Shadows

1
Shadows

• Soon Tee Teoh
• CS 116B

2
Motivation for Shadows

• Shadows make images more realistic
• Shadows give more depth perception

No shadow Where is the ball in relation to the
plane?
3
Shadow position hints at object position
4
How to Make Shadows?

• Can use ray-tracing methods (well cover ray
  tracing later in the semester) but ray tracing is
  slow
• Arbitrary casting of shadows among different
  objects is difficult to calculate. Need to search
  for all objects and calculate projections for
  each pair.
• For this class, we only cover simple shadows A
  triangle casts a shadow on the ground plane.

5
Simple shadow on a ground plane

• Assume directional light (sun is at an infinite
  distance away, so sunlight rays are parallel)
• Then, the model is as follows

Sun
Object
Shadow
Ground
6
Simple Shadow From Sun

• Suppose that the ground is on the x-z plane, that
  is at y0.
• Suppose that the light rays are in the direction
  (xL,yL,zL)
• Suppose that a point on the object is at
  (x0,y0,z0)
• Then, the projected (shadow) point on the ground
  (xS,yS,zS) would be (x0,y0,z0) k(xL,yL,zL)
• But yS 0. Therefore, y0 kyL 0.
• Re-arranging, k -y0/yL
• Then, substituting into xS x0 kxL, we get xS
  x0 y0xL/yL
• Similarly for zS.
• In this way, we get the following transformation
  matrix

xS 1 -xL/yL 0 0 x0
yS 0 0 0 0
y0

zS 0 -zL/yL 1 0 z0
1 0 0 0 1
1
7
OpenGL Code for Directional Light Source
void display( void ) float m16
1,0,0,0, 0,0,0,0, 0,0,1,0, 0,0.01,0,1
glMatrixMode(GL_MODELVIEW)
glLoadIdentity() glEnable(GL_LIGHTING)
glLightfv(GL_LIGHT0, GL_POSITION,
light_position) glEnable(GL_LIGHT0)
// Draw Actual Object

glMaterialfv() // make object color
glPushMatrix() glTranslatef(0.0,0.5,0.0)
glutSolidSphere(0.2,10.0,10.0)
glPopMatrix() // Draw the
Shadow
glMaterialfv() // make shadow color
glPushMatrix() m4 - light_position0/li
ght_position1 m6 - light_position2/l
ight_position1 glMultMatrixf(m)
glTranslatef(0.0,0.5,0.0)
glutSolidSphere(0.2,10.0,10.0)
glPopMatrix() glutSwapBuffers()
8
Simple Shadow from a Point Light Source to the
Ground Plane

• If the light source is a point light source, all
  we need to do is to change the projection from
  parallel projection to something similar to
  perspective projection.
• We first translate everything so that the light
  source is at the origin.
• Then we perform the perspective-like projection.
• Then we translate back.

y
y
Light
Light (xL,yL,zL)
x
Object
Translate by (-xL,-yL,-zL)
Object (x0,y0,z0)
Ground Plane
Ground Plane
x
Projected Point
9
Simple Shadow from a Point Light Source to the
Ground PlaneThe Mathematics behind the projection
y
Let the projected point be (xP,yP,zP)
Light
x
yP -yL xP x0(-yL)/y0
x0
yL
Object
zP z0(-yL)/y0
xp
Ground Plane
Therefore, transformation matrix is
Projected Point
1 0 0 0 x0
0 1 0 0 y0
How to derive this Using law of similar
triangles, xp/yp x0/y0 gt xp x0(-yL)/y0
0 0 1 0 z0
0 -1/yL 0 0 1
10
Finally, we translate back
y
y
Light (xL,yL,zL)
Light
x
x0
yL
Object
Object (x0,y0,z0)
Translate by (xL,yL,zL)
xp
Ground Plane
Ground Plane
x
Projected Point
Projected Point
11
OpenGL Code for Point Light Source
void sideGlutDisplay( void ) float m16
1,0,0,0, 0,1,0,0, 0,0,1,0, 0,0.01,0,0
glMatrixMode(GL_MODELVIEW)
glLoadIdentity() glEnable(GL_LIGHTING)
glLightfv(GL_LIGHT0, GL_POSITION,
light_position) glEnable(GL_LIGHT0)
// the object
glMaterialfv() //
object color glPushMatrix()
glTranslatef(0.0,0.5,0.0)
glutSolidSphere(0.2,10.0,10.0)
glPopMatrix() // the
shadow
glMaterialfv() // shadow color
glPushMatrix() m7 -1.0/light_position1
glTranslatef(light_position0,light_posi
tion1,light_position2)
glMultMatrixf(m) glTranslatef(-light_positio
n0,-light_position1,-light_position2)
glTranslatef(0.0,0.5,0.0)
glutSolidSphere(0.2,10.0,10.0)
glPopMatrix() glutSwapBuffers()