Post on 18-Jan-2016
168 471 Computer Graphics, KKU. Lecture 4
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The Computer Programming Laws
• Any given program, when running, is obsolete.
• Any given program costs more and takes longer.
• If a program is useful, it will have to be changed.
• If a program is useless, it will have to be documented.
• Any given program will expand to fill all available memory.
• The value of a program is proportional to the weight of its output.
• Program complexity grows until it exceeds the capability of the programmer who must maintain it.
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Points
• 3One of the fundamental objects in D space th athas posi t i on.
• TTT TTTTT TTT TTT TTTTTTTTT TT TTTTTTTT TTTTTTcs.
• Distinguishedf r omeach ot her wi t h i t s posi t i on.• Drawn as dots.• TTTTTTT TT TTTTTTTTTTT TTTT TTTTTTT TTTT TT
P TTT Q.
• TTT TTTTTTTTT TTTTTTTTTT TTTTTT TT TTTTTTT TT TTTTTTTTT , points componentwise by utilizing parenthesis
e.g(x,y,z).
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Trigonometry
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Polar Coordinates
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Parametric Functions
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Parametric Lines
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Parametric Circles
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Plotting Parametric Functions
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• The fundamental 3D space objects like points.• Hasbot h magni t ude and di r ect i on.• Butposi t i onl ess.• Drawn as lines segments with arrow.• Denoted by lower case letters with an arrow ab
TTT TTTT TT • we represent vectors componentwise by utilizin
TTT T.< u,v,w>.• TTT TTTT TTTT TT TTTTT P TTT Q TTTTT TTTTTT ,
T TTTTTT TTTTTT TTTT TTTT
Vectors
v
v w
and
v P - Q
v
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Vectors
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Properties of Vectors
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Properties of Vectors (Cont.)
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“Multiplying” Vectors
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Vector Projection
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Planes
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Planes (Cont.)
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Matrices
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Matrix Multiplication
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Matrices (Cont.)
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Matrix Inversion
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• Cartesian coordinate system• Frame-based coordinate system• Device space• Screen Space• Image Space• Object Space• World Space
Coordinate Systems
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Cartesian coordinate system
Right-hand system
z
x y
Right-hand system
x
z
y
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Cartesian coordinate system(Cont.)
• In the computer graphics community• In computer graphics
Right-hand system
z
x
y
Right-hand system
x
z
y
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Frame-based coordinate system
• Consists of an origin and linearly independent vectors
( , , , )F u v w O
u
v
w
O
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• The actual coordinate system by which points are plotted on the graphics device.
• Measured in pixels• On SGI machines
– (0,0) is the lower left corner of a window.– X value increases to the right.– Y value increases upward.
• On Window platform– (0,0) is the upper left corner of a window.– X value increases to the right.– Y value increase downward.
Device Space
x
x
y
y
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Screen Space
• Defined to be the square on the xy plane.
• TTTTTT TT TTTTTT TTTTT TT TTTT TTTTTTT TTT T peci fi c gr aphi cs devi ces
1 , 1x y
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Image Space
• Used in 3-D system to allow for depth calculations.
• Defined by• Points in this space are mapped onto screen
space by projecting them onto the plane .• For example, any point in image space
will be protected to the center of the display screen.
1 , , 1x y z
0z (0,0, )z
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World Space
• The coordinate system of the scene • Must be projected into image space for
viewing.
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Object Space
• Localized system of an object being created.• After objects are created, about the world
space they can be– transformed– duplicated– moved– .etc
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Affine Space
• Made up of points and vectors• In affine space,
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Affine Space (Cont.)
• Vectors can be added, but points cannot.• Vectors can be scaled, but point cannot.
That is
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What is Affine Combination?
If are points and are scalars such that
then
is defined to be the point
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Affine Combination of 2 Points
TTT P1
TTT P2
TTTTTT TT TT TTTTTT TTTTTT TTTTTTTT, the expression
This P point represents a point on the line that passes
through P1 and P2
T TTT TTTT TT then P is somewhere on the line
segment joining P1 and P2 .
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Affine Combination of 3 Points
Consider three points P1 P2 and P3 that form a triangle,a point P can be defined by
or
then the point will be within (or on the boundary) of the triangle.
If
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Convex VS Non-Convex
Convex Set Non-Convex Set
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Coordinates of a point in the frame
Given a frame , a point P can be written
as Where O is an origin of the frame and
so the point can be written uniquely as
The set of values are the coordinates TT P relative to the frame .
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Cartesian Frames
The 2D standard Cartesian frame, denoted as , has A point P = will has the coordinate (x,y) relative to this frame
A Cartesian frame with any dimension consists of(0,0,…..,0) as an origin<1,0,…,0>, <0,1,…,0>,…, <0,0,…,1> as
basis vectors
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Examples
Given the frame which has the origin (0,0) and the vectors u = <1,0> and v = <1,1>, the point P that has its coordinate (5,3) in this frame, can be written as
which is the Cartesian coordinate (8,3).
or
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Matrix Representation of Points
T TTTT T TT TT TTTTTT TTTTT T T T , can write a point P uniquely as
or
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Matrix Representation of Vectors
Given a frame in an affine space , we can write a vector v uniquely as
or
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Converting between Frames
Suppose that we have the point P , with coordinates (3,2) in the frame
. What are the coordinates o
f the point in the frame ?
Consider the two frames and _ where
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