Glossary

Affine Transformation

A mathematical function such as a translation, rotation, scaling, or shearing that transforms a line into a line or a plane into another plane. Using a trick, such functions, which are usually difficult to combine due to the nonlinear behavior of the translation, can be implemented with just one matrix multiplication: the number of coordinates is increased by one, so-called homogenous coordinates are used (see also below).

Alpha-Channel

Usually images are defined with three color values: red, green, and blue (when printing: yellow, magenta, cyan, and most often black). In computer graphics, additionally an alpha channel is needed to define the transparency of the pixel values. This is needed when a small image is used as a texture (see below) during rendering. The alpha channel is a separate layer that defines the level of transparency per pixel. Rather than restricting pixels to opaque or transpar­ent, an alpha channel allows pixels to be defined as partly opaque (e. g., 75% opaque).

Attractor

A final state a system reaches for a large number of initial conditions. The term is often used in chaos theory, but is also common in biology. See also inhibitor.

Bezier Curve

A mathematically defined curve, defined by polynomials for each coordinate. These polynomials are defined in a special way from a set of control points that are partly interpolated and partly approximated. By moving the points, the curve changes its form intuitively which makes it suitable for geometric mod­eling purposes. With standard polynomials this is almost not possible.

Bump-Mapping

In computer graphics, the realism of a surface is emphasized by applying a tex­ture, such as a photograph of a surface. For example, if a photograph of bricks is applied to a square, a wall can be simulated without having to model all de­tails, i. e., grooves, joints, ridges, and surface bumps of the bricks. Aside from changing the color, the texture can also modify other surface characteristics. A so called bump-map changes the direction of the surface normal and thereby is able to simulate a structured surface.

Cn-Continuity

Often individual components of a plant must be fitted, such as small branches to a main branch. The smoothness of the geometrical transitions is expressed by the Cn – continuity: the higher the value n, the smoother the transition. In mathematics Cn-continuity expresses that the derivations on both sides of the transition are in accordance up to the n-th order.

Coherence (Spatial/Temporal)

If an object is displayed in an image sequence or is maneuvred interactively on the screen, a continuous change of the shape without visible artifacts is de­sirable. If a smooth change of object transitions is achieved, we speak of a temporal coherence, in interactive systems also of a spatial coherence.

CSG Tree (Constructive Solid Geometry)

A data structure for the definition of complex volumes. The leaves of the data structure (nodes without successors) are geometric primitives such as cubes, balls, cylinders, or half-spaces; the inner nodes are Boolean set operations such as intersection, subtraction, or unification.

Display List

A display list includes geometric data for the display on the computer screen, such as triangles for the definition of a surface. The display list is assigned to the graphics hardware and can be called up very quickly. A display list is espe­cially useful when the same geometry is to be displayed in different places. In this case the list is defined and transferred once, the graphics unit then is able to rapidly draw several copies at different locations.

Finite Element Method

A procedure for the numeric computation of complex problems. A problem is discretized into a finite set of small simple components that are interacting. The reactions of these components to external influences are linear to the force of the influences. This process allows for the definition of the system over exten­sive matrix equations that can, however, often be solved efficiently.

Generalized Cylinder Appendix в

An extended cylinder model constructed from a surface, such as a square that is Glossary

moved along a path in a space. The resulting enclosed volume (resulting from the moving) of the surface forms the generalized cylinder.

Homogeneous Coordinates

An extended mathematical model for the definition of positions in a space. In­stead of three coordinates x, y, z, four coordinates (x, y, z, w) are used and the actual position is determined through a special projection of the four coordi­nate position (projective geometry). This permits the formulation of all affine transformations (see above) using 4 x 4-matrices, thereby considerably simpli­fying the computation.

Image Processing

Manipulation and analysis of images. simple image processing encompasses the modification of colors or retouching of image contents. The image analysis in general attempts to recognize and to understand the content of an image.

Inhibitor

A final state that a dynamic system avoids for a large number of initial condi­tions. The term is also used for substances that inhibit or decrease the effects of other substances in chemical or biological reactions (see also attractor).

