9 Transformations
9.001 I can't get transformations to
work. Where can I learn more about matrices?
A thorough explanation of basic matrix math
and linear algebra is beyond the scope of this FAQ. These
concepts are taught in high school math classes in the United
States.
If you understand the basics, but just get
confused (a common problem even for the experienced!), read
through Steve Baker's review
of matrix concepts and his article
on Euler angles.
Delphi
code for performing basic vector, matrix, and quaternion
operations can be found here.
9.005 Are OpenGL matrices column-major
or row-major?
Column-major versus row-major is purely a
notational convention. Note that post-multiplying with column-major
matrices produces the same result as pre-multiplying with row-major
matrices. The OpenGL Specification and the OpenGL Reference
Manual both use column-major notation. You can use any
notation, as long as it's clearly stated.
For programming purposes, OpenGL matrices are
16-value arrays with base vectors laid out contiguously in
memory.
9.010 What are OpenGL coordinate
units?
The short answer: Anything you want them to
be.
Depending on the contents of your geometry
database, it may be convenient for your application to treat
one OpenGL coordinate unit as being equal to one millimeter
or one parsec or anything in between (or larger or smaller).
OpenGL also lets you specify your geometry
with coordinates of differing values. For example, you may
find it convenient to model an airplane's controls in
centimeters, its fuselage in meters, and a world to fly
around in kilometers. OpenGL's ModelView matrix can then
scale these different coordinate systems into the same eye
coordinate space.
It's the application's responsibility to
ensure that the Projection and ModelView matrices are
constructed to provide an image that keeps the viewer at an
appropriate distance, with an appropriate field of view, and
keeps the zNear and zFar clipping planes at
an appropriate range. An application that displays molecules
in micron scale, for example, would probably not want to
place the viewer at a distance of 10 feet with a 60 degree
field of view.
9.020 How do I transform only one
object in my scene or give each object its own transform?
OpenGL provides matrix stacks specifically
for this purpose. In this case, use the ModelView matrix
stack.
A typical OpenGL application first sets the
matrix mode with a call to glMatrixMode(GL_MODELVIEW) and
loads a viewing transform, perhaps with a call to gluLookAt().More information is available on gluLookAt().
Then the code renders each object in the
scene with its own transformation by wrapping the rendering
with calls to glPushMatrix() and glPopMatrix(). For example:
glPushMatrix();
glRotatef(90., 1., 0., 0.);
gluCylinder(quad,1,1,2,36,12);
glPopMatrix();
The above code renders a cylinder rotated 90
degrees around the X-axis. The ModelView matrix is restored
to its previous value after the glPopMatrix() call. Similar
call sequences can render subsequent objects in the scene.
9.030 How do I draw 2D controls over
my 3D rendering?
The basic strategy is to set up a 2D
projection for drawing controls. You can do this either on
top of your 3D rendering or in overlay planes. If you do so
on top of a 3D rendering, you'll need to redraw the controls
at the end of every frame (immediately before swapping
buffers). If you draw into the overlay planes, you only need
to redraw the controls if you're updating them.
To set up a 2D projection, you need to change
the Projection matrix. Normally, it's convenient to set up
the projection so one world coordinate unit is equal to one
screen pixel, as follows:
glMatrixMode (GL_PROJECTION);
glLoadIdentity ();
gluOrtho2D (0, windowWidth, 0, windowHeight);
gluOrtho2D() sets up a Z range of -1 to 1, so
you need to use one of the glVertex2*() functions to ensure
your geometry isn't clipped by the zNear or zFar
clipping planes.
Normally, the ModelView matrix is set to the
identity when drawing 2D controls, though you may find it
convenient to do otherwise (for example, you can draw
repeated controls with interleaved translation matrices).
If exact pixelization is required, you might
want to put a small translation in the ModelView matrix, as
shown below:
glMatrixMode (GL_MODELVIEW);
glLoadIdentity ();
glTranslatef (0.375, 0.375, 0.);
If you're drawing on top of a 3D-depth
buffered image, you'll need to somehow disable depth testing
while drawing your 2D geometry. You can do this by calling
glDisable(GL_DEPTH_TEST) or glDepthFunc (GL_ALWAYS).
Depending on your application, you might also simply clear
the depth buffer before starting the 2D rendering. Finally,
drawing all 2D geometry with a minimum Z coordinate is also a
solution.
