Drawing Geometric Objects

After reading this chapter, you'll be able to do the following:

- Clear the window to an arbitrary color

Draw with any geometric primitive - points, lines, and polygons - in
two or three dimensions

Control the display of those primitives - for example, draw dashed lines
or outlined polygons

Specify normal vectors at appropriate points on the surface of solid
objects

Force any pending drawing to complete

At the highest level of abstraction, there are three basic drawing operations: clearing the window, drawing a geometric object, and drawing a raster object. Raster objects, which include such things as two-dimensional images, bitmaps, and character fonts, are covered in Chapter 8 . In this chapter, you learn how to clear the screen and to draw geometric objects, including points, straight lines, and flat polygons.

You might think to yourself, "Wait a minute. I've seen lots of computer graphics in movies and on television, and there are plenty of beautifully shaded curved lines and surfaces. How are those drawn, if all OpenGL can draw are straight lines and flat polygons?" Even the image on the cover of this book includes a round table and objects on the table that have curved surfaces. It turns out that all the curved lines and surfaces you've seen are approximated by large numbers of little flat polygons or straight lines, in much the same way that the globe on the cover is constructed from a large set of rectangular blocks. The globe doesn't appear to have a smooth surface because the blocks are relatively large compared to the globe. Later in this chapter, we show you how to construct curved lines and surfaces from lots of small geometric primitives.

This chapter has the following major sections:

- "A Drawing Survival Kit" explains how to clear the window and force
drawing to be completed. It also gives you basic information about controlling
the color of geometric objects and about hidden-surface removal.

"Describing Points, Lines, and Polygons" shows you what the set of primitive
geometric objects is and how to draw them.

"Displaying Points, Lines, and Polygons" explains what control you have
over the details of how primitives are drawn - for example, what diameter
points have, whether lines are solid or dashed, and whether polygons are
outlined or filled.

"Normal Vectors" discusses how to specify normal vectors for geometric
objects and (briefly) what these vectors are for.

"Some Hints for Building Polygonal Models of Surfaces" explores the issues and techniques involved in constructing polygonal approximations to surfaces.

At this point, you might be wondering why we keep talking about *clearing*
the window - why not just draw a rectangle of the appropriate color that's
large enough to cover the entire window? First, a special command to clear
a window can be much more efficient than a general-purpose drawing command.
In addition, as you'll see in Chapter 3 , OpenGL allows you to set the
coordinate system, viewing position, and viewing direction arbitrarily,
so it might be difficult to figure out an appropriate size and location
for a window-clearing rectangle. Also, you can have OpenGL use hidden-surface
removal techniques that eliminate objects obscured by others nearer to
the eye; thus, if the window-clearing rectangle is to be a background,
you must make sure that it's behind all the other objects of interest.
With an arbitrary coordinate system and point of view, this might be difficult.
Finally, on many machines, the graphics hardware consists of multiple buffers
in addition to the buffer containing colors of the pixels that are displayed.
These other buffers must be cleared from time to time, and it's convenient
to have a single command that can clear any combination of them. (All the
possible buffers are discussed in Chapter 10 .)

As an example, these lines of code clear the window to black:

glClearColor(0.0, 0.0, 0.0, 0.0); glClear(GL_COLOR_BUFFER_BIT);The first line sets the clearing color to black, and the next command clears the entire window to the current clearing color. The single parameter to

Chapter 5 and Chapter 10 talk about how other buffers are used. For now, all you need to know is that clearing them is simple. For example, to clear both the color buffer and the depth buffer, you would use the following sequence of commands:

glClearColor(0.0, 0.0, 0.0, 0.0); glClearDepth(0.0); glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);In this case, the call to

Sets the current clearing color for use in clearing color buffers in
RGBA mode. For more information on RGBA mode, see Chapter 5 . The **red**,
**green**, **blue**, and **alpha** values are clamped if necessary
to the range [0,1]. The default clearing color is (0, 0, 0, 0), which is
black.

void **glClear**(GLbitfield **mask**);

Clears the specified buffers to their current clearing values. The *mask*
argument is a bitwise-ORed combination of the values listed in Table 2-1
.

Buffer | Name | Reference |
---|---|---|

Color buffer | GL_COLOR_BUFFER_BIT | Chapter 5 |

Depth buffer | GL_DEPTH_BUFFER_BIT | Chapter 10 |

Accumulation buffer | GL_ACCUM_BUFFER_BIT | Chapter 10 |

Stencil buffer | GL_STENCIL_BUFFER_BIT | Chapter 10 |

Before issuing a command to clear multiple buffers, you have to set
the values to which each buffer is to be cleared if you want something
other than the default color, depth value, accumulation color, and stencil
index. In addition to the **glClearColor()** and **glClearDepth()**
commands that set the current values for clearing the color and depth buffers,
**glClearIndex()**,
**glClearAccum()**, and **glClearStencil()** specify the color index,
accumulation color, and stencil index used to clear the corresponding buffers.
See Chapter 5 and Chapter 10 for descriptions of these buffers and their
uses.

OpenGL allows you to specify multiple buffers because clearing is generally a slow operation, since every pixel in the window (possibly millions) is touched, and some graphics hardware allows sets of buffers to be cleared simultaneously. Hardware that doesn't support simultaneous clears performs them sequentially. The difference between

glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);and

glClear(GL_COLOR_BUFFER_BIT); glClear(GL_DEPTH_BUFFER_BIT);is that although both have the same final effect, the first example might run faster on many machines. It certainly won't run more slowly.

For example, the pseudocode

set_current_color(red); draw_object(A); draw_object(B); set_current_color(green); set_current_color(blue); draw_object(C);draws objects A and B in red, and object C in blue. The command on the fourth line that sets the current color to green is wasted.

Coloring, lighting, and shading are all large topics with entire chapters or large sections devoted to them. To draw geometric primitives that can be seen, however, you need some basic knowledge of how to set the current color; this information is provided in the next paragraphs. For details on these topics, see Chapter 5 and Chapter 6 .

