CS3162 Introduction to Computer Graphics
Helena Wong, 2000
5. Three Dimensional Transformations
Methods for geometric transforamtions and object modelling in 3D are extended from 2D methods by including
the considerations for the z coordinate.
Basic geometric transformations are: Translation, Rotation, Scaling
5.1 Basic Transformations
Translation
We translate a 3D point by adding translation distances, tx, ty, and tz, to the original coordinate
position (x,y,z):
x' = x + tx, y' = y + ty, z' = z + tz
Alternatively, translation can also be specified by the transformation matrix in the following formula:
 x ' 1
 y' 0
 =
 z '  0
  
 1  0
Exercise:
0 0 tx 
1 0 t y 
0 1 tz 

0 0 1
x
 y
 
z
 
1
translate a triangle with vertices at original coordinates (10,25,5), (5,10,5), (20,10,10) by
tx=15, ty=5,tz=5. For verification, roughly plot the x and y values of the original and
resultant triangles, and imagine the locations of z values.
Scaling With Respect to the Origin
We scale a 3D object with respect to the origin by setting the scaling factors sx, sy and sz, which are
multiplied to the original vertex coordinate positions (x,y,z):
x' = x * sx, y' = y * sy, z' = z * sz
Alternatively, this scaling can also be specified by the transformation matrix in the following formula:
 x ' s x
 y'  0
 =
 z'  0
  
1   0
Exercise:
0
sy
0
0
0
sz
0
0
0
0
0

1
x
 y
 
z
 
1
Scale a triangle with vertices at original coordinates (10,25,5), (5,10,5), (20,10,10) by
sx=1.5, sy=2, and sz=0.5 with respect to the origin. For verification, roughly plot the x and
y values of the original and resultant triangles, and imagine the locations of z values.
Scaling with respect to a Selected Fixed Position
Exercise:
What are the steps to perform scaling with respect to a selected fixed position? Check
your answer with the text book.
Exercise:
Scale a triangle with vertices at original coordinates (10,25,5), (5,10,5), (20,10,10) by
sx=1.5, sy=2, and sz=0.5 with respect to the centre of the triangle. For verification, roughly
plot the x and y values of the original and resultant triangles, and imagine the locations of
z values.
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
Coordinate-Axes Rotations
A 3D rotation can be specified around any line in space. The easiest rotation axes to handle are the
coordinate axes.
Z-axis rotation:
x' = x cos ? - y sin ?,
y' = x sin ? + y cos ?, and
z' = z
or:
 x ' cos θ − sin θ
 y'  sin θ cos θ
 =
 z'  0
0
  
0
1   0
X-axis rotation:
0 0  x 
0 0  y
1 0  z 
 
0 1  1 
y' = y cos ? - z sin ?,
z' = y sin ? + z cos ?, and
x' = x
or:
0
 x ' 1
 y'  cos θ − sin θ
 =
 z '  0 sin θ cos θ
  
0
0
 1  0
Y-axis rotation:
0
0
0

1
x
 y
 
z
 
1
z' = z cos ? - x sin ?,
x' = z sin ? + x cos ?, and
y' = y
or:
sin θ 0  x 
 x '  cos θ
 y'  − sin θ
cos θ 0  y
 =
 z'  0
0  z 
  