After the 2D projection is established as
above, you can render normal OpenGL primitives to the screen,
specifying their coordinates with XY pixel addresses (using
OpenGL-centric screen coordinates, with (0,0) in the lower
left).
9.040 How do I bypass OpenGL matrix
transformations and send 2D coordinates directly for
rasterization?
There isn't a mode switch to disable OpenGL
matrix transformations. However, if you set either or both
matrices to the identity with a glLoadIdentity() call,
typical OpenGL implementations are intelligent enough to know
that an identity transformation is a no-op and will act
accordingly.
More detailed information on using OpenGL as
a rasterization-only API is in the OpenGL
Game Developer’s FAQ.
9.050 What are the pros and cons of
using absolute versus relative coordinates?
Some OpenGL applications may need to render
the same object in multiple locations in a single scene.
OpenGL lets you do this two ways:
1) Use “absolute coordinates".
Maintain multiple copies of each object, each with its own
unique set of vertices. You don't need to change the
ModelView matrix to render the object at the desired location.
2) Use “relative coordinates". Keep
only one copy of the object, and render it multiple times by
pushing the ModelView matrix stack, setting the desired
transform, sending the geometry, and popping the stack.
Repeat these steps for each object.
In general, frequent changes to state, such
as to the ModelView matrix, can negatively impact your
application’s performance. OpenGL can process your
geometry faster if you don't wrap each individual primitive
in a lot of changes to the ModelView matrix.
However, sometimes you need to weigh this
against the memory savings of replicating geometry. Let's say
you define a doorknob with high approximation, such as 200 or
300 triangles, and you're modeling a house with 50 doors in
it, all of which have the same doorknob. It's probably
preferable to use a single doorknob display list, with
multiple unique transform matrices, rather than use absolute
coordinates with 10-15K triangles in memory.
As with many computing issues, it's a trade-off
between processing time and memory that you'll need to make
on a case-by-case basis.
9.060 How can I draw more than one
view of the same scene?
You can draw two views into the same window
by using the glViewport() call. Set glViewport() to the area
that you want the first view, set your scene’s view, and
render. Then set glViewport() to the area for the second view,
again set your scene’s view, and render.
You need to be aware that some operations don't
pay attention to the glViewport, such as SwapBuffers and
glClear(). SwapBuffers always swaps the entire window.
However, you can restrain glClear() to a rectangular window
by using the scissor rectangle.
Your application might only allow different
views in separate windows. If so, you need to perform a
MakeCurrent operation between the two renderings. If the two
windows share a context, you need to change the scene’s
view as described above. This might not be necessary if your
application uses separate contexts for each window.
9.070 How do I transform my objects
around a fixed coordinate system rather than the object's local
coordinate system?
If you rotate an object around its Y-axis,
you'll find that the X- and Z-axes rotate with the object. A
subsequent rotation around one of these axes rotates around
the newly transformed axis and not the original axis. It's
often desirable to perform transformations in a fixed
coordinate system rather than the object’s local
coordinate system.
The OpenGL
Game Developer’s FAQ contains information on using
quaternions to store rotations, which may be useful in
solving this problem.
The root cause of the problem is that OpenGL
matrix operations postmultiply onto the matrix stack, thus
causing transformations to occur in object space. To affect
screen space transformations, you need to premultiply. OpenGL
doesn't provide a mode switch for the order of matrix
multiplication, so you need to premultiply by hand. An
application might implement this by retrieving the current
matrix after each frame. The application multiplies new
transformations for the next frame on top of an identity
matrix and multiplies the accumulated current transformations
(from the last frame) onto those transformations using
glMultMatrix().
You need to be aware that retrieving the
ModelView matrix once per frame might have a detrimental
impact on your application’s performance. However, you
need to benchmark this operation, because the performance
will vary from one implementation to the next.
9.080 What are the pros and cons of
using glFrustum() versus gluPerspective()? Why would I want to
use one over the other?
glFrustum() and gluPerspective() both produce
perspective projection matrices that you can use to transform
from eye coordinate space to clip coordinate space. The
primary difference between the two is that glFrustum() is
more general and allows off-axis projections, while
gluPerspective() only produces symmetrical (on-axis)
projections. Indeed, you can use glFrustum() to implement
gluPerspective(). However, aside from the layering of
function calls that is a natural part of the GLU interface,
there is no performance advantage to using matrices generated
by glFrustum() over gluPerspective().