To set a color, use the command **glColor3f()**. It takes three parameters,
all of which are floating-point numbers between 0.0 and 1.0. The parameters
are, in order, the red, green, and blue components of the color. You can
think of these three values as specifying a "mix" of colors: 0.0 means
don't use any of that component, and 1.0 means use all you can of that
component. Thus, the code

glColor3f(1.0, 0.0, 0.0);makes the brightest red the system can draw, with no green or blue components. All zeros makes black; in contrast, all ones makes white. Setting all three components to 0.5 yields gray (halfway between black and white). Here are eight commands and the colors they would set:

glColor3f(0.0, 0.0, 0.0); black glColor3f(1.0, 0.0, 0.0); red glColor3f(0.0, 1.0, 0.0); green glColor3f(1.0, 1.0, 0.0); yellow glColor3f(0.0, 0.0, 1.0); blue glColor3f(1.0, 0.0, 1.0); magenta glColor3f(0.0, 1.0, 1.0); cyan glColor3f(1.0, 1.0, 1.0); whiteYou might have noticed earlier that when you're setting the color to clear the color buffer,

In addition, the application might be running on more than one machine. For example, suppose that the main program is running elsewhere (on a machine called the client), and that you're viewing the results of the drawing on your workstation or terminal (the server), which is connected by a network to the client. In that case, it might be horribly inefficient to send each command over the network one at a time, since considerable overhead is often associated with each network transmission. Usually, the client gathers a collection of commands into a single network packet before sending it. Unfortunately, the network code on the client typically has no way of knowing that the graphics program is finished drawing a frame or scene. In the worst case, it waits forever for enough additional drawing commands to fill a packet, and you never see the completed drawing.

For this reason, OpenGL provides the command **glFlush()**, which
forces the client to send the network packet even though it might not be
full. Where there is no network and all commands are truly executed immediately
on the server, **glFlush()** might have no effect. However, if you're
writing a program that you want to work properly both with and without
a network, include a call to **glFlush()** at the end of each frame
or scene. Note that **glFlush()** doesn't wait for the drawing to complete
- it just forces the drawing to begin execution, thereby guaranteeing that
all previous commands execute in finite time even if no further rendering
commands are executed.

A few commands - for example, commands that swap buffers in double-buffer
mode - automatically flush pending commands onto the network before they
can occur. void **glFlush**(void);

Forces previously issued OpenGL commands to begin execution, thus guaranteeing that they complete in finite time.

If **glFlush()** isn't sufficient for you, try **glFinish()**.
This command flushes the network as **glFlush()** does and then waits
for notification from the graphics hardware or network indicating that
the drawing is complete in the framebuffer. You might need to use **glFinish()**
if you want to synchronize tasks - for example, to make sure that your
three-dimensional rendering is on the screen before you use Display PostScript
to draw labels on top of the rendering. Another example would be to ensure
that the drawing is complete before it begins to accept user input. After
you issue a **glFinish()** command, your graphics process is blocked
until it receives notification from the graphics hardware (or client, if
you're running over a network) that the drawing is complete. Keep in mind
that excessive use of **glFinish()** can reduce the performance of your
application, especially if you're running over a network, because it requires
round-trip communication. If **glFlush()** is sufficient for your needs,
use it instead of **glFinish()**.void **glFinish**(void);

Forces all previously issued OpenGL commands to complete. This command doesn't return until all effects from previous commands are fully realized.

while (1) { get_viewing_point_from_mouse_position(); glClear(GL_COLOR_BUFFER_BIT); draw_3d_object_A(); draw_3d_object_B(); }it might be that for some mouse positions, object A obscures object B, and for others, the opposite relationship might hold. If nothing special is done, the preceding code always draws object B second, and thus on top of object A, no matter what viewing position is selected.

The elimination of parts of solid objects that are obscured by others
is called *hidden-surface removal*. (Hidden-line removal, which does
the same job for objects represented as wireframe skeletons, is a bit trickier,
and it isn't discussed here. See "Hidden-Line Removal," for details.) The
easiest way to achieve hidden-surface removal is to use the depth buffer
(sometimes called a z-buffer). (Also see Chapter 10 .)

A depth buffer works by associating a depth, or distance from the viewpoint,
with each pixel on the window. Initially, the depth values for all pixels
are set to the largest possible distance using the **glClear()** command
with GL_DEPTH_BUFFER_BIT, and then the objects in the scene are drawn in
any order.

Graphical calculations in hardware or software convert each surface that's drawn to a set of pixels on the window where the surface will appear if it isn't obscured by something else. In addition, the distance from the eye is computed. With depth buffering enabled, before each pixel is drawn, a comparison is done with the depth value already stored at the pixel. If the new pixel is closer to the eye than what's there, the new pixel's color and depth values replace those that are currently written into the pixel. If the new pixel's depth is greater than what's currently there, the new pixel would be obscured, and the color and depth information for the incoming pixel is discarded. Since information is discarded rather than used for drawing, hidden-surface removal can increase your performance.

To use depth buffering, you need to enable depth buffering. This has to be done only once. Each time you draw the scene, before drawing you need to clear the depth buffer and then draw the objects in the scene in any order.

To convert the preceding program fragment so that it performs hidden-surface removal, modify it to the following:

glEnable(GL_DEPTH_TEST); ... while (1) { glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT); get_viewing_point_from_mouse_position(); draw_3d_object_A(); draw_3d_object_B(); }The argument to

For example, one difference comes from the limitations of computer-based calculations. In any OpenGL implementation, floating-point calculations are of finite precision, and they have round-off errors. Consequently, the coordinates of OpenGL points, lines, and polygons suffer from the same problems.

Another difference arises from the limitations of a bitmapped graphics display. On such a display, the smallest displayable unit is a pixel, and although pixels might be less than 1/100th of an inch wide, they are still much larger than the mathematician's infinitely small (for points) or infinitely thin (for lines). When OpenGL performs calculations, it assumes points are represented as vectors of floating-point numbers. However, a point is typically (but not always) drawn as a single pixel, and many different points with slightly different coordinates could be drawn by OpenGL on the same pixel.

**Advanced**

OpenGL works in the homogeneous coordinates of three-dimensional projective
geometry, so for internal calculations, all vertices are represented with
four floating-point coordinates (*x*, *y*, *z*, *w*).
If *w* is different from zero, these coordinates correspond to the
euclidean three-dimensional point (*x/w, y/w, z/w*). You can specify
the *w* coordinate in OpenGL commands, but that's rarely done. If
the *w* coordinate isn't specified, it's understood to be 1.0. For
more information about homogeneous coordinate systems, see Appendix G .

