Description of a virtual situation

Surfaces define the object

Object defined as having mass

Ephemeral phenomena

Mathmatical study of shapes

Location in space

Straight line is the shortest distance between two points–possesses length

Flat surface–possesses length and width

Points that define shapes

Flat surface with many sides or edges

Creating models with polygons

Using polygons to approximate curves

Efficient reduction of polygons in a model in a way that preserves the polygonal approximation to the maximum extent possible

Ability to manipulate situations on the screen without any noticeable delay

The angle between lines coming perpendicularly off polygons at the shared vertex. Or, the angle between intersecting polygons -180° .

Adding polygons while retaining a shape (usually to prepare for some type of deformation–see deformation)

Smoothing or rounding sharp edges by adding polygons

Effecting everything (on an object, or a world)

Restricted area

Adding/subtracting vertices individually with the mouse

Grid of polygons

Two directions of organization of polygons in a mesh.

Structure of a surface

List of points and their connections to one another

A true curve is a mathematical formula instead of a description of connected points. NOTE: This is a key way to see the difference between bitmapped and vector-based computer images.

Series of straight lines intended to approximate a curve.

A type of curve created using control points or control veritices to bend a line into a desired shape. Comes from the shipbuilding trade where posts were driven into the ground and boards bent between them. Draftspersons have used flexible strips of wood or metal as aids in drawing curves as well. Most computer graphic splines are cubic, meaning they are derived by a certain type of mathmatical formula that raises variables to the third power, or cube.

Network of straight lines connecting the control vertices. Can often be manipulated as a way of manipulating the curve.

Curve passes directly through the control points. A variation called the cardinal spline has the curve passing through all but the first and last points.

Curve passes near the control points making them more forgiving for creating smooth curves made up of many points.

B-spline (B for "basis") has the curve starting near the second control point and ending near the second-to-last.

Bézier spline adds tangent vectors (represented by arrows) that have both direction and length. These handles allow you to change the direction of the curve my changing the direction of the arrow, and the amount of area affected by the point is changed by changing the vector’s length. These are powerful and wonderful ways of creating curves and are found in every type of computer program in every discipline.

Discontinuity or cusp creates a "kink" in a curve by breaking the two tangent vectors from there usual straight-line configuration so that each can be pointed in a different direction.

NURBS curve, or Non-Uniform Rational (describes mathmatical properties) B-Spline curve has the curve passing through both end points but not the intermediate points. Uses the best features of the interpolating and the approximating curve in that you have precision and smoothness. This is achieved by the existence of edit points on the curve itself (that were not used to create it, but rather appear afterward) and of course the control points used to make it. The control points allow gross adjustment retaining smoothness, the edit points allow precise fine adjustment. These always have a direction created when you make them (first to last) and changeable in many packages later.

Parameterized curves is a name sometimes given to spline curves that have a direction because it allows you to locate a specific place on the curve by its distance from the start point.

Surface made by pushing one curve out along another. The path this traces makes the surface. It has a U direction: the direction of the first curve, and a V direction, the direction of the second curve.

Surface made from two splines

Surface made of two cubic splines

Dividing up space using a grid. Handy for locating things as precisely as the grid resolution allows.

Direction, indicated by a line, of each dimension or variable of the coordinate system (horizontal/ vertical or up/down/back are simple examples. They have a positive and negative direction based on the origin (below).

Center of the grid; location=0 on all axes.

Subdivisions of a coordinate system created using combinations of positive and/or negative halves of axes (i.e. upper left, lower right). In a 2D system this is one of 4 sections.

The system of mutually perpendicular axes devised by French mathematician/philosopher René Descartes. There are two standards: the right-handed and left-handed coordinate systems--named so by the direction your thumb points to define the positive direction of the X axis when using your hand as a model. Try this using the illustration in the book.

Windows used to construct and view objects and worlds in a 3D computer animation or modeling environment

A view of the world similar to the one we see with our eyes. Parallel lines converge toward a vanishing point etc.

A perspective view from a defined camera

Views in which the effect of perspective is absent. Parallel lines remain parallel forever. These are the views typically used for design and construction drawing where it is important that all proportions remain in scale as they would on a real object (e.g. all edges of a cube are represented as being the same length).

There are top, side, and front views in some software packages adjustable to their opposites (e.g. right side instead of left side.) In these views we are looking along a single axis and the perpendicular axes are parallel to the computer screen (e.g. looking down the Z axis at a "flat" representation of the X and Y axes). These windows are critical for construction and alignment of virtual objects and worlds.

The most basic "building blocks" of virtual space--easily derived (mathematically) and very useful for modeling. 3D software supplies them as separate entities that appear fully realized.

They have adjustable parameters or can be used with their default values.

When created, the primitives will be placed in relation to a construction plane. In some software, this is can be defined by the user, in others, it is always the same and objects must be moved after they are created.

A face is any flat surface that has been modeled.

These can be relative: translate, rotate, or scale the object in relation to what it was before the transformation.

Or absolute: in relation to the coordinate system, its origin and axes.

Right-handed or left-handed rotation is determined by the direction you fingers close in around an axis represented by your extended thumb. Oddly, the more common is the right-handed, though many find that counter intuitive because it is counter clockwise. Try this using the illustration in the book.

Proportional or uniform scale mean the object retains its proportions while changing it overall size

Nonproportional or nonuniform scale means that the object is scaled differently on its different axes.

The point about which an object rotates also the reference point of the object for other transformations

The Cartesian system defined above

The objects own X, Y and Z system. When you transform an object, you are moving its local within the global.

