2 Rk: (2.11)

The family .pi;j /iD0:::3;jD0:::3 characterizes f . Moreover :

Proposition 2.6.1. Let us be given 16 points pi;j , i D 0; : : :3, j D 0; : : : 3.The function f : 0; 1 0; 1! Rk defined by (2.11) :

.u; v/ 7! f .u; v/ D3X

iD0

3XjD0

Bi.u/Bj .v/pi;j


is the unique bicubic function with the 16 following properties :

f .0; 0/ D p0;0; f .1; 0/ D p3;0; f .0; 1/ D p0;3; f .1; 1/ D p3;3fu.0; 0/ D 3.p1;0 p0;0/; fu.1; 0/ D 3.p3;0 p2;0/fu.0; 1/ D 3.p1;3 p0;3/; fu.1; 1/ D 3.p3;3 p2;3/fv.0; 0/ D 3.p0;1 p0;0/; fv.0; 1/ D 3.p0;3 p0;2/fv.1; 0/ D 3.p3;1 p3;0/; fv.1; 1/ D 3.p3;3 p3;2/

9>>>>>=>>>>>;

(2.12)

fu;v.0; 0/ D 9.p1;1 p1;0 p0;1 C p0;0/fu;v.1; 0/ D 9.p3;1 p3;0 p2;1 C p2;0/fu;v.0; 1/ D 9.p1;3 p1;2 p0;3 C p0;2/fu;v.1; 1/ D 9.p3;3 p3;2 p2;3 C p2;2/

9>>>=>>>;

(2.13)

Proof. Only uniqueness needs a proof. If f is bicubic, null at the four vertices of

the unit square, with first partial derivatives and cross second derivatives null at

these points, then f is null.

Unlike what is sometimes written, and due to this proposition, determining the

vectors pi;j is easy if the data are the value of f and the above partial derivatives

at the four vertices of the unit square. Indeed, from the relations (2.12) we obtain

immediately the pi;j s at the four vertices. The eight following equations in (2.12)

give the pi;j s on the edges, and the four equations (2.13) give the four remaining

vectors p1;1, p1;2, p2;1, p2;2.

2.6.2 POV-Ray bicubic patches

If we denote by Pij the point pi;j of proposition 2.6.1, the code for a

bicubic_patch is

bicubic_patch {

type T

flatness F

u_steps U v_steps V

P00, P10, P20, P30

P01, P11, P21, P31

P02, P12, P22, P32

P03, P13, P23, P33

texture{MyTexture}

}

2.6. BEZIER BICUBIC SURFACES 53

where T, F, U, V are values chosen by the user which modify POV-Rays

rendering. Their use is explained in section 3.4.5.2.1 Bicubic patches, and we

give here some precisions on these rather short explanations...

POV-RAy forces the user to write the parameter type which may take

two values 0 and 1 (otherwise, error message). If you dont specify the three

other parameters flatness , a_steps and v_steps , their value is implicitly

supposed to be 0.

Suppose you have written

bicubic_patch {

type 0

flatness 0

u_steps U v_steps V

....

with values of U and V positive integers. Then POV-Ray replaces the image

of the parametrization f by its triangulated approximation in the following way.

Consider a grid of the domain 0; 1 0; 1 of 2U equal intervals for the firstvariable u, and 2V equal intervals for the second variable v, each small rectangle

of the grid being decomposed in the canonical way into two triangles. Then,

these small triangles triangulate 0; 1 0; 1, and POV-Ray replaces the imageof f restricted to each of them by the image of the affine mapping coinciding

with f at the vertices of the small rectangle. Figure (2.24) shows POV-Rays

approximation for the same 16 base and control points according to the values

U D V chosen. A checker has been calibrated onto the surface to display thetriangulation. When the user decides type 1 , according to flatness value,

POV-Ray make simplifications and records intermediate results to win time and

economize memory in the ray-tracing step, but it doesnt seem to change the

geometric result, so we shall not go further into these details. You have just to

remember that a good value of U and V is in general U D V D 3 or at most4 Indeed, U D 3 gives 64 rectangles (a chessboard), and 128 triangles for eachbicubic patch.