**Figure 2-1 : **Two Connected Series of Line Segments

In general, polygons can be complicated, so OpenGL makes some strong
restrictions on what constitutes a primitive polygon. First, the edges
of OpenGL polygons can't intersect (a mathematician would call this a *simple
polygon*). Second, OpenGL polygons must be convex, meaning that they
cannot have indentations. Stated precisely, a region is convex if, given
any two points in the interior, the line segment joining them is also in
the interior. See Figure 2-2 for some examples of valid and invalid polygons.
OpenGL, however, doesn't restrict the number of line segments making up
the boundary of a convex polygon. Note that polygons with holes can't be
described. They are nonconvex, and they can't be drawn with a boundary
made up of a single closed loop. Be aware that if you present OpenGL with
a nonconvex filled polygon, it might not draw it as you expect. For instance,
on most systems no more than the convex hull of the polygon would be filled,
but on some systems, less than the convex hull might be filled.

**Figure 2-2 : **Valid and Invalid Polygons

For many applications, you need nonsimple polygons, nonconvex polygons,
or polygons with holes. Since all such polygons can be formed from unions
of simple convex polygons, some routines to describe more complex objects
are provided in the GLU. These routines take complex descriptions and *tessellate*
them, or break them down into groups of the simpler OpenGL polygons that
can then be rendered. (See Appendix C for more information about the tessellation
routines.) The reason for OpenGL's restrictions on valid polygon types
is that it's simpler to provide fast polygon-rendering hardware for that
restricted class of polygons.

Since OpenGL vertices are always three-dimensional, the points forming
the boundary of a particular polygon don't necessarily lie on the same
plane in space. (Of course, they do in many cases - if all the *z*
coordinates are zero, for example, or if the polygon is a triangle.) If
a polygon's vertices don't lie in the same plane, then after various rotations
in space, changes in the viewpoint, and projection onto the display screen,
the points might no longer form a simple convex polygon. For example, imagine
a four-point quadrilateral where the points are slightly out of plane,
and look at it almost edge-on. You can get a nonsimple polygon that resembles
a bow tie, as shown in Figure 2-3 , which isn't guaranteed to render correctly.
This situation isn't all that unusual if you approximate surfaces by quadrilaterals
made of points lying on the true surface. You can always avoid the problem
by using triangles, since any three points always lie on a plane.

**Figure 2-3 : **Nonplanar Polygon Transformed to Nonsimple Polygon

void

Draws the rectangle defined by the corner points (*x1, y1*) and
(*x2, y2*). The rectangle lies in the plane *z*=0 and has sides
parallel to the *x*- and *y*-axes. If the vector form of the
function is used, the corners are given by two pointers to arrays, each
of which contains an (*x, y*) pair.

Note that although the rectangle begins with a particular orientation
in three-dimensional space (in the *x-y* plane and parallel to the
axes), you can change this by applying rotations or other transformations.
See Chapter 3 for information about how to do this.

**Figure 2-4 : **Approximating Curves

Even though curves aren't geometric primitives, OpenGL does provide some direct support for drawing them. See Chapter 11 for information about how to draw curves and curved surfaces.

Specifies a vertex for use in describing a geometric object. You can
supply up to four coordinates (*x, y, z, w*) for a particular vertex
or as few as two (*x, y*) by selecting the appropriate version of
the command. If you use a version that doesn't explicitly specify *z*
or *w*, *z* is understood to be 0 and *w* is understood
to be 1. Calls to **glVertex*()** should be executed between a **glBegin()**
and **glEnd()** pair.

Here are some examples of using **glVertex*()**:

glVertex2s(2, 3); glVertex3d(0.0, 0.0, 3.1415926535898); glVertex4f(2.3, 1.0, -2.2, 2.0); GLdouble dvect[3] = {5.0, 9.0, 1992.0}; glVertex3dv(dvect);The first example represents a vertex with three-dimensional coordinates (2, 3, 0). (Remember that if it isn't specified, the

On some machines, the vector form of **glVertex*()** is more efficient,
since only a single parameter needs to be passed to the graphics subsystem,
and special hardware might be able to send a whole series of coordinates
in a single batch. If your machine is like this, it's to your advantage
to arrange your data so that the vertex coordinates are packed sequentially
in memory.

glBegin(GL_POLYGON); glVertex2f(0.0, 0.0); glVertex2f(0.0, 3.0); glVertex2f(3.0, 3.0); glVertex2f(4.0, 1.5); glVertex2f(3.0, 0.0); glEnd();

**Figure 2-5 : **Drawing a Polygon or a Set of Points

If you had used GL_POINTS instead of GL_POLYGON, the primitive would
have been simply the five points shown in Figure 2-5 . Table 2-2 in the
following function summary for **glBegin()** lists the ten possible
arguments and the corresponding type of primitive.void **glBegin**(GLenum
*mode*);

Marks the beginning of a vertex list that describes a geometric primitive.
The type of primitive is indicated by *mode*, which can be any of
the values shown in Table 2-2 .

Value | Meaning |
---|---|

GL_POINTS | individual points |

GL_LINES | pairs of vertices interpreted as individual line segments |

GL_POLYGON | boundary of a simple, convex polygon |

GL_TRIANGLES | triples of vertices interpreted as triangles |

GL_QUADS | quadruples of vertices interpreted as four-sided polygons |

GL_LINE_STRIP | series of connected line segments |

GL_LINE_LOOP | same as above, with a segment added between last and first vertices |

GL_TRIANGLE_STRIP | linked strip of triangles |

GL_TRIANGLE_FAN | linked fan of triangles |

GL_QUAD_STRIP | linked strip of quadrilaterals |

void **glEnd**(void);

Marks the end of a vertex list.

Figure 2-6 shows examples of all the geometric primitives listed in Table 2-2 . The paragraphs that follow the figure give precise descriptions of the pixels that are drawn for each of the objects. Note that in addition to points, several types of lines and polygons are defined. Obviously, you can find many ways to draw the same primitive. The method you choose depends on your vertex data.

**Figure 2-6 : **Geometric Primitive Types

As you read the following descriptions, assume that *n* vertices
(v0, v1, v2, ... , vn-1) are described between a **glBegin()** and **glEnd()**
pair.

- GL_POINTS

Draws a point at each of the

GL_LINES

Draws a series of unconnected line segments. Segments are drawn between v0 and v1, between v2 and v3, and so on. If

GL_POLYGON

Draws a polygon using the points v0, ... , vn-1 as vertices.

GL_TRIANGLES

Draws a series of triangles (three-sided polygons) using vertices v0, v1, v2, then v3, v4, v5, and so on. If

GL_LINE_STRIP

Draws a line segment from v0 to v1, then from v1 to v2, and so on, finally drawing the segment from vn-2 to vn-1. Thus, a total of

GL_LINE_LOOP

Same as GL_LINE_STRIP, except that a final line segment is drawn from vn-1 to v0, completing a loop.