 
0
0
1  1 
1   0
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
3D Rotations About an Axis Which is Parallel to an Axis
Step 1. Translate the object so that the rotation axis coincides with the parallel coordinate axis.
Step 2. Perform the specified rotation about that axis.
Step 3. Translate the object so that the rotation axis is moved back to its original position.
General 3D Rotations
Step 1.
Step 2.
Step 3.
Step 4.
Step 5.
Translate the object so that the rotation axis passes through the coordinate origin.
Rotate the object so that the axis of rotation coincides with one of the coordinate axes.
Perform the specified rotation about that coordinate axis.
Rotate the object so that the rotation axis is brought back to its original orientation.
Translate the object so that the rotation axis is brought back to its original position.
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
6. Coloring, Images, and File Formats
6.1 Color Models
A color model is a method for explaining the properties or behaviour of color within some particular
context.
Properties of Light and Colors
Light or colors are from a narrow frequency band within the electromagnetic spectrum:
Hue (Color): The dominant frequency reflected by the object
Brightness: The perceived intensity of the light. This is related to the luminance of the source.
Purity (Saturation): How pure the color of the light appears.
Chromaticity: collectively refer to purity and hue.
Complementary colors: eg. Red and Cyan, Green and Magenta, Blue and Yellow, which combine and
form white light.
Primary colors: eg. r,g,b - starting from these colors, a wide range of other colors can be formed.
Shades of a color: created by adding black pigment to the color.
Tints of a color: created by adding white pigment to the color.
Tones of a color: created by adding black or white pigments to the color.
RGB Model
Used by RGB monitors which separate signals
for the red, green, blue components of an
image.
Based on the vision system of our eyes, that
we have 3 visual pigments in the cones of the
retina, for red, green, and blue.
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
YIQ Model
Used by the NTSC televison monitors:
Y - Contain the luminance (brightness) information
I - Contain the orange-cyan hue information
Q - Contain the green-magenta hue information
CMY Model
Useful for describing color output to hard-copy devices.
These devices produce a color picture by coating a paper
with color pigments. We see the colors by reflected light,
which is a subtractive process.
CMYK means using the ink of Cyan, Magenta, Yellow, and
Black.
Consider that,
- Magenta ink indeeds subtracts the green component from incident light, so the remaining red and
blue components are seen by us, as a resultant color of magenta.
- Cyan ink indeeds subtracts the red component from incident light, so the remaining green and
blue components are seen by us, as a resultant color of cyan.
- If we mix the ink of magenta and cyan, then, this ink subtracts the green and red component from
the incident light, and the remaining blue component is seen by us, as a resultant color of blue.
HSV Model
In this model, users select a spectral color and
apply the amounts of white or black that are to
be added to obtain different shades, tints, and
tones.
HSV model is derived from the RGB cube.
H: Hue
S: Saturation
V: Value
Saturation and value are measured as the
horizontal and vertical axes. And a degree (0 360) describes the hue. Complementary colors
are 180 degrees apart.
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
Interesting facts about human eyes:
- can distinguish about 128 different hues
- can distinguish about 130 different tints (saturation levels)
- can distinguish about 23 shades for yellow and about 16 different shades for blue colors
- hence, can distinguish about 128 x 130 x 23 = 82720 different colors.
HSV Model
Used by Tektronix.
H: Hue
L: Lightness
S: Saturation
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
6.2 Light Intensities
Values of intensity calculated by an illumination model must be converted to one of the allowable
intensity levels for the particular graphics system in use.
We have 2 issues to consider:
1. Human perceive relative light intensities on a logarithmic scale.
Eg. We perceive the difference between intensities 0.20 and 0.22 to be the same as the difference
between 0.80 and 0.88. Therefore, to display successive intensity levels with equal perceived
differences of brightness, the intensity levels on the monitor should be spaced so that the ratio of
successive intensities is constant:
I1/I0 = I2/I1 = I3/I2 = … = a constant.
2. The intensities produced by display devices are not linear with the electron-gun voltage.
This is solved by applying a gamma correction for video lookup correction:
Voltage for intensity Ik is computed as:
Vk = (Ik / a )1 / ?
Where a is a constant and ? is an adjustment factor controlled by the user.
For example, the NTSC signal standard is ?=2.2.
6.3 Halftone Patterns and Dithering
Halftoning is used when an output device has a limited intensity range, but we want to create an
apparent increase in the number of available intensities.
Example: The following shows an original picture and the display of it in output devices of limited