2.6.3 Joining bicubic patches

To obtain a C1 surface by juxtaposition of bicubic patches forces contact

conditions fortunately simple and natural. Figure (2.25) shows immediately the

conditions : the rods should be the same at the common edge, and the transversal


Figure 2.24: POV-Ray bicubic patch with U D V D 1 and U D V D 2(Bicubic-Patch-Steps1.pov and Bicubic-Patch-Steps2.pov)

Figure 2.25: A not-G1 and a G1 joining of two bicubic patches (Bicubic-Patch-

Show.pov and Bicubic-Patch-Show2.pov)

rods should be aligned on this common articulation. It will be a corollary of

proposition 2.6.1.

A shorter reformulation of proposition 2.6.1 simplifies things. For four vectors

p0, p1, p2, p3, we denote by Bp1; p2; p3; p4 the one variable vectorial function

t 7!P3iD0Bi.t /pi .Proposition 2.6.2. Given points pi;j , i D 0; : : : 3, j D 0; : : : 3, there exists aunique bicubic fonction f on 0; 1 0; 1 bicubique such that :

f .; 0/ D Bp0;0; p1;0; p2;0; p3;0f .; 1/ D Bp0;3; p1;3; p2;3; p3;3f .0; / D Bp0;0; p0;1; p0;2; p0;3f .1; / D Bp3;0; p3;1; p3;2; p3;3

9>>>=>>>;

(2.14)

2.7. ISOSURFACES 55

fv.; 0/ D 3Bp0;1 p0;0; p1;1 p1;0; p2;1 p2;0; p3;1 p3;0fv.; 1/ D 3Bp0;3 p0;2; p1;3 p1;2; p2;3 p2;2; p3;3 p3;2fu.0; / D 3Bp1;0 p0;0; p1;1 p0;1; p1;2 p0;2; p1;3 p0;3fu.1; / D 3Bp3;0 p2;0; p3;1 p2;1; p3;2 p2;2; p3;3 p3;2

9>>>=>>>;

(2.15)

Proof. The first four formulas are immediate. Afterwards, it suffices to derive

partially the formula (2.11) for example with respect to v and put v D 0 to obtainthe desired formula for fv.u; 0/. The same for the other ones.

Then the 16 points pi;j data dtermine entirely f by :

its restriction to the boundary of the square ; the restriction of fv to the horizontal sides of the square ; the restriction of fu to the vertical sides of the square.This gives the condition to obtain a C1 surface by juxtaposition of bicubic

patches :

Corollaire 2.6.3. Let f be a function defined on the rectangle 0; n 0; m (nandm entiers), whose restriction to each ; C1 ; C1 ( and entiers) is,after integer translation, a bicubic patch associated to a family .p;i;j /iD0:::3;jD0::3.

A necessary and sufficient condition so that f be of class C1 is that, for all ,

, between evident bounds, and all i , j the follwing relations are verified :

p;3;j D pC1;0;j ; p;i;3 D p;C1i;0

p;3;j p;2;j D pC1;1;j pC1;0;j

p;i;3 p;i;2 D p;C1i;1 p;C1i;0

9>=>; (2.16)

A C2 joining is beyond the scope of these notes and standard POV-Ray.

2.7 Isosurfaces

2.7.1 Volume by isosurface

Consider a compact domainD bounded by the surface B , for example a box

(bounded by a rectangle parallelipiped), or a ball bounded a sphere. Consider also

a function f : R3 ! R, .x; y; z/ 7! f .x; y; z/, and choose an arbitrary number a.Then isosurface represents precisely the set V of< x; y; z >2 D (in POV-Raynotations and coordinates) such that f .x; y; z/ a.

The minimal code is


isosurface{

f(x,y,z)

threshold a

contained_by D

max_gradient m

[open]

}

where you notice a very important parameter max_gradient whose value m

is written by the user, which is roughly an upper bound (not necessary the least)

of the norm of the gradient of f in the domain D. You have to know it a priori,

since POV-Ray really uses this value of m to draw the volume. This point will be

discussed latter in this subsection.

For example the code

isosurface{function{y-cos(2*x)*cos(2*z)}

threshold 0

max_gradient MaxGradient

contained_by{box{

}}

}

gives figure (2.26) on the left, and when we replace f .x; y; z/ D y cos 2x cos 2z by f .x; y; z/, we obtain figure (2.26) on the right. The valueof max_gradient has been calculated by another software (Mathematica or

Maple). If you forget it, POV-Ray assigns an arbitrary general value which leads

to holes like in figure (2.27). A first solution is to put a rough approximation of

max_gradient , and to increase it until the holes and visual artefacts disappear.