GL_QUADS

Draws a series of quadrilaterals (four-sided polygons) using vertices v0, v1, v2, v3, then v4, v5, v6, v7, and so on. If

GL_QUAD_STRIP

Draws a series of quadrilaterals (four-sided polygons) beginning with v0, v1, v3, v2, then v2, v3, v5, v4, then v4, v5, v7, v6, and so on. See Figure 2-6 .

GL_TRIANGLE_STRIP

Draws a series of triangles (three-sided polygons) using vertices v0, v1, v2, then v2, v1, v3 (note the order), then v2, v3, v4, and so on. The ordering is to ensure that the triangles are all drawn with the same orientation so that the strip can correctly form part of a surface. Figure 2-6 should make the reason for the ordering obvious.

GL_TRIANGLE_FAN

Same as GL_TRIANGLE_STRIP, except that the vertices are v0, v1, v2, then v0, v2, v3, then v0, v3, v4, and so on. Look at Figure 2-6 .

Command | Purpose of Command | Reference |
---|---|---|

glVertex*() | set vertex coordinates | Chapter 2 |

glColor*() | set current color | Chapter 5 |

glIndex*() | set current color index | Chapter 5 |

glNormal*() | set normal vector coordinates | Chapter 2 |

glEvalCoord*() | generate coordinates | Chapter 11 |

glCallList(), glCallLists() | execute display list(s) | Chapter 4 |

glTexCoord*() | set texture coordinates | Chapter 9 |

glEdgeFlag*() | control drawing of edges | Chapter 2 |

glMaterial*() | set material properties | Chapter 6 |

No other OpenGL commands are valid between a **glBegin()** and **glEnd()**
pair, and making any other OpenGL call generates an error. Note, however,
that only OpenGL commands are restricted; you can certainly include other
programming-language constructs. For example, the following code draws
an outlined circle:

#define PI 3.1415926535897; GLint circle_points = 100; glBegin(GL_LINE_LOOP); for (i = 0; i < circle_points; i++) { angle = 2*PI*i/circle_points; glVertex2f(cos(angle), sin(angle)); } glEnd();This example isn't the most efficient way to draw a circle, especially if you intend to do it repeatedly. The graphics commands used are typically very fast, but this code calculates an angle and calls the

Unless they are being compiled into a display list, all **glVertex*()**
commands should appear between some **glBegin()** and **glEnd()**
combination. (If they appear elsewhere, they don't accomplish anything.)
If they appear in a display list, they are executed only if they appear
between a **glBegin()** and a **glEnd()**.

Although many commands are allowed between **glBegin()** and **glEnd()**,
vertices are generated only when a **glVertex*()** command is issued.
At the moment **glVertex*()** is called, OpenGL assigns the resulting
vertex the current color, texture coordinates, normal vector information,
and so on. To see this, look at the following code sequence. The first
point is drawn in red, and the second and third ones in blue, despite the
extra color commands:

glBegin(GL_POINTS); glColor3f(0.0, 1.0, 0.0); /* green */ glColor3f(1.0, 0.0, 0.0); /* red */ glVertex(...); glColor3f(1.0, 1.0, 0.0); /* yellow */ glColor3f(0.0, 0.0, 1.0); /* blue */ glVertex(...); glVertex(...); glEnd();You can use any combination of the twenty-four versions of the

Sets the width in pixels for rendered points; *size* must be greater
than 0.0 and by default is 1.0.

The actual collection of pixels on the screen that are drawn for various point widths depends on whether antialiasing is enabled. (Antialiasing is a technique for smoothing points and lines as they're rendered. This topic is covered in detail in "Antialiasing." ) If antialiasing is disabled (the default), fractional widths are rounded to integer widths, and a screen-aligned square region of pixels is drawn. Thus, if the width is 1.0, the square is one pixel by one pixel; if the width is 2.0, the square is two pixels by two pixels, and so on.

With antialiasing enabled, a circular group of pixels is drawn, and the pixels on the boundaries are typically drawn at less than full intensity to give the edge a smoother appearance. In this mode, nonintegral widths aren't rounded.

Most OpenGL implementations support very large point sizes. A particular
implementation, however, might limit the size of nonantialiased points
to its maximum antialiased point size, rounded to the nearest integer value.
You can obtain this floating-point value by using GL_POINT_SIZE_RANGE with
**glGetFloatv()**.

Sets the width in pixels for rendered lines; *width* must be greater
than 0.0 and by default is 1.0.

The actual rendering of lines is affected by the antialiasing mode,
in the same way as for points. (See "Antialiasing." ) Without antialiasing,
widths of 1, 2, and 3 draw lines one, two, and three pixels wide. With
antialiasing enabled, nonintegral line widths are possible, and pixels
on the boundaries are typically partially filled. As with point sizes,
a particular OpenGL implementation might limit the width of nonantialiased
lines to its maximum antialiased line width, rounded to the nearest integer
value. You can obtain this floating-point value by using GL_LINE_WIDTH_RANGE
with **glGetFloatv()**.

Keep in mind that by default lines are one pixel wide, so they appear wider on lower-resolution screens. For computer displays, this isn't typically an issue, but if you're using OpenGL to render to a high-resolution plotter, one-pixel lines might be nearly invisible. To obtain resolution-independent line widths, you need to take into account the physical dimensions of pixels.

**Advanced**

With nonantialiased wide lines, the line width isn't measured perpendicular
to the line. Instead, it's measured in the *y* direction if the absolute
value of the slope is less than 1.0; otherwise, it's measured in the *x*
direction. The rendering of an antialiased line is exactly equivalent to
the rendering of a filled rectangle of the given width, centered on the
exact line. See "Polygon Details," for a discussion of the rendering of
filled polygonal regions.

glLineStipple(1, 0x3F07); glEnable(GL_LINE_STIPPLE);void

Sets the current stippling pattern for lines. The *pattern* argument
is a 16-bit series of 0s and 1s, and it's repeated as necessary to stipple
a given line. A 1 indicates that drawing occurs, and 0 that it does not,
on a pixel-by-pixel basis, beginning with the low-order bits of the pattern.
The pattern can be stretched out by using *factor*, which multiplies
each subseries of consecutive 1s and 0s. Thus, if three consecutive 1s
appear in the pattern, they're stretched to six if *factor* is 2.
*factor*
is clamped to lie between 1 and 255. Line stippling must be enabled by
passing GL_LINE_STIPPLE to **glEnable()**; it's disabled by passing
the same argument to **glDisable()**.

