Cornell CS465 Spring 2003

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FAQ for Homework 4a
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NOTE: the current version of the handout is version 4, revised Oct. 25.  The
most recent change was to correct a transpose that should have been an inverse
in Problem 2.  The previous change was to specify that in part 1.3
transformations can be about arbitrary points *and axes*.


Problem 1:

Q. What do you mean, transformations can be centered around arbitrary points?

A. Mathematically, this means that if M was an elementary transformation in
   part 2 and T is a translation, then T * M * T^-1 is an elementary
   transformation for part 3.


Q. What do you mean, transformations can be about arbitrary axes?

A. Mathematically, this means that if M was an elementary transformation in
   part 2, and R is any rotation, then R * M * R^-1 is an elementary
   transformation for part 3.


Q. How should I approach figuring out the answers?

A. To me the easiest way is to look at how the unit square moves and think
   about what transformations would do that.  One hint for part 1.3 is that
   if a transformation is centered at a point p then it does not move p.


Q. Are the answers unique for part 1.2?

A. No -- there will generally be several reasonable ways (some probably
   simpler than others) to write each transform using the elementary
   transforms.


Problem 2:

Q. Where it says M R M^T, is that right?  Doesn't M have a translation
   component so that the transpose isn't even an affine transformation at all?

A. You're right, it's wrong -- and v.4 of the handout corrects it to M R M^-1.


Problem 3:

Q. What are the viewing and projection matrices?

A. The viewing matrix is a rigid motion (that is, a matrix with just rotation
   and translation) that transforms the scene so that the camera is in its
   canonical position (at the origin facing -z, with +y up).  The projection
   matrix is the one that maps 3D points into 2D points in the image; it is a
   3x4 matrix, since it takes 3D homogeneous points to 2D homogeneous points.


Q. In part 6, does the field of view change?

A. Yes, the box always fills the whole image, so the camera has to zoom in as
   it moves back.