15462 Computer Graphics I
Assignment 5  Camera Movement
Overview
The best way we've seen to keep track of your camera coordinate
systems is by Ken Sloan, summarized below. If you didn't keep track of
your coordinate system from position to position along the spline, then
your up vector would point in an arbitratry direction each time. This
might cause your camera to rotate unpredictably as it moved along the
spline.
Camera Coordinates
So we'll keep track of the coordinate system by using the previous
coordinate system that you defined in order to generate your new one.
For every point P along the spline, you will need to define 3 vectors, T
(tangent), N (normal), and B (binormal).
The first thing you need to do is generate a starting set of axes
at P0. Since you already have the T0 vector which is just the tangent
vector of the spline at that point. You can pick some arbitrary vector
M. Then make N0=unit(T0xM) and B0=unit(T0xN0). Now that you have T0,
N0, and B0 you have the coordinate system for P0. Here write write
unit(v) for the normalized vector v/v.
Next you have to calculate a coordinate system for
P1. Since you already have T1, N1=unit(B0xT1) and B1=unit(T1xN1).
This process is repeated along every point along the spline so that
your coordinate system will always stay consistent.
Applications
There are two areas of the lab where this comes into use. The first
is obviously the camera, you can use the coordinate system to orient the
camera properly. The second use is when deciding the orientation of the
cross sections when drawing them. To orient a cross section you
need to create a transformation matrix which will move your tie to the
correct position. Such a matrix mat can be constructed from T, N, B, and P.
Just as a reminder, OpenGL reads matrices in a transposed form as
described in the OpenGL primer. You would want to add this matrix to
the transformation before drawing your cross section rail. For example,
glPushMatrix();
glMultMatrixf(mat);
drawCross();
glPopMatrix();
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Frank Pfenning
