CHAPTER 12
Height Maps, Hidden Surface
Removal, Clipping and Level
of Detail Algorithms
© 2008 Cengage Learning EMEA
LEARNING OBJECTIVES
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In this chapter you will learn about:
– Height maps
– Creating and rendering a height map
– Hidden surface removal
– The Z-buffer algorithm
– The painter’s algorithm
– Clipping
– Two-dimensional line clipping
– Cohen-Sutherland clipping
– Liang-Barsky clipping
– The clipping of polygons
– Three-dimensional clipping
– Level of Detail (LOD) algorithms
HEIGHT MAPS
Height maps, also called height fields, are predominantly
used in computer games for realistic terrain generation.
 A height map is stored as a two-dimensional array
consisting of x- and z-values.
 Each x- and z-coordinate is assigned a certain value (a ycoordinate) representing the height of a specific point on
a 3D surface.
 Height maps can thus be described as simple twodimensional arrays containing height values, sampled at
evenly spaced intervals.
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HEIGHT MAPS
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Height maps, as a function of x- and zcoordinates, can be represented using the
following equation:
HEIGHT MAPS
Creating and Rendering a Height
Map
[see the textbook and online source code for a detailed
discussion and examples].
HIDDEN SURFACE REMOVAL
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Culling is the process of comparing the position and
orientation of polygons against the view volume’s field of
view, with polygons facing away from the camera being
eliminated.
This elimination minimized the amount of computational
overhead involved with hidden surface removal.
Culling is basically a test determining the visibility of an
object, and based on this test the object can be removed
if not visible – a process known as hidden surface
removal.
A culling test is based on a simple vector calculation,
specifically, the dot product between a polygon’s normal
and the line-of-sight vector (the vector from the center
of projection to the polygon).
If the result of the dot product is positive then we can
flag the polygon as visible (with its normal facing
towards the viewer).
HIDDEN SURFACE REMOVAL
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There are two main hidden surface [see the textbook for a detailed
example and discussion].
removal algorithm classes:
– object space algorithms
– image space algorithms.
The Z-Buffer Algorithm
The most common hidden surface removal
algorithm is called the z-buffer algorithm.
 This image space algorithm was developed in
1975 by Edwin Catmull, current president of
Walt Disney and Pixar Animation Studios.
 Z-buffering, sometimes referred to as depthbuffering, stores the depth values for each pixel
being drawn in the frame buffer’s depth buffer.
 This stored depth data, or z-values, are then
used by the algorithm to determine the pixels
that should be drawn.
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[see the textbook for a detailed
example and discussion].
The Painter’s Algorithm
The painter’s algorithm orders polygons based
on their distance from the viewer.
 This distance is calculated from the polygon’s
centroid to the point of view.
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The Painter’s Algorithm
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It is also possible to draw the objects of a scene in a front-to-back
manner, resulting in the reverse painter’s algorithm.
There is one serious flaw in the painter’s hidden surface removal
algorithm – the overlapping of three or more polygons can cause it
to fail.
Another problem is encountered when two or more polygons
intersect each other.
CLIPPING
Clipping is an optimization operation resulting in the
display of only visible objects.
 Clipping can be performed at the image space level,
object space level or at both.
 Image space clipping, as with image space hidden
surface removal, occurs during the rasterization phase
and is done on a per-pixel basis.
 It defines a clipping filter operating on a pixel-by-pixel
basis, determining whether a particular pixel lies within
the specified viewport or not.
 Object space clipping is on the other hand either planebased in object space or boundary-based in screen
space.
 When performing plane-based clipping, we have to
convert our view space boundaries to an object space
clipping-plane, subsequently testing for the visibility of
vertices inside this clipping-plane.
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CLIPPING
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Boundary-based clipping clips against
screen boundaries while in screen space by
flagging the intersection of objects with
planes adjacent to the x-, y- and z-axes.
Two-Dimensional Line Clipping
Two-dimensional line clipping or line-segment
clipping determines the primitives, or sections of
primitives, visible to the viewer.
 All lines located within the predefined view
volume pass the clipping test and are sent to the
rasterizer for display.
 Primitives not located within the view volume
are culled or eliminated.
 The lines, or primitives, partially located within
the view volume are clipped so that only their
visible sections are sent to the rasterizer for
display.
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Two-Dimensional Line Clipping
The subsequent figure illustrates the following four
possible clipping situations:
A The line is completely visible within the predefined view
volume, thus passing the clipping test.