Since glFrustum() is more general than
gluPerspective(), you can use it in cases when gluPerspective()
can't be used. Some examples include projection shadows, tiled
renderings, and stereo views.
Tiled rendering uses multiple off-axis
projections to render different sections of a scene. The
results are assembled into one large image array to produce
the final image. This is often necessary when the desired
dimensions of the final rendering exceed the OpenGL
implementation's maximum viewport size.
In a stereo view, two renderings of the same
scene are done with the view location slightly shifted. Since
the view axis is right between the “eyes”, each
view must use a slightly off-axis projection to either side
to achieve correct visual results.
9.085 How can I make a call to
glFrustum() that matches my call to gluPerspective()?
The field of view (fov) of your glFrustum()
call is:
fov*0.5 = arctan ((top-bottom)*0.5 / near)
Since bottom == -top for
the symmetrical projection that gluPerspective() produces,
then:
top = tan(fov*0.5) * near
bottom = -top
The left and right parameters
are simply functions of the top, bottom, and aspect:
left = aspect * bottom
right = aspect * top
The OpenGL Reference Manual (where do I get this?)
shows the matrices produced by both functions.
9.090 How do I draw a full-screen
quad?
This question usually means, "How do I
draw a quad that fills the entire OpenGL viewport?"
There are many ways to do this.
The most straightforward method is to set the
desired color, set both the Projection and ModelView matrices
to the identity, and call glRectf() or draw an equivalent GL_QUADS
primitive. Your rectangle or quad's Z value should be in the
range of –1.0 to 1.0, with –1.0 mapping to the zNear
clipping plane, and 1.0 to the zFar clipping
plane.
As an example, here's how to draw a full-screen
quad at the zNear clipping plane:
glMatrixMode (GL_MODELVIEW);
glPushMatrix ();
glLoadIdentity ();
glMatrixMode (GL_PROJECTION);
glPushMatrix ();
glLoadIdentity ();
glBegin (GL_QUADS);
glVertex3i (-1, -1, -1);
glVertex3i (1, -1, -1);
glVertex3i (1, 1, -1);
glVertex3i (-1, 1, -1);
glEnd ();
glPopMatrix ();
glMatrixMode (GL_MODELVIEW);
glPopMatrix ();
Your application might want the quad to have
a maximum Z value, in which case 1 should be used for the Z
value instead of -1.
When painting a full-screen quad, it might be
useful to mask off some buffers so that only specified
buffers are touched. For example, you might mask off the
color buffer and set the depth function to GL_ALWAYS, so only
the depth buffer is painted. Also, you can set masks to allow
the stencil buffer to be set or any combination of buffers.
9.100 How can I find the screen
coordinates for a given object-space coordinate?
You can use the GLU library gluProject()
utility routine if you only need to find it for a few
vertices. For a large number of coordinates, it can be more
efficient to use the Feedback mechanism.
To use gluProject(), you'll need to provide
the ModelView matrix, projection matrix, viewport, and input
object space coordinates. Screen space coordinates are
returned for X, Y, and Z, with Z being normalized (0 <= Z
<= 1).
9.110 How can I find the object-space
coordinates for a pixel on the screen?
The GLU library provides the gluUnProject()
function for this purpose.
You'll need to read the depth buffer to
obtain the input screen coordinate Z value at the X,Y
location of interest. This can be coded as follows:
GLdouble z;
glReadPixels (x, y, 1, 1, GL_DEPTH_COMPONENT, GL_DOUBLE, &z);
Note that x and y are
OpenGL-centric with (0,0) in the lower-left corner.
You'll need to provide the screen space X, Y,
and Z values as input to gluUnProject() with the ModelView
matrix, Projection matrix, and viewport that were current at
the time the specific pixel of interest was rendered.
9.120 How do I find the coordinates
of a vertex transformed only by the ModelView matrix?
It's often useful to obtain the eye
coordinate space value of a vertex (i.e., the object space
vertex transformed by the ModelView matrix). You can obtain
this by retrieving the current ModelView matrix and
performing simple vector / matrix multiplication.
9.130 How do I calculate the object-space
distance from the viewer to a given point?