intensity ranges (4 colors, 8 colors, 16 colors):
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
If we view a very small area from a sufficiently large
viewing distance, our eyes average fine details within the
small area and record only the overall intensity of the area.
By halftoning, each small resolution unit is imprinted with a
circle of black ink whose area is proportional to the
blackness of the area in the original photograph.
Graphics output devices can approximate the variable-area
circles of halftone reproduction by incorporating multiple
pixel positions into the display of each intensity value.
A 2 x 2 pixel grid used to display 5 intensity levels (I) on a bilevel system:
A 3 x 3 pixel grid used to display 10 intensities on a bilevel system:
A 2x2 pixel grid used to display 13 intensities on a 4-level system:
An RGB 2x2 pixel-grid pattern:
Each r,g,b color has 4 phosphor dots in the pattern, which allows 5
possible settings per color. This gives a total of 125 different color
combinations.
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
Dithering
The above approach, however, needs a higher resolution output device to display a picture in the same physical
dimensions. So, in reality, we have to refine this approach so that it does not require for higher resolution.
Dithering generally means to approximate halftone without this requirement. Interested students may find
further discussion on dithering in many text books.
Below are two examples of dithering results, using 4 and 2 colors respectively.
6.4 Anti-Aliasing
On dealing with integer pixel positions, jagged or stairstep appearances happen very usually. This
distortion of information due to undersampling is called aliasing. A number of antialiasing methods
have been developed to compensate this problem.
One way is to display objects at higher resolution. However there is a limit to how big we can make
the frame buffer and still maintaining acceptable refresh rate.
Other methods modify pixel intensities by varying them along the boundaries of primitives =>
smoothing the edges. These include supersampling, area sampling, and pixel phasing.
Supersampling
In supersampling, intensity information is obtained from multiple points that contribute to the overall
intensity of a pixel.
For straight lines with zero widths:
3 intensity values are used to paint the line
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
For straight lines with finite widths:
9 intensity values are used to paint the line
Pixel-Weighting Masks: Supersampling can be implemented by giving more weight to sub-pixels near
the center of a pixel area.
1 2 1
2
4
2
1
2
1
Filtering Technique: Similar to pixel-weighting. Instead of using the grid of weighting values, we
imagine a continuous weighting surface covering the pixel:
Area Sampling
In area sampling, we set each pixel intensity proportional to the area of overlap of the pixel.
Pixel Phasing
Move the electron beam to more nearly approximate positions (Micropositioning).
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
Compensation for Line Intensity Differences
Diagonal lines normally appear less bright than horizontal line. Some compensation should be done
to compensate for this effect by adjusting the intensity of each line according to its slope.
Antialiasing Area Boundaries
To smooth area outlines, we may adjust each pixel intensity at the boundary positions according to the
percent of pixel area that is inside the boundary.
Temporal Aliasing
Aliasing problems also appear in temporal cases:
-
The wheel is in fact rotating clockwisely. But it appears rotating counter-clockwisely.
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
6.6 Graphic Files
Bitmap Images and Vector Images
Graphic file formats differ in the way they represent graphic information— either as vector drawings
or as bitmap images. Some formats contain only vector drawings or bitmap images, but many include
both in the same file.
1. Bitmap images
Most image-editing applications generate bitmap images, also called raster images.
A bitmap image is resolution-dependent—that is, it contains a fixed number of pixels to represent its
image data. As a result, a bitmap image can lose detail and appear jagged if viewed at a high
magnification on-screen or printed at too low a resolution.
Bitmap images are the best choice for representing gradations of shades and color—for example, in
photographs or painted images.
Common imaging applications provide a collection of tools to improve or transform bitmap images,
eg. Sharpening filters, Blurring filters (normal blurring, Gaussian blur, Motion blur, Unsharp Mask)
2. Vector graphics
Some drawing applications create vector graphics, made of lines and curves defined by mathematical
objects called vectors. Vectors describe graphics according to their geometric characteristics.
A vector graphic is resolution-independent—that is, it can be scaled to any size and printed on any
output device at any resolution without losing its detail or clarity. As a result, vector graphics are the
best choice for graphics that must retain crisp lines when scaled to various sizes—for example, logos.
Graphic File Sizes
Generally graphic file size is proportional to the pixel dimensions of the image. Images with more
