CS U540 Computer Graphics

1
CS U540Computer Graphics

• Prof. Harriet Fell
• Spring 2007
• Lecture 19 February 22, 2007

2
Adding R, G, and B Values
http//en.wikipedia.org/wiki/RGB
3
RGB Color Cube
4
RGB Color Cube The Dark Side
5
Doug Jacobson's RGB Hex Triplet Color Chart
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Making Colors Darker
(1, 0, 0)
(0, 0, 0)
(.5, 0, 0)
(0, 1, 0)
(0, 0, 0)
(0, .5, 0)
(0, 0, 1)
(0, 0, 0)
(0, 0, .5)
(1, 1, 0)
(0, 0, 0)
(0, .5, .5)
(1, 0, 1)
(0, 0, 0)
(.5, 0, .5)
(0, 0, 0)
(.5, .5, 0)
(1, 1, 0)
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Getting Darker, Left to Right

• for (int b 255 b gt 0 b--)
• c new Color(b, 0, 0) g.setPaint(c)
• g.fillRect(8003(255-b), 50, 3, 150)
• c new Color(0, b, 0) g.setPaint(c)
• g.fillRect(8003(255-b), 200, 3, 150)
• c new Color(0, 0, b) g.setPaint(c)
• g.fillRect(8003(255-b), 350, 3, 150)
• c new Color(0, b, b) g.setPaint(c)
• g.fillRect(8003(255-b), 500, 3, 150)
• c new Color(b, 0, b) g.setPaint(c)
• g.fillRect(8003(255-b), 650, 3, 150)
• c new Color(b, b, 0) g.setPaint(c)
• g.fillRect(8003(255-b), 800, 3, 150)

8
Gamma Correction
half black half red
(127, 0, 0)

• Gamma

9
Making Pale Colors
(1, 0, 0)
(1, 1, 1)
(1, .5, .5)
(0, 1, 0)
(1, 1, 1)
(.5, 1, .5)
(0, 0, 1)
(1, 1, 1)
(.5, .5, 1)
(1, 1, 0)
(1, 1, 1)
(.5, 1, 1)
(1, 0, 1)
(1, 1, 1)
(1, .5, 1)
(1, 1, 1)
(1, 1, .5)
(1, 1, 0)
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Getting Paler, Left to Right

• for (int w 0 w lt 256 w)
• c new Color(255, w, w) g.setPaint(c)
• g.fillRect(3w, 50, 3, 150)
• c new Color(w, 255, w) g.setPaint(c)
• g.fillRect(3w, 200, 3, 150)
• c new Color(w, w, 255) g.setPaint(c)
• g.fillRect(3w, 350, 3, 150)
• c new Color(w, 255, 255) g.setPaint(c)
• g.fillRect(3w, 500, 3, 150)
• c new Color(255,w, 255) g.setPaint(c)
• g.fillRect(3w, 650, 3, 150)
• c new Color(255, 255, w) g.setPaint(c)
• g.fillRect(3w, 800, 3, 150)

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Portable Pixmap Format (ppm)

• A "magic number" for identifying the file type.
• A ppm file's magic number is the two characters
  "P3".
• Whitespace (blanks, TABs, CRs, LFs).
• A width, formatted as ASCII characters in
  decimal.
• Whitespace.
• A height, again in ASCII decimal.
• Whitespace.
• The maximum color value again in ASCII decimal.
• Whitespace.
• Width height pixels, each 3 values between 0
  and maximum value.
• start at top-left corner proceed in normal
  English reading order
• three values for each pixel for red, green, and
  blue, resp.
• 0 means color is off maximum value means color
  is maxxed out
• characters from "" to end-of-line are ignored
  (comments)
• no line should be longer than 70 characters

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ppm Example

• P3
• feep.ppm
• 4 4
• 15
• 0 0 0 0 0 0 0 0 0 15 0 15
• 0 0 0 0 15 7 0 0 0 0 0 0
• 0 0 0 0 0 0 0 15 7 0 0 0
• 15 0 15 0 0 0 0 0 0 0 0 0