All the transformations of an object combined into a single transformation of an object from it’s original state. So if you moved an object nine times since you created it, the program will concatenate or combine those into one transformation value. On the X axis, move an object +3, -4, +2, -6: the concatenated value is a movement of -5.

The standard in 3D programs: a grid with three rows labeled translate, rotate and scale, and three columns labeled X, Y and Z.

Identity transformations are the ones the object had at the time of it’s creation: "0" values for translate and rotate, "1" for its scale.

An efficient method of representing and rendering multiples of an object without duplicating it’s geometry. Literally, the program is taking one set of geometry and expressing in a variety of transformations so that it is unnecessary to calculate the same relationships more than once. This create multiple instances of the object.

The result of rotating a curve around an axis. Think of a table leg, or some object turned on a wood lathe. Even bowls can be done this way--that's what the name comes from.

Synonymous with Lathed surface.

Synonymous with Lathed surface.

The original curve, used to then rotate around an axis, or perform other modeling techniques.

Pushing a curve through space along an axis or along another curve

Or an extrusion is the surface creating by the above.

In a closed curve the beginning and end points are the same, in an open or unclosed curve they are different.

An extrusion in which the generating curve is modified over the course of the extrusion; e.g. scaled.

A surface created between two curves to connect them together. This is a very common way to model: create one curve, duplicate it, transform and/or translate it, then use the software to create a lofted surface between them.

Most common in patch-oriented modeling, this technique creates a surface using curves that represent its edges.

The curves used to generate boundary patches; two to four of them are used.

Ways to change the shape of a model that function on the whole model instead of selected vertices--like scaling but more exotic: i.e. bend, twist, taper.

A way of organizing models or situations with many parts so that they can move as one when you want, or individually as before.

In this case, a diagram illustrating the heirarchy.

An element of the heirarchy

A node above, or on a higher hierarchical level than the node below: it’s child. The parent controls the child. See! Proof that the virtual world represents a fundamental escape from reality!

The lower, or controlled node.

An alternative to the schematic representation, and one that gives useful transformation data for each node, for each transformation.

Curly brackets indicate groupings: nodes withing the curly brackets of other nodes are the children.

A push is the point where a curly bracket begins a cascade of transformation, a pop is where the brackets close and elements below will not be effected.

Ways of joining groups of models to form a single new surface--named after George Boole who invented much of the logic for these operations. This is easier with polygonal modeling than with patch modeling, or NURBS modeling because the UV direction of surfaces is difficult to resolve during the operations.

Boolean operations dealing with computer graphics

Boolean operations used in solid modeling systems (those that see models as having mass, as opposed to surface modeling systems which is what most 3D animation packages are).

Creating a surface that is the combination of the previous group of volumes.

Creating a surface that is the overlap of one surface or group of surfaces subracted from the other.

Modifying one surface, usually a patch, by trimming it to a new curve placed on its surface by projection or some other techinique.

The "line" along which two surfaces, or a curve and a surface meet.

A branch of mathmatics refined and named by Benoit Mandelbrot. It allows geometries with fractional dimensions as opposed to the usual 1, 2, or 3 dimensions. The important thing for us is that the graphical depictions of these create rich, intricate patterns reminiscent of natural growth or geological structures.

No matter how much you zoom in or out, the type of shapes you see remain the same. Shapes are micro or macro versions of each other.

The triangular result of making a line vary from its straight path in order to detour 60 degrees to a single point, then —120 degrees back to continue in its original direction. It looks like the line had to go over a bump the shape of an equalateral triangle.

The original pattern that then subdivides to yield an increasingly complex pattern.

Turning all that virtual space into a 2D picture.

"Wire"edges and isoparms are all that are visible. This is the fastest, simplest type of rendering.

You can tumble, zoom, as well as transform objects and see the feedback instantly on the screen. This requires faster and faster computers as your the rendering becomes more complete (with shading, color, texture, etc.)

The foreward-most surfaces, or those closest to the camera hide the ones behind them with solid planes of various shades of gray.

Surfaces have value modulations that simulate the effects of light, or on higher end machines, approximate the effects of the lighting you’ve placed in the scene. Computers now handle texture mapping and some handle shadows–all with Real-time interaction.

The camera through which we choose the view of scenes in computer graphics.

The point in space where the virtual camera is located

The point in space where the camera is looking.

The direction the camera in looking in.

All camera movements that keep the camera location stationary while moving the camera interest.

All camera movements that keep the center of interest stationary while moving the camera.

Changing both the center of interest and the camera location simultaneously.

Changing the camera location by moving into or out of the image (along the Z axis). This comes from the use of a real dolly under a film camera to do the same thing.

Rotating the camera around the X axis so the image moves up or down.

Rotating the camera around the Y axis so the image goes side to side.

Rotating the camera on the Z axis so that the image "tilts" as if you cocked you head to the side. NOTE: This is a confusing departure from the same word as applied to use of a film or video camera where a tilt means rotating the camera on the X axis like the Pitch defined above.

A way to specify the Tilt as a value: by setting the precise direction the top of the camera is oriented toward.

Changing the virtual focal length of the lens to be shorter or longer, producing wider or tighter shots respectively.

The rectangular area of view that is visible on the screen. This is the same as the Picture Plane in the language of traditional art.

The cone-shaped volume of space that defines what is visible to your eye in the "real" world.

The pyramid-shaped volume of space that defines what is visible on the computer screen.