CS G140 Graduate Computer Graphics

1
CS G140Graduate Computer Graphics

• Prof. Harriet Fell
• Spring 2006
• Lecture 2 - January 23, 2006

2
Todays Topics

• Ray Tracing
• Ray-Sphere Intersection
• Light Diffuse Reflection
• Shadows
• Phong Shading
• More Math
• Matrices
• Transformations
• Homogeneous Coordinates

3
Ray Tracinga World of Spheres
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What is a Sphere

• Vector3D center // 3 doubles
• double radius
• double R, G, B // for RGB colors between 0 and
  1
• double kd // diffuse coeficient
• double ks // specular coefficient
• int specExp // specular exponent 0 if ks 0
• (double ka // ambient light coefficient)
• double kgr // global reflection coefficient
• double kt // transmitting coefficient
• int pic // gt 0 if picture texture is used

5

• -.01 .01 500 800 // transform theta phi mu
  distance
• 1 // antialias
• 1 // numlights
• 100 500 800 // Lx, Ly, Lz
• 9 // numspheres
• //cx cy cz radius R G B ka kd ks specExp kgr
  kt pic
• -100 -100 0 40 .9 0 0 .2 .9 .0 4 0
  0 0
• -100 0 0 40 .9 0 0 .2 .8 .1 8 .1
  0 0
• -100 100 0 40 .9 0 0 .2 .7 .2 12 .2
  0 0
• 0 -100 0 40 .9 0 0 .2 .6 .3 16 .3
  0 0
• 0 0 0 40 .9 0 0 .2 .5 .4 20 .4
  0 0
• 0 100 0 40 .9 0 0 .2 .4 .5 24 .5
  0 0
• 100 -100 0 40 .9 0 0 .2 .3 .6 28 .6
  0 0
• 100 0 0 40 .9 0 0 .2 .2 .7 32 .7
  0 0
• 100 100 0 40 .9 0 0 .2 .1 .8 36 .8
  0 0

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World of Spheres

• Vector3D VP // the viewpoint
• int numLights
• Vector3D theLights5 // up to 5 white lights
• double ka // ambient light coefficient
• int numSpheres
• Sphere theSpheres20 // 20 sphere max
• int ppmT3 // ppm texture files
• View sceneView // transform data
• double distance // view plane to VP
• bool antialias // if true antialias

7
Simple Ray Tracing for Detecting Visible Surfaces

• select window on viewplane and center of
  projection
• for (each scanline in image)
• for (each pixel in the scanline)
• determine ray from center of projection
• through pixel
• for (each object in scene)
• if (object is intersected and
• is closest considered thus far)
• record intersection and object name
• set pixels color to that of closest object
  intersected

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Ray Trace 1Finding Visible Surfaces
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Ray-Sphere Intersection

• Given
• Sphere
• Center (cx, cy, cz)
• Radius, R
• Ray from P0 to P1
• P0 (x0, y0, z0) and P1 (x1, y1, z1)
• View Point
• (Vx, Vy, Vz)
• Project to window from (0,0,0) to (w,h,0)

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Sphere Equation
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Ray Equation

• P0 (x0, y0, z0) and P1 (x1, y1, z1)
• The ray from P0 to P1 is given by
• P(t) (1 - t)P0 tP1 0 lt t lt 1
• P0 t(P1 - P0)

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Intersection Equation

• P(t) P0 t(P1 - P0) 0 lt t lt 1
• is really three equations
• x(t) x0 t(x1 - x0)
• y(t) y0 t(y1 - y0)
• z(t) z0 t(z1 - z0) 0 lt t lt 1
• Substitute x(t), y(t), and z(t) for x, y, z,
  respectively in

13
Solving the Intersection Equation
is a quadratic equation in variable t. For a
fixed pixel, VP, and sphere, x0, y0, z0, x1,
y1, z1, cx, cy, cz, and R eye pixel
sphere are all constants. We solve for t using
the quadratic formula.
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The Quadratic Coefficients
Set dx x1 - x0 dy y1 - y0 dz z1 - z0
Now find the the coefficients
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Computing Coefficients
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The Coefficients
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Solving the Equation
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• The intersection nearest P0 is given by
• To find the coordinates of the intersection
  point

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First Lighting Model

• Ambient light is a global constant.
• Ambient Light ka (AR, AG, AB)
• ka is in the World of Spheres
• 0 ? ka ? 1
• (AR, AG, AB) average of the light sources
• (AR, AG, AB) (1, 1, 1) for white light
• Color of object S (SR, SG, SB)
• Visible Color of an object S with only ambient
  light
• CS ka (AR SR, AG SG, AB SB)
• For white light
• CS ka (SR, SG, SB)

20
Visible SurfacesAmbient Light
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Second Lighting Model

• Point source light L (LR, LG, LB) at (Lx, Ly,
  Lz)
• Ambient light is also present.
• Color at point p on an object S with ambient
  diffuse reflection
• Cp ka (AR SR, AG SG, AB SB) kd kp(LR SR, LG SG,
  LB SB)
• For white light, L (1, 1, 1)
• Cp ka (SR, SG, SB) kd kp(SR, SG, SB)
• kp depends on the point p on the object and (Lx,
  Ly, Lz)
• kd depends on the object (sphere)
• ka is global
• ka kd ? 1

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Diffuse Light
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Lambertian Reflection ModelDiffuse Shading

• For matte (non-shiny) objects
• Examples
• Matte paper, newsprint
• Unpolished wood
• Unpolished stones
• Color at a point on a matte object does not
  change with viewpoint.