pixels may produce more detail at a given printed size, but they require more disk space to store and
may be slower to edit and print.
Eg. a 1-inch-by-1-inch 200-dpi (dot per inch) image contains 200 x 200 pixels, which is four times as
many pixels as a 1-inch-by-1-inch 100-dpi image and so has four times the file size.
To determine the image resolution to use, one should consider the medium of final distribution for the
image. For on-screen display, the image resolution only needs to match the typical monitor resolution
(72 or 96 ppi). For printing on laser printer, a resolution of at least 150 dpi is preferred.
Note that, using too low a resolution for a printed image results in pixelation— output with large,
coarse-looking pixels. Using too high a resolution (pixels smaller than what the output device can
produce) increases the file size and slows the printing of the image, and some printer may not have
enough internal memory to cope with the image.
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
Color Modes
To cope with different requirement and characteristics of image files, most applications can handle
multiple coloring modes.
1. Bitmap mode
Uses one of two color values (black or white) to represent the pixels in an image. Images in Bitmap
mode are called bitmapped, or 1-bit, images because they have a bit depth of 1.
2. Grayscale mode
Uses up to 256 shades of gray. Every pixel of a grayscale image has a brightness value ranging from 0
(black) to 255 (white).
3. Indexed color mode
Uses at most 256 colors. When converting a true-color image to indexed color, a color lookup table is
built to stores and indexes the colors in the image. If a color in the original image does not appear in
the table, the software chooses the closest one or simulates the color using available colors.
By limiting the palette of colors, indexed color can reduce file size while maintaining visual quality.
4. Multichannel mode
Uses 256 levels in each multiple channel.
File Compression
Many image file formats use compression techniques to reduce the storage space required by bitmap
image data. Compression techniques are distinguished by whether they remove detail and color from
the image. Lossless techniques compress image data without removing detail; lossy techniques
compress images by removing detail.
-
Run Length Encoding (RLE) is lossless. This method scans the bitmap row by row. For each
row, it divides the line of colors into run-lengths according to the change of colors, then record for
each color how many pixels are to be painted.
-
Lemple-Zif-Welch (LZW) is a lossless. Similarly it archive by identifying repetitions of data.
But this is a more complex approach which builds a table of tokens as it encodes data of the
whole image, so that if it identifies one chunk of data and then encounters it again, it can store the
same token into the file. This technique is most useful in compressing images that contain large
areas of single color, such as screenshots or simple paint images.
-
Joint Photographic Experts Group (JPEG) is lossy. It achieves data compression through
sampling techniques in the context of digital signal processing. It is best for continuous-tone
images, such as photographs. You can manipulate the compression parameters to choose between
greater compression or greater accuracy.
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CS3162 Introduction to Computer Graphics
Helena Wong, 2000
Common File Formats
1. BMP
BMP is the standard Windows image format on DOS and Windows-compatible computers. The BMP
format supports RGB, indexed-color, grayscale, and Bitmap color modes.
2. GIF
The Graphics Interchange Format (GIF) is the file format commonly used to display indexed-color
graphics and images. GIF uses a LZW-compressed format. Transparent color is supported.
3. JPEG format
The Joint Photographic Experts Group (JPEG) format is commonly used to display photographs and
other continuous-tone images. The JPEG format supports CMYK, RGB, and grayscale color modes.
4. PCX
The PCX format supports RGB, indexed-color, grayscale, and Bitmap color modes. PCX supports the
RLE compression method. Images can have a bit depth of 1, 4, 8, or 24.
5. PDF
Portable Document Format (PDF) is used by Adobe Acrobat, Adobe’s electronic publishing software.
PDF files can represent both vector and bitmap graphics, and can contain electronic document search
and navigation features such as electronic links.
6. Raw
The Raw format is a flexible file format for transferring files between applications and computer
platforms. Raw format consists of a stream of bytes describing the color information in the file. Each
pixel is described in binary format, with 0 equalling black and 255 white (for images with 16-bit
channels, the white value is 65535).
7. TIFF
The Tagged-Image File Format (TIFF) is used to exchange files between applications and computer
platforms. TIFF is a flexible bitmap image format supported by virtually all paint, image-editing, and
page-layout applications. Also, virtually all desktop scanners can produce TIFF images.
The TIFF format supports CMYK, RGB, and grayscale files with alpha channels, and Lab, indexedcolor, and Bitmap files without alpha channels. TIFF also supports LZW compression.
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