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Introduction to Open GL

By

Rama Satish KV

Visiting Professor, WASE

satishkvr@gmail.com

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Class Objective

WINDOW TO VIEWPORT TRANSFORMATION

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WINDOW~TO-VIEWPORT

TRANSFORMATION

Some graphics packages allow the programmer to

specify output primitive coordinates in a floating-point

world coordinate system.

The term world is used because the application programis representing a world that is being interactively created

or displayed to the user.

Given that output primitives are specified in world

coordinates, the graphics subroutine package must betold how to map world coordinates onto screen

coordinates.

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Coordinate mapping

We could do this mapping by having the application

programmer provide the graphics package with a

transformation matrix to effect the mapping.

Another way is to have the application programmer specify arectangular region in world coordinates, called the world

coordinate window, and at corresponding rectangular region

in screen coordinates, called the viewport. into which the

world-coordinate window is to be mapped.

The transformation that maps the window into the viewport is

applied to all of the output primitives in world coordinates,

thus mapping them into screen coordinates. Figure 5.10

shows this concept.

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ViewPort

As seen in this figure, if the window and viewport do not have

the same height to width ratio, a non uniform scaling occurs.

If the application program changes the window or viewport,

then new output primitives drawn onto the screen will be

affected by the change.

Existing output primitives are not affected by such a change.

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The application program would be able to change the window or

the viewport at any time, in which case subsequently specified

output primitive would be subjected to a new transformation.

If the change included a different viewport, then the new output

primitive would be located on the canvas in positions different

from those of the old ones, as shown in Fig. 5.11.

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Output primitives in world coordinates are clipped against

the window. Those that remain are displayed in the

viewport

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CLIPPING IN A RASTER WORLD

Clipping can be done analytically on thefly

during scan conversion, or as part of a

copyPixel with the desired clip rectangle from

a canvas storing unclipped primitive to the

destination canvas.

Combining clipping and scan conversion,

sometimes called scissoring, is easy to dofor filled or thick primitives as part of span

arithmetic

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CLIPPING LINES This section treats analytical clipping of lines against

rectangles.

Circles and Ellipses may be piece wise linearly

approximated with a sequence of very short lines, so that

boundaries can be treated as a single polyline or polygon forboth clipping and scan conversion.

Clipping a rectangle against a rectangle results in at most a

single rectangle.

Clipping a convex polygon against a rectangle results in atmost a single convex polygon, but clipping a concave

polygon may produce more than one concave polygon.

Clipping a circle or ellipse against a rectangle results in as

many as four arcs.

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Lines intersecting a rectangular clip region

are always clipped to a single line segment;

lines lying on the clip rectangle's border are

considered inside and hence are displayed.

Figure 3.38 shows several examples of

clipped lines.

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Fig 3.38

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Clipping Endpoints

Let's look at the simpler problem of clipping individual points.

If the x coordinate boundaries of the clip rectangle are at xmin

and Xmaxand the y coordinate boundaries are at Yminand

Ymaxthen four inequalities must be satisfied for a point (x. y)to be inside the clip rectangle:

If any of the four inequalities does not hold, the point isoutside the clip rectangle.

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Clipping the end points

continued

To clip a line, we need to consider only its endpoints, not itsinfinitely many interior points.

If both endpoints of a line lie inside the clip rectangle (e.g.,AB inFig. 3.38), the entire line lies inside the clip rectangle and can be

triviallyaccepted. If one endpoint lies inside and one out side (e.g., CD in the

figure), the line intersects the clip rectangle and we mustcompute the intersection point. lf both endpoints are outside theclip rectangle, the line may (or may not) intersect with the cliprectangle (EF. GH, and IJ in the figure), and

we need to perform further calculations to determine whetherthere are any intersections, and if there are, where they occur.

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The Cohen-Sutherland Line

Clipping Algorithm

The more efficient Cohen-Sutherland algorithm performs initialtests on a line to determine whether intersection calculations canbe avoided.

First. endpoint pairs are checked for trivial acceptance.

If the line can not be trivially accepted. region checks are done. For instance, two simple comparisons onx show that both

endpoints of line EF in Fig. 3.38 have anx coordinate less thanxminand thus lie in the region to the left of the clip rectangle (l.e.,in the outside half plane defined by the left edge) therefore linesegment EF can be trivially rejected and needs to be neitherclipped nor displayed.

Similarly, we can trivially reject lines with both endpoints inregions to the right of Xmaxbelow Yminand above Ymax.

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To perform trivial accept and reject tests, we extend

the edges of the clip rectangle to divide the plane of

the clip rectangle into nine regions (see Fig. 3.39 ).

Each region is assigned a 4-bit code, determined bywhere the region lies with respect to the outside half

planes of the clip-rectangle edges.