Keyframing

Concept for the definition of computer animations. The animation is specified using a set of essential key frames or scene states. The computer calculates the intermediate parts through interpolation of all values that define the key frames.

Level-of-Detail (LOD)

Method for the dynamic representation of geometric data. For the display of data no fixed representation is used, but rather a definition that is modified in its complexity relative to the visual size of the object on the screen. A distant object consists of less data than an object closer to the virtual viewer. We differ­entiate between discrete and continuous LoD. For the discrete LoD, several fixed object definitions with varying details are used which are blended into each other. In the continuous LoD, the object is represented individually for each frame, without the viewer noticing that the model changes.

Morphing Algorithm

Method for the distortion of images or objects. At defined points in the image or the object surface, displacement vectors are defined. The deformation of the remaining points is computed by interpolation.

Polygon

A description of a surface in space. The triangle is the most simple form of polygon; it is defined by the coordinates of the three vertices. A general poly­gon can have arbitrarily many vertices. A volume element that is defined by polygonal surfaces is called a polyhedron.

Postprocessing

When creating computer animations, the images supplied by the rendering al­gorithm are usually processed further, i. e., colors are adjusted or different im­age layers are combined. This process is called postprocessing.

Pseudocode

An English-language expression for a program code that notes only the essen­tial processes without distracting the reader by presenting too many details. The implementation in a concrete programming language can look different, but can be translated easily from the pseudocode.

Real-Time

A non-trivial term from informatics. Real-time operations mean that tasks are solved by a program so that the results can be used continuously in a running system, such as a technical installation or an interactive system. The growth simulation of a tree in true real-time is not interesting, since it pertains to the time span that the tree actually uses to grow. Therefore, real-time describes a human rather than a machine or absolute sense of time.

Rendering

The process of the image production that transforms data input in the form of geometries and lighting parameters into an image using a computer program that simulates the light transport between the objects. There also exists sound rendering, which similarly produces sound.

Texture

To enhance realism in computer graphics, pictures or photographs are projected onto a surface. These projected images are called textures. For example, when modeling trees, textures are used for the bark and the leaves. This makes it possible to only use a square per leaf onto which a picture of a real leaf is pro­jected. In this process it is necessary to assign to each pixel of the texture a so-called alpha value. This allows the user to project also ragged – or irregular­shaped leaves onto a square (see also bump-mapping and alpha channel).

Virtual-Reality System (VR)

A computer system for the simulation of a synthetic environment. This often takes place using special input devices such as head-mounted displays and data gloves that allow the user to interact with the virtual world. An example of a VR system is a flight simulator that is equipped with a realistic cockpit, sight and sound system as well as force feedback.

[1]A definition that refers back to the approach introduced by Halle and Oldeman [80].

[2] 1 360° n 360°

– = 1 resp. ————— = 360°——— ^ = 222.4922…° (3.10)

т т2 т т2

[3]Here it must be noted, that in [172] no description is given of the actual recursive procedure, but only a rough motivation with the calculation of the respective parameters.

[4] CSG stands for constructive solid geometry.

[5]In the extreme case just one triangle can be used; however, that means that the leaf cannot be bent. This is a disadvantage, since it often affects the visual appearance of a plant.

[6]In this case, only tubes are permitted as primitives.

[7]For the in appendix described PC-based variant of the modeling system recursions are not permitted.

[8]Bandlimitation emerges in connection with the Fourier transform: if the frequency spectrum of a function has only local support, then the function is bandlimited.

[9] Select a tile from the tile set with a south side that corresponds to the north side of the leftmost tile in the line above.

[10]This is only an approximate indication, since the size of the matrix depends on different fac­tors. The order of magnitude, however, is correct.

[11]The triangle strips used in many cases to decrease the number of vertices can most often not be applied in foliage, which consists of many small isolated surfaces.

[12] For example, it must be taken into consideration that the lines can overlap, and thereby produce a smaller overall grayscale value.

[13]see http://www. artworks. co. uk

[14]with “Organic Modeller Deluxe” ArtWorks offers a commercial variant of the principles im­plemented in Mutator

[15]The first version issued 1997 at the ZKM Karlsruhe under the name “Morphogenesis”.