With the preceding example and the pattern 0x3F07 (which translates
to 0011111100000111 in binary), a line would be drawn with 3 pixels on,
then 5 off, 6 on, and 2 off. (If this seems backward, remember that the
low-order bits are used first.) If *factor* had been 2, the pattern
would have been elongated: 6 pixels on, 10 off, 12 on, and 4 off. Figure
2-7 shows lines drawn with different patterns and repeat factors. If you
don't enable line stippling, drawing proceeds as if *pattern* were
0xFFFF and *factor* 1. (Use **glDisable()** with GL_LINE_STIPPLE
to disable stippling.) Note that stippling can be used in combination with
wide lines to produce wide stippled lines.

**Figure 2-7 : **Stippled Lines

One way to think of the stippling is that as the line is being drawn,
the pattern is shifted by one bit each time a pixel is drawn (or *factor*
pixels are drawn, if *factor* isn't 1). When a series of connected
line segments is drawn between a single **glBegin()** and **glEnd()**,
the pattern continues to shift as one segment turns into the next. This
way, a stippling pattern continues across a series of connected line segments.
When **glEnd()** is executed, the pattern is reset, and - if more lines
are drawn before stippling is disabled - the stippling restarts at the
beginning of the pattern. If you're drawing lines with GL_LINES, the pattern
resets for each independent line.

Example 2-1 illustrates the results of drawing with a couple of different stipple patterns and line widths. It also illustrates what happens if the lines are drawn as a series of individual segments instead of a single connected line strip. The results of running the program appear in Figure 2-8 .

**Figure 2-8 : **Wide Stippled Lines

**Example 2-1 : **Using Line Stipple Patterns: lines.c

#include <GL/gl.h> #include <GL/glu.h> #include "aux.h" #define drawOneLine(x1,y1,x2,y2) glBegin(GL_LINES); \ glVertex2f ((x1),(y1)); glVertex2f ((x2),(y2)); glEnd(); void myinit (void) { /* background to be cleared to black */ glClearColor (0.0, 0.0, 0.0, 0.0); glShadeModel (GL_FLAT); } void display(void) { int i; glClear (GL_COLOR_BUFFER_BIT); /* draw all lines in white */ glColor3f (1.0, 1.0, 1.0); /* in 1st row, 3 lines, each with a different stipple */ glEnable (GL_LINE_STIPPLE); glLineStipple (1, 0x0101); /* dotted */ drawOneLine (50.0, 125.0, 150.0, 125.0); glLineStipple (1, 0x00FF); /* dashed */ drawOneLine (150.0, 125.0, 250.0, 125.0); glLineStipple (1, 0x1C47); /* dash/dot/dash */ drawOneLine (250.0, 125.0, 350.0, 125.0); /* in 2nd row, 3 wide lines, each with different stipple */ glLineWidth (5.0); glLineStipple (1, 0x0101); drawOneLine (50.0, 100.0, 150.0, 100.0); glLineStipple (1, 0x00FF); drawOneLine (150.0, 100.0, 250.0, 100.0); glLineStipple (1, 0x1C47); drawOneLine (250.0, 100.0, 350.0, 100.0); glLineWidth (1.0); /* in 3rd row, 6 lines, with dash/dot/dash stipple, */ /* as part of a single connected line strip */ glLineStipple (1, 0x1C47); glBegin (GL_LINE_STRIP); for (i = 0; i < 7; i++) glVertex2f (50.0 + ((GLfloat) i * 50.0), 75.0); glEnd (); /* in 4th row, 6 independent lines, */ /* with dash/dot/dash stipple */ for (i = 0; i < 6; i++) { drawOneLine (50.0 + ((GLfloat) i * 50.0), 50.0, 50.0 + ((GLfloat)(i+1) * 50.0), 50.0); } /* in 5th row, 1 line, with dash/dot/dash stipple */ /* and repeat factor of 5 */ glLineStipple (5, 0x1C47); drawOneLine (50.0, 25.0, 350.0, 25.0); glFlush (); } int main(int argc, char** argv) { auxInitDisplayMode (AUX_SINGLE | AUX_RGBA); auxInitPosition (0, 0, 400, 150); auxInitWindow (argv[0]); myinit (); auxMainLoop(display); }

Controls the drawing mode for a polygon's front and back faces. The
parameter *face* can be GL_FRONT_AND_BACK, GL_FRONT, or GL_BACK; *mode*
can be GL_POINT, GL_LINE, or GL_FILL to indicate whether the polygon should
be drawn as points, outlined, or filled. By default, both the front and
back faces are drawn filled.

For example, you can have the front faces filled and the back faces outlined with two calls to this routine:

glPolygonMode(GL_FRONT, GL_FILL); glPolygonMode(GL_BACK, GL_LINE);See the next section for more information about how to control which faces are considered front-facing and which back-facing.

Suppose you've consistently described a model of an orientable surface
but that you happen to have the clockwise orientation on the outside. You
can swap what OpenGL considers the back face by using the function **glFrontFace()**,
supplying the desired orientation for front-facing polygons.void **glFrontFace**(GLenum
*mode*);

Controls how front-facing polygons are determined. By default, *mode*
is GL_CCW, which corresponds to a counterclockwise orientation of the ordered
vertices of a projected polygon in window coordinates. If *mode* is
GL_CW, faces with a clockwise orientation are considered front-facing.

**Advanced**

In more technical terms, the decision of whether a face of a polygon is front- or back-facing depends on the sign of the polygon's area computed in window coordinates. One way to compute this area is

where *x*i and *y*i are the *x* and *y* window coordinates
of the *i*th vertex of the *n*-vertex polygon and:

Assuming that GL_CCW has been specified, if *a*>0, the polygon
corresponding to that vertex is considered to be front-facing; otherwise,
it's back-facing. If GL_CW is specified and if *a*<0, then the
corresponding polygon is front-facing; otherwise, it's back-facing.

In a completely enclosed surface constructed from polygons with a consistent
orientation, none of the back-facing polygons are ever visible - they're
always obscured by the front-facing polygons. In this situation, you can
maximize drawing speed by having OpenGL discard polygons as soon as it
determines that they're back-facing. Similarly, if you are inside the object,
only back-facing polygons are visible. To instruct OpenGL to discard front-
or back-facing polygons, use the command **glCullFace()** and enable
culling with **glEnable()**. void **glCullFace**(GLenum *mode*);

Indicates which polygons should be discarded (culled) before they're
converted to screen coordinates. The mode is either GL_FRONT, GL_BACK,
or GL_FRONT_AND_BACK to indicate front-facing, back-facing, or all polygons.
To take effect, culling must be enabled using **glEnable()** with GL_CULL_FACE;
it can be disabled with **glDisable()** and the same argument.