But for a too high value of max_gradient , execution time and memory size

increases drastically. To help users estimation of max_gradient , POV-Ray

offers an option evaluate . You will find its use in the Help.

2.7.2 Surface by isosurface

With the open option, POV-Ray represents only the (infinitely thin) surface

of points < x; y; z >2 D such that f .x; y; z/ D a, see figure (2.28).A natural question (to which the answer is no so clear in the documentation) is

2.7. ISOSURFACES 57

Figure 2.26: Volumes f .x; y; z/ a and f .x; y; z/ a by isosurface. (Isosur-face1.pov and Isosurface2.pov)

Figure 2.27: Isosurface without max_gradient control (Isosurface3.pov)

Figure 2.28: Isosurface with option open (Isosurface4.pov)


whether the image obtained is the set V of< x; y; z >2 D (in POV-Ray notationsand coordinates) such that f .x; y; z/ a or its boundary @V , that is the union of :

the set of points< x; y; z > (in POV-Ray coordinates) such that f .x; y; z/ Da ;

the set of points < x; y; z > of the boundary B such that f .x; y; z/ 0.To see that without the option open it is in fact the volume, it suffices to

assign it a texture of glass with refraction index, and to push in it a stick of wood.

If it is a volume, refraction will bend the image of the stick see figure (2.29).

Figure 2.29: An isosurface is a volume. (Isosurface5.pov)

It may happen that, for CSG, you have to thicken a bit the infinitely thin

surface obtained with the open option. A simple mean to do this, which

we shall find again in algebraic surfaces, is to suppress open option, and

to draw the intersection of V D f< x; y; z >2 D = f .x; y; z/ ag withV 0 D f< x; y; z >2 D = f .x; y; z/ a C "g, for an arbitrary small value of". In figure (2.30), " has been taken not too small to show the thickening.

2.8 Algebraic surfaces

2.8.1 Surface with poly and polynomial

POV-Rays treatment of surfaces S : f .x; y; z/ D a by isosurface iscomplicated, based on the gradient of f near the points of S , and requiring

control of a delicate parameter max_gradient . POV-Ray users are interested in

drawing with isosurface very irregular (even if smooth) surfaces.

For mathematicians, it gives good looking surfaces for a non-polynomial

surfaces, as seen in the above examples.

2.8. ALGEBRAIC SURFACES 59

Figure 2.30: Slightly thickened isosurface (Isosurface6.pov)

POV-Ray can benefit from the algebra when the equation f .x; y; z/ D 0 ofS is polynomial, since in this case, it usees a very different algorithm without

the risk of artefacts of the above methods. For example the code for Hadamards

double hyperboloid, of equation

..x 2/2 C z2 y2 1/..x C 2/2 C z2 y2 1/ 2 D 0;

presented in the introduction is

object{

poly{4,

scale < 2,2,2>

texture{...}

clipped_by{box{ }}

no_shadow

}

In general the surface S is defined by the code

poly{N

clipped_by{...]}

}


where N is a degree chosen by us and between < and > are written the

coefficients of P which must have a total degree inferior or equal to N , in a very

precise order which we shall describe now.

For the three variables x, y, z (in this order), the chosen order for the

monomials xiyj zk is the decreasing lexicographic order. It means that

.i; j; k/ precedes .i 0; j 0; k0/,

8 i 0/ or

.i D i 0 and j > j 0/ or

.i D i 0 and j D j 0 and k > k0/

But an essential point is that all coefficients, even the null ones, for monomials of

degree N must be present in the list. The number of coefficients depends onlyon the chosen number N , and POV-Ray rejects the polynomial if the number of

coefficients between < and > is notNC3

2

D .N C 1/.N C 2/.N C 3/6

.

This order may be visualized by grouping the terms as follows. Once chosen

N (recall that w have to choose N ), we write our polynomial P in the form

P D xN C xN1P1.y; z/xN2P2.y; z/C : : : ;

where we attach to Pq the number q, and we think that the coefficients will be

written in this order. Next, each Pq is written in the form

Pq D yqPq;0.y; z/C yq1Pq;1.z/C yq2Pq;2.z/C : : :

Finally, each Pq;r is written in decreasing degrees of z.