Each bit in the outcode is set to either 1(true) or 0

(false); the 4 bits in the code correspond to thefollowing conditions:

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Compute OutCode

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Cohen Sutherland Line Clip

Algorithm

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A Parametric Line-Clipping

Algorithm

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CLIPPING CIRCLES AND

ELLIPSES

To clip a circle against a rectangle, we can first do a trivialaccept/reject test by intersecting the circle's extent (a squareof the size of the circle diameter) with the clip rectangle,

Using the algorithm in the next section for polygon clipping.

If the circle intersects the rectangle, we divide it intoquadrants and do the trivial accept/reject test for each.

These tests may lead in turn to tests for octants.

We can then compute the intersection of the circle and the

edge analytically by solving their equations simultaneously,and then scan convert the resulting arcs using theappropriately initialized algorithm with the calculated (andsuitably rounded) starting and ending points.

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CLIPPING CIRCLES AND

ELLIPSESs

If scan conversion is fast or if the circle is not

too large, it s probably more efficient to

scissor on a pixel-by-pixel basis, testing each

boundary pixel against the rectangle boundsbefore it iswritten.

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Viewing in 3D

The 3D viewing process is inherently more complex than is the

2D viewing process. In 2D we simply specify a window on the 2D world and a

viewport on the 2D view surface.

Conceptually objects in the world are dipped against the

window and are then transformed into the viewport for display. The extra complexity of 3D viewing is caused in part by the

added dimension and in part by the fact that display devices are

only 2D.

The solution to the mismatch between 3D objects and 2Ddisplays is accomplished by Introdudngprojections, which

transform 3D objects onto a 2D projection plane.

Much of this chapter is devoted to projections: what they are,

their mathematics, and how they are used in a current graphicssubroutine acka e.

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Conceptual model of 3D

viewing process

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In 3D viewing, we specify a view volume in the world, a

projection onto a projection planeand a viewport on the

view surface.

Conceptually, objects in the 3D world are clipped againstthe 3D view volume and are then projected.

The contents of the projection of the view volume onto

the projection plane. called the window, are then

transformed (mapped) into the viewport for display.

Figure 6.1 shows this conceptual model of the 3D

viewing process, which is the model presented to the

users of numerous 3D graphics subroutine packages.

Just as with 2D viewing, a variety of strategies can be

used to implement the viewing process.

Conceptual model of 3D

viewing process

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PROJECTIONS

In general, projections transform points in a coordinate systemof dimension n into points in a coordinate system of dimension

less than n.

The projection of a 3D object is defined by straight projection

rays (calledprojectors) emanating from a center of projection,passing through each point of the objects and intersecting a

projection plane to form the projection.

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Planar geometr ic p roject ions

The class of projections we deal with here is known asplanar

geometric projections, because the projection is onto a planerather than some curved surface and uses straight rather than

curved projectors.

Many cartographic projections are either non planar or non

geometric. Similarly, the Omnimax film format requires a nongeometric projection.

Planar geometric projections. hereafter referred to simply as

projections, can be divided into two basic classes :perspective

and parallel.

The distinction is in the relation of the center of projection to

the projection plane.

If the distance from the one to the other is finite, then the

projection is perspective; If the distance is infinite, the

projection is parallel.

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Perspective Projections

The perspective projections of any set of parallel lines

that are not parallel to the projection plane converge to a

vanishing point.

ln 3D, the parallel lines meet only at infinity, so thevanishing point can be thought of as the projection of a

point at infinity.

There is of course an infinity of vanishing points, one for

each of the infinity of directions in which a line can be

oriented.

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Perspective projections are categorized by their number

of principal vanishing points and therefore by the number

of axes the projection plane cuts.

Figure 6.3 shows two different one-point perspectiveprojections of a cube.

It is clear that they are one-point projections because

lines parallel to thex and y axis do not converge; only

lines parallel to the z axis do so.

Figure 6.4 shows the construction of a one-point

perspective with some of the projectors and with the

projection plane cutting only the z axis.

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Fig. 6.4 Construction

of one-point

perspective

projection of cube

onto plane cuttingthe z axis. Projection-

plane normal is

parallel to z axis.

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Figure 6.5 shows the construction of a two-point

perspective.

Notice that lines parallel to the y axis do not converge in

the projection. Two-point perspective is commonly used in architectural,

engineering, industrial design, and in advertising

drawings.

Three-point perspectives are used less frequently. sincethey add little realism beyond that afforded by the two-

point perspective.

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Fig. 6.5 Two-pointperspective projection of a

cube. The projection plane

cuts thex and z axes.

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Parallel Projections

page4

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Next Class

Viewing in 2D/3D

Clipping in 2D/3D

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