Defines the current stipple pattern for filled polygons. The argument
*mask*
is a pointer to a 32 × 32 bitmap that's interpreted as a mask of 0s and
1s. Where a 1 appears, the corresponding pixel in the polygon is drawn,
and where a 0 appears, nothing is drawn. Figure 2-9 shows how a stipple
pattern is constructed from the characters in *mask*. Polygon stippling
is enabled and disabled by using **glEnable()** and **glDisable()**
with GL_POLYGON_STIPPLE as the argument. The interpretation of the **mask**
data is affected by the **glPixelStore*()** GL_UNPACK* modes. See "Controlling
Pixel-Storage Modes."

**Figure 2-9 : **Constructing a Polygon Stipple Pattern

In addition to defining the current polygon stippling pattern, you must enable stippling:

glEnable(GL_POLYGON_STIPPLE);Use

Figure 2-10 shows the results of polygons drawn unstippled and then with two different stippling patterns. The program is shown in Example 2-2 . The reversal of white to black (from Figure 2-9 to Figure 2-10 ) occurs because the program draws in white over a black background, using the pattern in Figure 2-9 as a stencil.

**Figure 2-10 : **Stippled Polygons

**Example 2-2 : **Using Polygon Stipple Patterns: polys.c

#include <GL/gl.h> #include <GL/glu.h> #include "aux.h" void display(void) { GLubyte fly[] = { 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x03, 0x80, 0x01, 0xC0, 0x06, 0xC0, 0x03, 0x60, 0x04, 0x60, 0x06, 0x20, 0x04, 0x30, 0x0C, 0x20, 0x04, 0x18, 0x18, 0x20, 0x04, 0x0C, 0x30, 0x20, 0x04, 0x06, 0x60, 0x20, 0x44, 0x03, 0xC0, 0x22, 0x44, 0x01, 0x80, 0x22, 0x44, 0x01, 0x80, 0x22, 0x44, 0x01, 0x80, 0x22, 0x44, 0x01, 0x80, 0x22, 0x44, 0x01, 0x80, 0x22, 0x44, 0x01, 0x80, 0x22, 0x66, 0x01, 0x80, 0x66, 0x33, 0x01, 0x80, 0xCC, 0x19, 0x81, 0x81, 0x98, 0x0C, 0xC1, 0x83, 0x30, 0x07, 0xe1, 0x87, 0xe0, 0x03, 0x3f, 0xfc, 0xc0, 0x03, 0x31, 0x8c, 0xc0, 0x03, 0x33, 0xcc, 0xc0, 0x06, 0x64, 0x26, 0x60, 0x0c, 0xcc, 0x33, 0x30, 0x18, 0xcc, 0x33, 0x18, 0x10, 0xc4, 0x23, 0x08, 0x10, 0x63, 0xC6, 0x08, 0x10, 0x30, 0x0c, 0x08, 0x10, 0x18, 0x18, 0x08, 0x10, 0x00, 0x00, 0x08}; GLubyte halftone[] = { 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55, 0xAA, 0xAA, 0xAA, 0xAA, 0x55, 0x55, 0x55, 0x55}; glClear (GL_COLOR_BUFFER_BIT); glColor3f (1.0, 1.0, 1.0); glRectf (25.0, 25.0, 125.0, 125.0); glEnable (GL_POLYGON_STIPPLE); glPolygonStipple (fly); glRectf (125.0, 25.0, 225.0, 125.0); glPolygonStipple (halftone); glRectf (225.0, 25.0, 325.0, 125.0); glDisable (GL_POLYGON_STIPPLE); glFlush (); } void myinit (void) { glClearColor (0.0, 0.0, 0.0, 0.0); glShadeModel (GL_FLAT); } int main(int argc, char** argv) { auxInitDisplayMode (AUX_SINGLE | AUX_RGBA); auxInitPosition (0, 0, 350, 150); auxInitWindow (argv[0]); myinit (); auxMainLoop(display); }As mentioned in "Display-List Design Philosophy," you might want to use display lists to store polygon stipple patterns to maximize efficiency.

OpenGL can render only convex polygons, but many nonconvex polygons
arise in practice. To draw these nonconvex polygons, you typically subdivide
them into convex polygons - usually triangles, as shown in Figure 2-11
- and then draw the triangles. Unfortunately, if you decompose a general
polygon into triangles and draw the triangles, you can't really use **glPolygonMode()**
to draw the polygon's outline, since you get all the triangle outlines
inside it. To solve this problem, you can tell OpenGL whether a particular
vertex precedes a boundary edge; OpenGL keeps track of this information
by passing along with each vertex a bit indicating whether that vertex
is followed by a boundary edge. Then, when a polygon is drawn in GL_LINE
mode, the nonboundary edges aren't drawn. In Figure 2-11 , the dashed lines
represent added edges.

**Figure 2-11 : **Subdividing a Nonconvex Polygon

By default, all vertices are marked as preceding a boundary edge, but
you can manually control the setting of the edge flag with the command
**glEdgeFlag*()**.
This command is used between **glBegin()** and
**glEnd()** pairs,
and it affects all the vertices specified after it until the next **glEdgeFlag()**
call is made. It applies only to vertices specified for polygons, triangles,
and quads, not to those specified for strips of triangles or quads. void
**glEdgeFlag**(GLboolean *flag*);

void **glEdgeFlagv**(const GLboolean ***flag**);

Indicates whether a vertex should be considered as initializing a boundary
edge of a polygon. If **flag** is GL_TRUE, the edge flag is set to TRUE
(the default), and any vertices created are considered to precede boundary
edges until this function is called again with **flag** being 0.

As an example, Example 2-3 draws the outline shown in Figure 2-12 .

**Figure 2-12 : **An Outlined Polygon Drawn Using Edge Flags

**Example 2-3 : **Marking Polygon Boundary Edges

glPolygonMode(GL_FRONT_AND_BACK, GL_LINE); glBegin(GL_POLYGON); glEdgeFlag(GL_TRUE); glVertex3fv(V0); glEdgeFlag(GL_FALSE); glVertex3fv(V1); glEdgeFlag(GL_TRUE); glVertex3fv(V2); glEnd();

An object's normal vectors define the orientation of its surface in space - in particular, its orientation relative to light sources. These vectors are used by OpenGL to determine how much light the object receives at its vertices. Lighting - a large topic by itself - is the subject of Chapter 6 , and you might want to review the following information after you've read that chapter. Normal vectors are discussed briefly here because you generally define normal vectors for an object at the same time you define the object's geometry.