Transform the point into eye-coordinate space
by multiplying it by the ModelView matrix. Then simply
calculate its distance from the origin. (If this doesn't work,
you may have incorrectly placed the view transform on the
Projection matrix stack.)
9.140 How do I keep my aspect ratio
correct after a window resize?
It depends on how you are setting your
projection matrix. In any case, you'll need to know the new
dimensions (width and height) of your window. How to obtain
these depends on which platform you're using. In GLUT, for
example, the dimensions are passed as parameters to the
reshape function callback.
The following assumes you're maintaining a
viewport that's the same size as your window. If you are not,
substitute viewportWidth and viewportHeight for windowWidth
and windowHeight.
If you're using gluPerspective() to set your
Projection matrix, the second parameter controls the aspect
ratio. When your program catches a window resize, you'll need
to change your Projection matrix as follows:
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
gluPerspective(fov, (float)windowWidth/(float)windowHeight, zNear, zFar);
If you're using glFrustum(), the aspect ratio
varies with the width of the view volume to the height of the
view volume. You might maintain a 1:1 aspect ratio with the
following window resize code:
float cx, halfWidth = windowWidth*0.5f;
float aspect = (float)windowWidth/(float)windowHeight;
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
/* cx is the eye space center of the zNear plane in X */
glFrustum(cx-halfWidth*aspect, cx+halfWidth*aspect, bottom, top, zNear, zFar);
glOrtho() and gluOrtho2D() are similar to
glFrustum().
9.150 Can I make OpenGL use a left-handed
coordinate space?
OpenGL doesn't have a mode switch to change
from right- to left-handed coordinates. However, you can
easily obtain a left-handed coordinate system by multiplying
a negative Z scale onto the ModelView matrix. For example:
glMatrixMode (GL_MODELVIEW);
glLoadIdentity ();
glScalef (1., 1., -1.);
/* multiply view transforms as usual… */
/* multiply model transforms as usual… */
9.160 How can I transform an object
so that it points at or follows another object or point in my
scene?
You need to construct a matrix that
transforms from your object's local coordinate system into a
coordinate system that faces in the desired direction. See this example code to see how this
type of matrix is created.
If you merely want to render an object so
that it always faces the viewer, you might consider simply
rendering it in eye-coordinate space with the ModelView
matrix set to the identity.
9.170 How do I render a mirror?
Render your scene twice, once as it is
reflected in the mirror, then once from the normal (non-reflected)
view. Example code demonstrates this
technique.
For axis-aligned mirrors, such as a mirror on
the YZ plane, the reflected scene can be rendered with a
simple scale and translate. Scale by -1.0 in the axis
corresponding to the mirror's normal, and translate by twice
the mirror's distance from the origin. Rendering the scene
with these transforms in place will yield the scene reflected
in the mirror. Use the matrix stack to restore the view
transform to its previous value.
Next, clear the depth buffer with a call to
glClear(GL_DEPTH_BUFFER_BIT). Then render the mirror. For a
perfectly reflecting mirror, render into the depth buffer
only. Real mirrors are not perfect reflectors, as they absorb
some light. To create this effect, use blending to render a
black mirror with an alpha of 0.05. glBlendFunc(GL_SRC_ALPHA,GL_ONE_MINUS_SRC_ALPHA)
is a good blending function for this purpose.
Finally, render the non-reflected scene.
Since the entire reflected scene exists in the color buffer,
and not just the portion of the reflected scene in the mirror,
you will need to touch all pixels to overwrite areas of the
reflected scene that should not be visible.
9.180 How can I do my own perspective
scaling?
OpenGL multiplies your coordinates by the
ModelView matrix, then by the Projection matrix to get clip
coordinates. It then performs the perspective divide to
obtain normalized device coordinates. It's the perspective
division step that creates a perspective rendering, with
geometry in the distance appearing smaller than the geometry
in the foreground. The perspective division stage is
accomplished by dividing your XYZ clipping coordinate values
by the clipping coordinate W value, such as:
Xndc = Xcc/Wcc
Yndc = Ycc/Wcc
Zndc = Zcc/Wcc
To do your own perspective correction, you
need to obtain the clipping coordinate W value. The feedback
buffer provides homogenous coordinates with XYZ in device
coordinates and W in clip coordinates. You might also
glGetFloatv(GL_CURRENT_RASTER_POSITION,…) and the W
value will again be in clipping coordinates, while XYZ are in
device coordinates.
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