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Babylon University
College of Info. Tech.
Software Dep. 3d Class
Ass. Lecturer. Wadhah R. Baiee
Geometric 3D Objects
THREE-DIMENSIONAL PRIMITIVES
In a three-dimensional world, we can have a far greater variety of geometric
objects than we can in two dimensions. When we worked in a two-dimensional
plane in Chapter 2, we considered objects that were simple curves, such as
line segments, and flat objects with well-defined interiors, such as simple
polygons. In three dimensions, we retain these objects, but they are no longer
restricted to lie in the same plane. Hence, curves become curves in space
(Figure 4.18), and objects with interiors can become surfaces in space (Figure
4.19). In addition, we can have objects with volumes, such as parallelepipeds
and ellipsoids (Figure 4.20).
Three features characterize three-dimensional objects that fit well with
existing graphics hardware and software:
1. The objects are described by their surfaces and can be thought of as
being hollow.
2. The objects can be specified through a set of vertices in three
dimensions.
3. The objects either are composed of or can be approximated by flat,
convex polygons.
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The first condition implies that we need only two-dimensional primitives to
model three-dimensional objects because a surface is a two- rather than a
three dimensional entity. The second condition is an extension of our
observations in Chapters 1 and 2 of your textbox. If an object is defined by
vertices, we can use a pipeline architecture to process these vertices at high
rates, and we can use the hardware to generate the images of the objects only
during rasterization. The final condition is an extension from our discussion of
two-dimensional polygons. Most graphics systems are optimized for the
processing of points and polygons. In three dimensions, a polygon can be
defined by an ordered list of vertices. However, if there are more than three
vertices, they do not have to lie in the same plane; if they do not, there is no
simple way to define the interior of the object .
MODELING A COLORED CUBE
We now have most of the basic conceptual and practical knowledge we need
to build three - dimensional graphical applications. We will use them to produce
a program that draws a rotating cube. One frame of an animation might be as
shown in Figure 4.28. However, before we can rotate the cube, we will
consider how we can model it efficiently.
Modeling the Faces
The cube is as simple a three-dimensional object as we might expect to
model and display. There are a number of ways, however, to model it. A CSG
system would regard it as a single primitive. At the other extreme, the hardware
processes the cube as an object defined by eight vertices. Our decision to use
surface-based models implies that we regard a cube either as the intersection
of six planes or as the six polygons, called facets, that define its faces. A
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carefully designed data structure should support both the high-level application
view of the cube and the low-level view needed for the implementation.
We start by assuming that the vertices of the cube are available through an
array of vertices; for example, we could use the following:
float vertices[8] [3] =
{{-1.0,-1.0,-1.0},{1.0,-1.0,-1.0},
{1.0,1.0,-1.0}, {-1.0,1.0,-1.0},
{-1.0,-1.0,1.0}, {1.0,-1.0,1.0},
{1.0,1.0,1.0}, {-1.0,1.0,1.0}};
We can then use the list of points to specify the faces of the cube. For
example, one face is
glBegin(GL_P0LYG0N);
glVertex3fv(vertices[0]);
glVertex3fv(vertices[3]);
glVertex3fv(vertices[2]);
glVertex3fv(vertices [1]);
glEndO;
And we can define the other five faces similarly.
Data Structures for Object Representation
We could now describe our cube through a set of vertex specifications. For
example, we could use glBegin(GL_P0LYG0N) six times, each time
followed by four vertices (via glVertex) and a glEnd, or we could use
glBegin(GL_QUADS) followed by 24 vertices and a glEnd. Both of these
methods work, but they both fail to capture the essence of the cube's
topology, as opposed to the cube's geometry. If we think of the cube as a
polyhedron, we have an object—the cube—that is composed of six faces. The
faces are each quadrilaterals that meet at vertices; each vertex is shared by
three faces. In addition, pairs of vertices define edges of the quadrilaterals;
each edge is shared by two faces.
The data specifying the location of the vertices contain the geometry and can
be stored as a simple list or array, such as in ver t ices [8]—the vertex list. The
toplevel entity is a cube; we regard it as being composed of six faces. Each
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face consists of four ordered vertices. Each vertex can be specified indirectly
through its index. This data structure is shown in Figure 4.30. One of the
advantages of this structure is that each geometric location appears only once,
instead of being repeated each time it is used for a facet. If, in an interactive
application, the location of a vertex is changed, the application needs to
change that location only once, rather than searching for multiple occurrences
of the vertex.
The Color Cube
We can use the vertex list to define a color cube. We assign the colors of the
vertices of the color solid of Chapter 2' (black, white, red, green, blue, cyan,
magenta, yellow) to the vertices. We define a function quad to draw
quadrilateral polygons specified by pointers into the vertex list. We assign a
color for the face using the index of the first vertex.
Finally, the colorcube specifies the six faces, taking care to make them all
outward-facing as follows:
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Sphere
In Cartesian coordinates, a spherical surface with radius r centered on the
coordinate origin is defined as the set of points (x, y, z) that satisfy the
equation.
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We can also describe the spherical surface in parametric form, using latitude
and longitude angles , We can describe it by the following three equations:
The parametric representation in Equations above provides a symmetric
range for the angular parameters
and . Alternatively, we could write the
parametric equations using standard spherical coordinates, where angle
is
specified as the colatitude. Then, . is defined over the range 0 < < π, and
is often taken in the range 0 < < 2π .
Fans and strips allow us to approximate many curved surfaces simply. For
example, one way to construct an approximation to a sphere is to use a set of
polygons defined by lines of longitude and latitude as shown in Figure of
sphere .
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We can do so very efficiently using either quad strips or triangle strips. If we
fix and draw curves as we change , we get circles of constant longitude. By
generating points at fixed increments of and , we can define quadrilaterals
as shown in Figure of sphere. Remembering that we must convert degrees to
radians for the standard trigonometric functions, the code for the quadrilaterals
corresponding to increments of 20 degrees in and to 20 degrees in is
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