B The line is completely outside the predefined view
volume and can be eliminated from the drawing process.
C The line extends from outside the clipping region, enters
the view volume and exits it again at another point.
Lines like these must be clipped at both ends of the view
volume.
D The line segment is only partially within the predefined
view volume. Its section outside of the clipping region
must thus be clipped.
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Two-Dimensional Line Clipping
Two-Dimensional Line Clipping
There are several line clipping algorithms,
such as the Cohen-Sutherland clipping
algorithm, Liang-Barsky clipping, and
Cyrus-Beck clipping.
 The core issue with
clipping algorithms is
determining the
intersection of two lines.
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Two-Dimensional Line Clipping
The clipping process determines these
intersections and can be described by the
following three steps:
1 Take the input coordinates of a line, (x1, y1) and
(x2, y2), as input.
2 Read the coordinates of the clipping rectangle.
3 Generate the clipped line coordinates, (x1’, y1’)
and (x2’, y2’), by comparing the input values of
the line to the coordinates of the clipping region.
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Two-Dimensional Line Clipping
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Extending the mathematical approach used for
calculating the intersection of two lines to
determine the point of intersection of a line and
clipping rectangle is relatively simple.
Cohen-Sutherland Clipping
The Cohen-Sutherland two-dimensional
line clipping algorithm can be considered a
brute-force approach to the clipping
problem.
 The algorithm basically divides the
clipping region into a number of sections
(specifically nine regions), each with its
own unique 4-bit binary number, called an
outcode.
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Cohen-Sutherland Clipping
[see the textbook for
a detailed discussion
and example].
Liang-Barsky Clipping
The Liang-Barsky two-dimensional line
clipping algorithm uses the parametric
form of a line and the clipping window to
determine the clipping coordinates of lines
intersecting the clipping volume.
 This algorithm is significantly more
efficient than Cohen-Sutherland clipping.
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Liang-Barsky Clipping
Liang-Barsky Clipping
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To summarize, the Liang-Barsky clipping
algorithm consists of the following steps:
1 Define all line segments in parametric form.
2 Set the minimum and maximum values for t (tmin ¼
0 and tmax ¼ 1).
3 Calculate all the t values located between tmin and
tmax.
4 Determine whether the arbitrary values of each
parametric line lie within the clipping window.
a If the line intersects, shorten the line segment via a floatingpoint division.
b If the line doesn’t intersect the clipping window, discard it.
5 Display all clipped line segments.
Clipping Polygons
Extending two-dimensional line clipping to
polygons involves the clipping of polygons
against a clipping rectangle or against other
polygons, as is the case with hidden surface
removal.
 Polygon clipping algorithms are derived from
simple line clipping algorithms.
 We can thus clip a polygon against the edges of
a clipping rectangle.
 There is, however, a problem with this approach
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– clipping concave polygons can result in several
additional polygons as shown in Figure 12-19.
Clipping Polygons
Clipping Polygons
The illustrated problem is not encountered when
working with convex polygons.
 The only solution to this problem is to consider
the clipped region as a single polygon or to
tessellate (divide) the polygon into a number of
convex polygons
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Clipping Polygons
Three-Dimensional Clipping
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Three-dimensional objects are clipped against a bounding
volume as opposed to a clipping rectangle.
Specifically, when clipping in 3D space we need to
consider the intersections of lines and planes or polygons
and planes as opposed to the clipping of lines against
lines, which is the case with 2D space.
A simple technique often employed to reject objects
completely outside the clipping volume and to accept
objects entirely inside it is to create a bounding sphere
for the object and to subsequently test its edges against
that of the view volume.
A bounding volume, or bounding box, is simply a volume
containing a geometric object.
Three-Dimensional Clipping
Three-Dimensional Clipping
INTRODUCTION TO LEVEL OF
DETAIL (LOD) ALGORITHMS
Level of Detail algorithms are concerned with
the complexity of geometric objects; basically
decreasing an object’s mesh complexity (the
number of polygons) as the distance between
the object and the viewer increases.
 The main reason for using these algorithms is to
increase the efficiency of a rendering operation
by lessening the workload on the graphics
system.
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INTRODUCTION TO LEVEL OF
DETAIL (LOD) ALGORITHMS
INTRODUCTION TO LEVEL OF
DETAIL (LOD) ALGORITHMS
INTRODUCTION TO LEVEL OF
DETAIL (LOD) ALGORITHMS
INTRODUCTION TO LEVEL OF
DETAIL (LOD) ALGORITHMS