The list of monomials is given, for low degree, in section 3.4.5.3.5 of the

Help. This ordering is also called lexicographic ordering in [C-L-OS] (Definition

3 page 55). They say (where their strict inequality< is our precedes) :

D .1; 2; 3/ < D .1; 2; 3/ if and only if, in the vector difference 2 Z3 the left-most nonzero entry is positive.

We shall see later that Mathematica can construct this list of coefficients

for us, given a traditional mathematical formula. Anyway, if you have forgotten

lexicographic order (or the little Mathematica tool), you may use poynomial ,

and write precisely the correspondence between indices and coefficients like in

the following code


object{

// Torus having major radius sqrt(40), minor radius sqrt(12)

polynomial { 4,

xyz(4,0,0):1,

xyz(2,2,0):2,

xyz(2,0,2):2,

xyz(2,0,0):-104,

xyz(0,4,0):1,

xyz(0,2,2):2,

xyz(0,2,0):56,

xyz(0,0,4):1,

xyz(0,0,2):-104,

xyz(0,0,0):784

}

texture{...}

clipped_by{box{ }}

no_shadow

}

which gives figure (2.31).

Figure 2.31: Drawing a polynomial (Draw-Polynomial.pov)

2.8.2 Singular surface with poly

There is no rose without a thorn. In mathematics, thorns are named cusps.

Some unpleasant artefact appears when drawing with poly the surface (with


POV-Ray conventions)

f .x; y; z/ D x2 C y2 z3:

See the thin vertical line in figure (2.32) on the left. Such an inconvenient does

not arise with a conical singularity, like in figure (2.32) on the left. The precise

Figure 2.32: Cuspidal and conical singularity with poly (Cuspidal-singularity-

poly.pov and Conical-singularity-poly.pov)

position of the artefact (a line testifying a problem of ray tracing in presence of a

cusp) can hardly be foreseen, since when there are two cusps, like in the famous

zitrus f .x; y; z/ D y2Cz2Cx3.x1/3 (figure (2.33)), there is only one unwantedline, and it does not pass necessarily at the cusp. It is not necessarily orthogonal

Figure 2.33: Zitrus drawn with poly (and rotated) and with isosurface (Zitrus-

poly.pov and Zitrus-isosurface.pov)

to the cuspidal tangent. Is is really undesirable. See figure (2.33) on the left.


Figure 2.34: Whitney umbrella drawn with poly and with isosurface (Whitney-

umbrella-poly.pov andWhitney-umbrella-isosurface.pov)

Such problems arise in non isolated singularity, like Whitney umbrella, see

figure (2.34).

We may think that, as usual, POV-Ray doesnt feel at ease with codimension

two. The reason is certainlymore technical, since isosurface (with a convenient

value of max_gradient ) solves quickly the problem, see figure (2.33) on the

right. The same is true for the cusp, perfectly represented by isosurface .

Another solution is to mask the artefact, by choosing well a clipping box,

or with a CSG technique like difference . All that to avoid to control

max_gradient ...

2.8.3 Complete intersection with poly

A CSG technique to enhance the complete intersection of the surface

f .x; y; z/ D 0 and g.x; y; z/ D 0 (in the domain D) is the following. First, wethicken each surface by defining F D f< x; y; z >2 D = " f .x; y; z/ "g,constructed as a difference , andG D f< x; y; z >2 D = " g.x; y; z/ "g.Then the (thickened) intersection curve will be rope F \ G, constructed withintersection .

A transverse intersection is well represented in this way, but we have to accept

variations of the section of the rope thus constructed. For example, the rope in

figure (2.35) is flattened on its highest part (on the left) since the angle between

the tangent planes of the two surfaces is rather small. When the tangent planes

make an angle of 90 degrees, the section of the rope is a square.