You use **glNormal*()** to set the current normal to the value of
the argument passed in. Subsequent calls to **glVertex*()** cause the
specified vertices to be assigned the current normal. Often, each vertex
has a different normal, which necessitates a series of alternating calls
like this:

glBegin (GL_POLYGON); glNormal3fv(n0); glVertex3fv(v0); glNormal3fv(n1); glVertex3fv(v1); glNormal3fv(n2); glVertex3fv(v2); glNormal3fv(n3); glVertex3fv(v3); glEnd();void

void

Sets the current normal vector as specified by the arguments. The nonvector
version (without the **v**) takes three arguments, which specify an
(*nx, ny, nz*) vector that's taken to be the normal. Alternatively,
you can use the vector version of this function (with the **v**) and
supply a single array of three elements to specify the desired normal.
The **b**, **s**, and **i** versions scale their parameter values
linearly to the range [-1.0,1.0].

There's no magic to finding the normals for an object - most likely, you have to perform some calculations that might include taking derivatives - but there are several techniques and tricks you can use to achieve certain effects. Appendix F explains how to find normal vectors for surfaces. If you already know how to do this, if you can count on always being supplied with normal vectors, or if you don't want to use OpenGL's lighting facility, you don't need to read this appendix.

Note that at a given point on a surface, two vectors are perpendicular
to the surface, and they point in opposite directions. By convention, the
normal is the one that points to the outside of the surface being modeled.
(If you get inside and outside reversed in your model, just change every
normal vector from (*x, y, z*) to (-*x, *-*y, *-*z*)).

Also, keep in mind that since normal vectors indicate direction only,
their length is mostly irrelevant. You can specify normals of any length,
but eventually they have to be converted to having a length of 1 before
lighting calculations are performed. (A vector that has a length of 1 is
said to be of unit length, or normalized.) In general, then, you should
supply normalized normal vectors. These vectors remain normalized as long
as your model transformations include only rotations and translations.
(Transformations are discussed in detail in Chapter 3 .) If you perform
irregular transformations (such as scaling or multiplying by a shear matrix),
or if you specify nonunit-length normals, then you should have OpenGL automatically
normalize your normal vectors after the transformations. To do this, call
**glEnable()** with GL_NORMALIZE as its argument. By default, automatic
normalization is disabled. Note that in some implementations of OpenGL,
automatic normalization requires additional calculations that might reduce
the performance of your application.

Constructing polygonal approximations to surfaces is an art, and there is no substitute for experience. This section, however, lists a few pointers that might make it a bit easier to get started.

- Keep polygon orientations consistent. Make sure that when viewed from
the outside, all the polygons on the surface are oriented in the same direction
(all clockwise or all counterclockwise). Try to get this right the first
time, since it's excruciatingly painful to fix the problem later.

When you subdivide a surface, watch out for any nontriangular polygons.
The three vertices of a triangle are guaranteed to lie on a plane; any
polygon with four or more vertices might not. Nonplanar polygons can be
viewed from some orientation such that the edges cross each other, and
OpenGL might not render such polygons correctly.

There's always a trade-off between the display speed and the quality
of the image. If you subdivide a surface into a small number of polygons,
it renders quickly but might have a jagged appearance; if you subdivide
it into millions of tiny polygons, it probably looks good but might take
a long time to render. Ideally, you can provide a parameter to the subdivision
routines that indicates how fine a subdivision you want, and if the object
is farther from the eye, you can use a coarser subdivision. Also, when
you subdivide, use relatively large polygons where the surface is relatively
flat, and small polygons in regions of high curvature.

For high-quality images, it's a good idea to subdivide more on the silhouette
edges than in the interior. If the surface is to be rotated relative to
the eye, this is tougher to do, since the silhouette edges keep moving.
Silhouette edges occur where the normal vectors are perpendicular to the
vector from the surface to the viewpoint - that is, when their vector dot
product is zero. Your subdivision algorithm might choose to subdivide more
if this dot product is near zero.

Try to avoid T-intersections in your models (see Figure 2-13 ). As shown, there's no guarantee that the line segments AB and BC lie on exactly the same pixels as the segment AC. Sometimes they do, and sometimes they don't, depending on the transformations and orientation. This can cause cracks to appear intermittently in the surface.

**Figure 2-13 : **Modifying an Undesirable T-intersection

- If you're constructing a closed surface, make sure to use exactly the
same numbers for coordinates at the beginning and end of a closed loop,
or you can get gaps and cracks due to numerical round-off. Here's a two-dimensional
example of bad code:

/* don't use this code */ #define PI 3.14159265 #define EDGES 30 /* draw a circle */ for (i = 0; i < EDGES; i++) { glBegin(GL_LINE_STRIP); glVertex2f(cos((2*PI*i)/EDGES), sin((2*PI*i)/EDGES); glVertex2f(cos((2*PI*(i+1))/EDGES), sin((2*PI*(i+1))/EDGES); glEnd(); }The edges meet exactly only if your machine manages to calculate the sine and cosine of 0 and of (2*PI*EDGES/EDGES) and gets exactly the same values. If you trust the floating-point unit on your machine to do this right, the authors have a bridge they'd like to sell you.... To correct the code, make sure that when

Finally, note that unless tessellation is very fine, any change is likely to be visible. In some animations, these changes are more visually disturbing than the artifacts of undertessellation.

**Example 2-4 : **Drawing an Icosahedron

#define X .525731112119133606 #define Z .850650808352039932 static GLfloat vdata[12][3] = { {-X, 0.0, Z}, {X, 0.0, Z}, {-X, 0.0, -Z}, {X, 0.0, -Z}, {0.0, Z, X}, {0.0, Z, -X}, {0.0, -Z, X}, {0.0, -Z, -X}, {Z, X, 0.0}, {-Z, X, 0.0}, {Z, -X, 0.0}, {-Z, -X, 0.0} }; static GLint tindices[20][3] = { {0,4,1}, {0,9,4}, {9,5,4}, {4,5,8}, {4,8,1}, {8,10,1}, {8,3,10}, {5,3,8}, {5,2,3}, {2,7,3}, {7,10,3}, {7,6,10}, {7,11,6}, {11,0,6}, {0,1,6}, {6,1,10}, {9,0,11}, {9,11,2}, {9,2,5}, {7,2,11} }; for (i = 0; i < 20; i++) { /* color information here */ glBegin(GL_TRIANGLES); glVertex3fv(&vdata[tindices[i][0]][0]); glVertex3fv(&vdata[tindices[i][1]][0]); glVertex3fv(&vdata[tindices[i][2]][0]); glEnd(); }The strange numbers

The line that mentions color information should be replaced by a command
that sets the color of the **i**th face. If no code appears here, all
faces are drawn in the same color, and it'll be impossible to discern the
three-dimensional quality of the object. An alternative to explicitly specifying
colors is to define surface normals and use lighting, as described in the
next section.