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private void saveImage() String outFileName
my.ppm" File outFile new File(outFileName)
int clrR, clrG, clrB try PrintWriter out
new PrintWriter(new BufferedWriter(new
FileWriter(outFile))) out.println("P3") ou
t.print(Integer.toString(xmax-xmin1))
System.out.println(xmax-xmin1) out.print("
") out.println(Integer.toString(ymax-ymin1))
System.out.println(ymax-ymin1) out.println("2
55") for (int y ymin y lt ymax
y) for (int x xmin x lt xmax x)
// compute clrR, clrG, clrB out.pri
nt(" ") out.print(clrR) out.print(" ")
out.print(clrG) out.print(" ")
out.println(clrB) out.close()
catch (IOException e) System.out.println(e.toS
tring())
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Vectors

• A vector describes a length and a direction.

a zero length vector
a
b
1
a unit vector
a b
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Vector Operations
b
ba
a
c-d
b
Vector Sum
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Cartesian Coordinates

• Any two non-zero, non-parallel 2D vectors form a
  2D basis.
• Any 2D vector can be written uniquely as a linear
  combination of two 2D basis vectors.
• x and y (or i and j) denote unit vectors parallel
  to the x-axis and y-axis.
• x and y form an orthonormal 2D basis.
• a xax yay
• a
  ( xa, ya) or
• x, y and z form an orthonormal 3D basis.

or a (ax,ay)
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Vector Length

• Vector a ( xa, ya )

a
ya
a
xa
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Dot Product

• Dot Product
• a ( xa, ya ) b ( xb, yb )
• a?b xa xb ya yb
• a?b ?a???b?cos(?)

xa ?a?cos(??) xb ?b?cos(?) xa
?a?sin(??) xb ?b?sin(?)
?
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Projection

• a ( xa, ya ) b ( xb, yb )
• a?b ?a???b?cos(?)
• The length of the projection of a onto b is
  given by

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3D Vectors

• This all holds for 3D vectors too.
• a ( xa, ya, za ) b ( xb, yb, zb )

a?b xa xb ya yb za zb a?b
?a???b?cos(?)
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Vector Cross Product
axb
axb is perpendicular to a and b.
Use the right hand rule to determine the
direction of axb.
b
?
a
Image from www.physics.udel.edu
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Cross Product and Area
axb
b
?
a
axb area pf the parallelogram.
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Computing the Cross Product
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Linear Interpolation

• LERP /lerp/, vi.,n.
• Quasi-acronym for Linear Interpolation, used as a
  verb or noun for the operation. Bresenham's
  algorithm lerps incrementally between the two
  endpoints of the line.
• p (1 t) a t b a t(b a)

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p (1 t) a t b a t(b a)
b
L
a
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Triangles
a
If (x, y) is on the edge ab, (x, y) (1 t) a
t b a t(b a). Similar formulas hold for
points on the other edges. If (x, y) is in the
triangle (x, y) ? a ? b ? c ? ? ?
1 (? , ? , ? ) are the Barycentric
coordinates of (x, y).
(x,y)
c
b
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Triangles
p a ?(b-a) ?(c-a)
? 2
? 1
? 0
? -1
? 2
p (1- ? - ?)a ?b ?c
? 1- ? - ? p p(?, ?, ?) ?a
?b ?c
? 1
c
c-a
? 0
b
b-a
a
Barycentric coordinates
? -1
? 0
? 1
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ComputingBarycentric Coordinates
a
b
c
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Barycentric Coordinates as Areas
a
(x,y)
where A is the area of the triangle.
? ? ? 1
c
b
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3D Triangles
a
This all still works in 3D.
(x,y,z)
where A is the area of the triangle.
? ? ? 1
c
b
But how do we find the areas of the triangles?
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3D Triangles - Areas
axb
B
b
C
?
?
a
A
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Triangle Assignment

• http//www.ccs.neu.edu/home/fell/CSU540/programs/C
  SU540ColorTriangle.html