The code of figure (refComplete-Intersection) (we give here only the drawing

of the black rope) is


#declare epsilon = 0.01;

object{

intersection{

difference{poly{2, }

poly{2, }

}

difference{poly{2, }

poly{2, }

}

}

pigment{Black}

no_shadow

}

where f .x; y; z/ D x2 C z2 x D 0 the blue cylinder) and g.x; y; z/ Dy x2 C z2 (the violet hyperboloid). Of course, at a point where the surfaces are

Figure 2.35: Complete intersection (Complete-Intersection.pov)

tangent, the thickened intersection will be thicker. We can play on ".

Essentially the same result would be obtained with isosurface instead of

poly .

2.9. PARAMETRIC SURFACES 65

2.9 Parametric surfaces

POV-Ray is able to draw parametrized surfaces, but, as we said above,

parametrized surfaces are not natural in ray tracing. They are faithfully repre-

sented, this is why they are evoked here, but rendering is very (very !) slow, even

with some special techniques proposed by POV-Ray to avoid useless or repeated

calculations.

As usual the code for a parametrized surface

< u; v >7!< X.u; v/; Y.u; v/; Z.u; v/ >(with POV-Ray notations) where < u; v > runs in umini ; umaxi vmini ; vmaxi is

parametric{

function {X(u,v)},

function {Y(u,v)},

function {Z(u,v)}

,

[contained_by {box or sphere}]

[max_gradient m]

[accuracy alpha]

[precompute N x,y,z]

texture{...}

}

where the brackets mean that these parameter lay be forgotten since POV-Ray

assigns a default value to them.

Take care that the domain is denoted (as usual for a box ) by the extreme

points of the diagonal of the box, and not a pair of intervals.

Default value of accuracy is 0.0001. The documentation does not define

precisely this parameter, and as rendering is really very slow, the users would

have helped, in an concrete and simple situation how its value could help loose

less time...

The discussion of max_gradient is the same as before (in isosurface

section), and as before, it is important to give it a reasonable value. Of course,

here the gradient has not the same meaning as in isosurface section, since the

surface is defined by a parametrization and not an equation. The documentation

says (in section 3.4.5.1.10) that : its the maximum magnitude of all six partial

derivatives over the specified ranges of u and v. You have just to estimate an


upper bound of their absolute values, not necessarily the least. At least, you know

what max_gradient means...

The precompute parameters are used by POV-Ray like this. To represent

the surface, POV-Ray triangulates the image, this triangulation being the image

of a regular canonical triangulation of the rectangular domain, the same

triangulation as in section 2.6. But now, the bicubic parametrization of section

2.6 is replaced by .X; Y; Z/ D itself. Each elementary triangle .a; b; c/ of thedomain triangulation defines a triangle ..a/; .b/; .c//, and POV-Ray has to

determine which light rays cut this last triangle. If the value of the parameter

precompute is N , the values taken on a grid of 2N small rectangles by each

chosen coordinate (among x, y, z) are memorized, to accelerate the calculations.

A good value of this parameter is 4.

We give a famous example : the graph of the function of Calvert and

Vamanamurthy polynomial with only one critical point, strict local minimum

but not global minimum (see figure (2.36))

y D f .x; z/ D x2.1C z/3 C z2:

The code is

Figure 2.36: Graph of Calvert and Vamanamurthy drawn with Parametric

(Calvert-Parametric.pov)

2.9. PARAMETRIC SURFACES 67

parametric{

function {u},

function {u*u*(1+v)*(1+v)*(1+v)+v*v},

function {v}

,

contained_by {box{,}}

accuracy 0.001

precompute 4 x,y,z

texture{...}

no_shadow

}

Bibliography

[C-L-OS] D. Cox, J. Little, D. OShea, Ideal, Varieties and Algorithms, UTM

Springer (1992)

[La-Si] R. Langevin and J-C Sifre. Foliations of S3 by Dupin cyclides, in Folia-

tions 2012, P. Walczak, J. A. Lopez, S. Hurder, R. Langevin, T. Suboi,

editors, Word Scientific (2013)

[La-Si2] R. Langevin and J-C Sifre. Foliations of S3 by Cyclides, preprint (2014)

[La-Si-Dru-Gar-Pa] R. Langevin, J-C. Sifre, L. Druoton , L. Garnier and

M. Paluszny. Gluing Dupin cyclides along circles, finding a cyclide

given three contact conditions, Preprint IMB, Dijon (2012).

69

pov-ray for geometry

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