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Physics of Lambertian Reflection

• Incoming light is partially absorbed and
  partially transmitted equally in all directions

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Geometry of Lamberts Law
90 - ?
?
dAcos(?)
90 - ?
Surface 2
Surface 1
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cos(?)N?L
Cp ka (SR, SG, SB) kd N?L (SR, SG, SB)
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Finding N
N (x-cx, y-cy, z-cz) (x-cx, y-cy, z-cz)
normal
(x, y, z)
radius
(cx, cy, cz)
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Diffuse Light 2
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Time for a Break
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Shadows on Spheres
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More Shadows
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Finding Shadows
Shadow Ray
P
Pixel gets shadow color
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Shadow Color

• Given
• Ray from P (point on sphere S) to L (light)
• P P0 (x0, y0, z0) and L P1 (x1, y1, z1)
• Find out whether the ray intersects any other
  object (sphere).
• If it does, P is in shadow.
• Use only ambient light for pixel.

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Shape of Shadows
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Different Views
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Planets
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Starry Skies
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Shadows on the Plane
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Finding Shadowson the Back Plane
Shadow Ray
P
Pixel in Shadow
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Close up
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On the Table
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Phong Highlight
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Phong Lighting Model
Light Normal Reflected View
The viewer only sees the light when ? is 0.
N
L
R
V
?
?
?
We make the highlight maximal when ? is 0, but
have it fade off gradually.
Surface
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Phong Lighting Model
cos(?) R?V We use cosn(?). The higher n is,
the faster the drop off.
N
L
R
V
For white light
?
?
?
Surface
Cp ka (SR, SG, SB) kd N?L (SR, SG, SB) ks
(R?V)n(1, 1, 1)
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Powers of cos(?)
cos10(?) cos20(?) cos40(?) cos80(?)
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Computing R
L R (2 L?N) N R (2 L?N) N - L
L
R
N
LR
R
L
?
?
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The Halfway Vector

• From the picture
• ? ? - ? ?
• So ? ?/2.

H L V L V Use H?N instead of
R?V. H is less expensive to compute than R.
N
H
This is not generally true. Why?
L
R
?
V
?
?
?
Surface
Cp ka (SR, SG, SB) kd N?L (SR, SG, SB) ks
(H?N)n (1, 1, 1)
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Varied Phong Highlights
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Varying Reflectivity
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More Math

• Matrices
• Transformations
• Homogeneous Coordinates

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Matrices

• We use 2x2, 3x3, and 4x4 matrices in computer
  graphics.
• Well start with a review of 2D matrices and
  transformations.

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Basic 2D Linear Transforms
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Scale by .5
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Scaling by .5
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General Scaling
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General Scaling
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Rotation
-sin(?)
cos(?)
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Rotation
?
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Reflection in y-axis
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Reflection in y-axis
y
x
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Reflection in x-axis
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Reflection in x-axis
y
y
x
x
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Shear-x
s
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Shear x
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Shear-y
s
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Shear y
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Linear Transformations

• Scale, Reflection, Rotation, and Shear are all
  linear transformations
• They satisfy T(au bv) aT(u) bT(v)
• u and v are vectors
• a and b are scalars
• If T is a linear transformation
• T((0, 0)) (0, 0)

68
Composing Linear Transformations

• If T1 and T2 are transformations
• T2 T1(v) def T2( T1(v))
• If T1 and T2 are linear and are represented by
  matrices M1 and M2
• T2 T1 is represented by M2 M1
• T2 T1(v) T2( T1(v)) (M2 M1)(v)

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Reflection About an Arbitrary Line (through the
origin)
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Reflection as a Composition
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Decomposing Linear Transformations

• Any 2D Linear Transformation can be decomposed
  into the product of a rotation, a scale, and a
  rotation if the scale can have negative numbers.
• M R1SR2

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Rotation about an Arbitrary Point
?
This is not a linear transformation. The origin
moves.
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Translation
(x, y)?(xa,yb)
y
y
(a, b)
x
x
This is not a linear transformation. The origin
moves.
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Homogeneous Coordinates
y
Embed the xy-plane in R3 at z 1. (x, y) ? (x,
y, 1)
y
x
x
z
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2D Linear Transformations as 3D Matrices

• Any 2D linear transformation can be represented
  by a 2x2 matrix

or a 3x3 matrix
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2D Linear Translations as 3D Matrices

• Any 2D translation can be represented by a 3x3
  matrix.

This is a 3D shear that acts as a translation on
the plane z 1.
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Translation as a Shear
y
y
x
x
z