In all the examples described in this section, unless the surface is to be drawn only once, you should probably save the calculated vertex and normal coordinates so that the calculations don't need to be repeated each time that the surface is drawn. This can be done using your own data structures or by constructing display lists (see Chapter 4 .)

**Example 2-5 : **Supplying Normals for an Icosahedron

GLfloat d1[3], d2[3], norm[3]; for (j = 0; j < 3; j++) { d1[j] = vdata[tindices[i][0]][j] - vdata[tindices[i][1]][j]; d2[j] = vdata[tindices[i][1]][j] - vdata[tindices[i][2]][j]; } normcrossprod(d1, d2, norm); glNormal3fv(norm);The function

**Example 2-6 : **Calculating the Normalized Cross Product of Two
Vectors

void normalize(float v[3]) { GLfloat d = sqrt(v[0]*v[1]+v[1]*v[1]+v[2]*v[2]); if (d == 0.0) { error("zero length vector"); return; } v[0] /= d; v[1] /= d; v[2] /= d; } void normcrossprod(float v1[3], float v2[3], float out[3]) { GLint i, j; GLfloat length; out[0] = v1[1]*v2[2] - v1[2]*v2[1]; out[1] = v1[2]*v2[0] - v1[0]*v2[2]; out[2] = v1[0]*v2[1] - v1[1]*v2[0]; normalize(out); }If you're using an icosahedron as an approximation for a shaded sphere, you'll want to use normal vectors that are perpendicular to the true surface of the sphere, rather than being perpendicular to the faces. For a sphere, the normal vectors are simple; each points in the same direction as the vector from the origin to the corresponding vertex. Since the icosahedron vertex data is for an icosahedron of radius 1, the normal and vertex data is identical. Here is the code that would draw an icosahedral approximation of a smoothly shaded sphere (assuming that lighting is enabled, as described in Chapter 6 ):

for (i = 0; i < 20; i++) { glBegin(GL_POLYGON); glNormal3fv(&vdata[tindices[i][0]][0]); glVertex3fv(&vdata[tindices[i][0]][0]); glNormal3fv(&vdata[tindices[i][1]][0]); glVertex3fv(&vdata[tindices[i][1]][0]); glNormal3fv(&vdata[tindices[i][2]][0]); glVertex3fv(&vdata[tindices[i][2]][0]); glEnd(); }

**Figure 2-14 : **Subdividing to Improve a Polygonal Approximation
to a Surface

Example 2-7 performs a single subdivision, creating an eighty-sided spherical approximation.

**Example 2-7 : **Single Subdivision

void drawtriangle(float *v1, float *v2, float *v3) { glBegin(GL_POLYGON); glNormal3fv(v1); glVertex3fv(v1); glNormal3fv(v2); glVertex3fv(v2); glNormal3fv(v3); glVertex3fv(v3); glEnd(); } void subdivide(float *v1, float *v2, float *v3) { GLfloat v12[3], v23[3], v31[3]; GLint i; for (i = 0; i < 3; i++) { v12[i] = v1[i]+v2[i]; v23[i] = v2[i]+v3[i]; v31[i] = v3[i]+v1[i]; } normalize(v12); normalize(v23); normalize(v31); drawtriangle(v1, v12, v31); drawtriangle(v2, v23, v12); drawtriangle(v3, v31, v23); drawtriangle(v12, v23, v31); } for (i = 0; i < 20; i++) { subdivide(&vdata[tindices[i][0]][0], &vdata[tindices[i][1]][0], &vdata[tindices[i][2]][0]); }Example 2-8 is a slight modification of Example 2-7 that recursively subdivides the triangles to the proper depth. If the depth value is 0, no subdivisions are performed, and the triangle is drawn as is. If the depth is 1, a single subdivison is performed, and so on.

**Example 2-8 : **Recursive Subdivision

void subdivide(float *v1, float *v2, float *v3, long depth) { GLfloat v12[3], v23[3], v31[3]; GLint i; if (depth == 0) { drawtriangle(v1, v2, v3); return; } for (i = 0; i < 3; i++) { v12[i] = v1[i]+v2[i]; v23[i] = v2[i]+v3[i]; v31[i] = v3[i]+v1[i]; } normalize(v12); normalize(v23); normalize(v31); subdivide(v1, v12, v31, depth-1); subdivide(v2, v23, v12, depth-1); subdivide(v3, v31, v23, depth-1); subdivide(v12, v23, v31, depth-1); }

To look at a more general solution to the problem of subdivision, consider
an arbitrary surface parameterized by two variables *u[0]* and *u[1]*.
Suppose that two routines are provided:

void surf(GLfloat u[2], GLfloat vertex[3], GLfloat normal[3]); float curv(GLfloat u[2]);If

Example 2-9 shows the recursive routine that subdivides a triangle either until the maximum depth is reached or until the maximum curvature at the three vertices is less than some cutoff.

**Example 2-9 : **Generalized Subdivision

void subdivide(float u1[2], float u2[2], float u3[2], float cutoff, long depth) { GLfloat v1[3], v2[3], v3[3], n1[3], n2[3], n3[3]; GLfloat u12[2], u23[2], u32[2]; GLint i; if (depth == maxdepth || (curv(u1) < cutoff && curv(u2) < cutoff && curv(u3) < cutoff)) { surf(u1, v1, n1); surf(u2, v2, n2); surf(u3, v3, n3); glBegin(GL_POLYGON); glNormal3fv(n1); glVertex3fv(v1); glNormal3fv(n2); glVertex3fv(v2); glNormal3fv(n3); glVertex3fv(v3); glEnd(); return; } for (i = 0; i < 2; i++) { u12[i] = (u1[i] + u2[i])/2.0; u23[i] = (u2[i] + u3[i])/2.0; u31[i] = (u3[i] + u1[i])/2.0; } subdivide(u1, u12, u31, cutoff, depth+1); subdivide(u2, u23, u12, cutoff, depth+1); subdivide(u3, u31, u23, cutoff, depth+1); subdivide(u12, u23, u31, cutoff, depth+1); }

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