Shadows via Projection

1
Shadows via Projection

• Glenn G. ChappellCHAPPELLG_at_member.ams.org
• U. of Alaska Fairbanks
• CS 381 Lecture Notes
• Wednesday, November 5, 2003

2
ReviewTwo Kinds of Interfaces

• Now, about that translation
• In our zoom pan, driving, the translation
  was done in the display function, not the saved
  matrix.
• But in flying the translation was in the saved
  matrix.
• Advanced interfaces can be divided into two
  categories
• View the world interfaces.
• We look at the scene.
• Rotation and scaling are centered on the point we
  are looking at.
• So the translation does not go in the saved
  matrix.
• Examples
• Driving from above.
• Object manipulation.
• Move in the world interfaces.
• In these, we are explicitly inside the scene.
• Rotation is centered on the camera.
• So the translation goes in the saved matrix.
• Examples
• Flying.
• Driving, where we appear to be inside the car.

3
ReviewMouse-Based Viewing 1/2

• Last time we will look at two mouse-based viewing
  interfaces.
• Click-and-drag rotation.
• View an object and rotate it in an intuitive way
  with the mouse (click-and-drag).
• Flying with the mouse.
• Fly forward on mouse-button press.
• Turn in the direction of the mouse position.
• The first is a view the world type of
  interface. The second is a move in the world
  type of interface.
• These facts have ramifications for how the
  interfaces are implemented. (See the previous
  slide.)
• Both of these involve rotation about a line
  perpendicular to a vector computed from mouse
  positions.
• First The vector is the difference between the
  current and previous mouse positions. (So we
  needed to save the mouse position.)
• Second The vector is the difference between the
  current mouse position and the center of the
  screen.

4
ReviewMouse-Based Viewing 2/2

• The following code was useful in both interfaces,
  for doing the perpendicular rotation.
• // perprot
• // Rotates about an axis in the x,y-plane,
  perpendicular to the given
• // vector (vx, vy). Rotation amount is
  proportional to the length of
• // the vector and to rotmultiplier.
• void perprot(double rotmultiplier, double vx,
  double vy)
• double len sqrt(vxvx vyvy)
• glPushMatrix()
• glRotated(rotmultiplierlen, -vy,vx,0.)
• glMultMatrixd(your_matrix_variable)
• glGetDoublev(GL_MODELVIEW_MATRIX,
  your_matrix_variable)
• glPopMatrix()
• glutPostRedisplay()

5
Shadows via ProjectionIntroduction 1/2

• Now that we know something about transformation
  matrices, we can use them to draw shadows.
• Covering shadows before covering lighting is a
  bit odd, of course.
• This topic is covered in section 5.10 of the
  (blue) text. My presentation differs somewhat
  from that in the text.
• What is a shadow?
• Answer 1 A region where light is blocked.
• Thats a great answer, but to use it, we need to
  know something about lighting. Maybe next week
• Answer 2 The projection of the shape of an
  object on another object.
• We say shape here, since we do not want to draw
  the shadow using the same colors as the original
  object.
• We know (more or less) how to project onto a
  plane. We can use this knowledge to draw the
  shadow of an object on a plane.

6
Shadows via ProjectionIntroduction 2/2

• We wish to find the matrix that projects onto a
  given plane from a given light position.
• This is essentially the synthetic-camera model.
  The light is at the center of projection, and the
  plane is the image plane.
• Does this mean that, with this method, objects
  will still cast shadows when they do not lie
  between the light and the plane? Yes, it does.

7
Shadows via ProjectionThe Math 1/2

• To compute the proper projection matrix, we need
  the light position and an equation for the plane.
• We represent the light position as a 4-D vector L
  in homogeneous form, as shown below.
• The equation of a plane in 3-D space can be
  written as Ax By Cz D 0. We represent
  this with another 4-D vector P, as shown below.
• A point V (expressed in homogeneous form) lies on
  the plane precisely when VP 0.

Plane EquationAx By Cz D 0P (A, B,
C, D)
Light PositionL (Lx, Ly, Lz , 1)
8
Shadows via ProjectionThe Math 2/2

• Now, let k PL be the dot product of P and L.
• The matrix that does the projecting is given
  below. On the left, we use matrix notation (which
  you may not be familiar with).
• What do we do with this matrix?
• It does projection, so we might be tempted to put
  it in OpenGLs projection transformation.
• However, it does not set camera properties it
  places objects in the world. (A shadow is an
  object!)
• So the matrix goes in model/view, after the
  viewing transformations, and before moving the
  object that casts the shadow.

9
Shadows via ProjectionSome Code

• Look at shadowproj.cpp, on the web page, for code
  that uses this method to draw a shadow.
• The light position is stored in the global
  lightpos.
• The plane is specified by putting the coordinates
  of points in the global planepts.
• Function findplane computes and returns the plane
  equation (A, B, C, D). This is called from
  init, where these values are stored in the global
  plane_eq.
• Function makeshadowmatrix computes and returns
  the shadow projection matrix. This is also called
  from init, where the matrix is stored in the
  global shadowmat.
• The object is drawn by drawobject.
• The parameter is true if the object is to be
  drawn in normal color, false if it is to be drawn
  in gray.
• This function is called twice once to draw the
  object, and once to draw the shadow.
• The shadow matrix is used in function display.

10
Shadows via ProjectionIssues

• If we draw the shadow on a polygon, then the
  depth test might give us trouble.
• One solution is to disable the depth test before
  drawing the shadow. This only works if no part of
  the shadow is hidden by any previously drawn
  object.
• The shadow, being an object, is not confined to
  the polygon it (supposedly) is cast on.
• We can solve this using stenciling, a method
  for restricting drawing to certain portions of
  the frame buffer.
• Shadows are not gray they are unlit.
• To do shadows properly, we draw the unshadowed
  portion with full lighting, and the shadowed
  portion using only ambient light. More on this
  when we cover lighting.
• Suppose we use this method to draw a shadow
  falling on a more complex object.
• For each polygon the shadow falls on, we would
  need to compute a separate matrix and draw the
  object casting the shadow.
• So this method is not recommended for casting
  shadows on complex objects. There are other
  shadowing methods more on these later.