Windows and viewport

2D Drawing
Filling Rectangle, Filling Polygon

Coordinate Systems

We have been using the coordinate system of the
screen window (in pixels).
The range is from 0 (left) to some value screenWidth
– 1 in x, and from 0 (usually top) to some value
screenHeight –1 in y.
 We can use only positive values of x and y.
 The values must have a large range (several hundred pixels) to
get a reasonable size drawing.

CONTINUE…

It may be much more natural to think in terms of x
varying from, say, -1 to 1, and y varying from –100.0
to 20.0.
We want to separate the coordinates we use in a
program to describe the geometrical object from the
coordinates we use to size and position the pictures
of the objects on the display.
Description is usually referred to as a modeling
task, and displaying pictures as a viewing task.

WORLD COORDINATES

The space in which objects are described is called
world coordinates (the numbers used for x and y
are those in the world, where the objects are
defined).
World coordinates use the Cartesian xy-coordinate
system used in mathematics, based on whatever
units are convenient.

world window

We define a rectangular world window in these
world coordinates.
The world window specifies which part of the world
should be drawn: whichever part lies inside the
window should be drawn, and whichever part lies
outside should be clipped away and not drawn.
OpenGL does the clipping automatically.

viewport

In addition, we define a rectangular viewport in the
screen window on the display.
A mapping (consisting of scalings [change size] and
translations [move object]) between the world
window and the viewport is established by OpenGL.
The objects inside the world window appear
automatically at proper sizes and locations inside the
viewport (in screen coordinates, which are pixel
coordinates on the display).

Coordinate Systems Example

 We want to graph

 Sinc(0)= 1 by
definition.
Interesting parts of
the function are in
-4.0 ≤ x ≤ 4.0.

Set WW

 The function setWindow sets the world window size:

void setWindow(GLdouble left, GLdouble right, GLdouble bottom, GLdouble top)
{
glMatrixMode(GL_PROJECTION);

glLoadIdentity();

gluOrtho2D(left, right, bottom, top);

}

Set VP

void setViewport(GLint left, GLint right, GLint bottom, GLint top)
{
glViewport(left, bottom, right - left, top - bottom);
}

 Calls: setWindow(-5.0, 5.0, -0.3, 1.0);
 setViewport(0, 640, 0, 480);

In myInit();

Windows and Viewports

We use natural coordinates for what we are drawing
(the world window).
OpenGL converts our coordinates to screen
coordinates when we set up a screen window and a
viewport. The viewport may be smaller than the
screen window. The default viewport is the entire
screen window.
The conversion requires scaling and shifting:
mapping the world window to the screen window
and the viewport.

Mapping From the Window to VIEWPORT

Windows are described by their left, top, right, and
bottom values, w.l, w.t, w.r, w.b.
Viewports are described by the same values: v.l, v.t,
v.r, v.b, but in screen window coordinates.

Window-to-Viewport Mapping (Transformation)

We want our mapping to be proportional: for
example, if x is ¼ of the way between the left and
right world window boundaries, then the screen x (sx)
should be ¼ of the way between the left and right
viewport boundaries.

CONTINUE….

Proportionality forces the mappings to have a linear
form:
sx= A * x + C,
sy = B * y + D
For some constants A, B, C and D. The constants A and B scale the x
and y coordinates, and C and D shift (or translate) them.

How can A, B, C, and D be determined?
Consider first the mapping for x.

After algebraic manipulation

Or
A = (V.r-V.l)/(W.r-W.l),
C = V.l – A*w.l

CONTINUE…

Similarly, proportionality in y dictates that

and writing sy as B y + D yields:

Example

Consider the window and viewport of Figure The
window has (W.l, W.r, W.b, W.t) = (0, 2.0, 0, 1.0)
and the viewport has (V.l, V.r, V.b, V.t) = (40, 400,
60, 300).

Solution:

 Using the above eq:

Thus for this example, the window to viewport
mapping is:

Check that this mapping properly maps various
points of interest, such as
Each corner of the window is indeed mapped to the
corresponding corner of the viewport. For example,
(2.0, 1.0) maps to (400, 300).

CONTINUE….

The center of the window (1.0, 0.5) maps to the
center of the viewport (220, 180 )

Practice Exercise

Find values of A, B, C, and D for the case of a world
window (10.0, 10.0, -6.0, 6.0) and a viewport (0,
600, 0, 400).

oGL Functions To Create the Map

Because OpenGL uses matrices to set up all its
transformations, the call to gluOrtho2D() must be
preceded by two setup functions:

glMatrixMode(GL_PROJECTION);
glLoadIdentity(); //to reset the matrix

Application:Tiling with Viewports

Windows and viewport

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