WWH5: Calculating a Rotation Matrix Based on Location/Target
by Paul Nettle

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WWH	Calculating a rotation matrix based on location/target
Text version	1.0
Written by	Paul Nettle (midnight@grafix3d.tzo.com)
Last Modified	May 15, 1997
Prerequisites	Basic vector and matrix mathematics

What

The purpose here is to describe how to obtain a proper matrix given a location and a target (or a vector), and an amount of roll.

This document will also emulate the specific nature of 3DS cameras, when it comes to the degenerate case (input vector points directly up).

The accompanying source code is in C++.

Why

This is useful when dealing with cameras in a 3D world, or when you need to orient an object based on a vector, rather than roll/pitch/yaw components.

How

Define

cVector - dx, dy, dz
cMatrix - 3x3 matrix class (consisting of 3 cVectors)

The input Vector must be a directional vector. So for location->target, calculate like this:

vector = target - camera;

There is a problem with creating rotation matricies out of direction vectors. There is a degenerate case when the [delta y] of the direction vector is anything but zero. 3D Studio handles this in a special way, and here's a solution to it.

Testing

Many people claim to have perfectly working code, and I have found that in over 50% of them, this was not the case.

To test this, simply view an object from all 6 directions (above, below, left, right, front, back). Make sure the view vector contains two 0 components and a 1 component (i.e. [0,0,1] or [0,-1,0]).

The degenerate cases are [0,1,0] and [0,-1,0]. Pay special attention to the degenerate cases.

This code based on the descriptions in _Computer Graphics Principles and Practice_ (page 222) by Foley, van Dam, Feiner and Hughes.

Source

cMatrix	cMatrix::generateMatrix( cVector &vector, const float rollDegrees )
{
  // Get our direction vector (the Z vector component of the matrix)
  // and make sure it's normalized into a unit vector

  cVector zAxis(vector);
  zAxis.normalize();

  // Build the Y vector of the matrix (handle the degenerate case
  // in the way that 3DS does) -- This is not the TRUE vector, only
  // a reference vector.

  cVector yAxis;

  if (!zAxis.dx && !zAxis.dz)
    yAxis = cVector(-zAxis.dy, 0.0f, 0.0f );
  else
    yAxis = cVector(0.0f, 1.0f, 0.0f);

  // Build the X axis vector based on the two existing vectors

  cVector xAxis = yAxis.cross( zAxis );
  xAxis.normalize();

  // Correct the Y reference vector

  yAxis = xAxis.cross( zAxis );
  yAxis.normalize();
  yAxis = -yAxis;

  // Generate rotation matrix without roll included

  cMatrix rot(xAxis, yAxis, zAxis);

  // Generate the Z rotation matrix for roll (bank)

  cMatrix roll(MATRIX_Z, rollDegrees);

  // Concatinate them for a complete rotation matrix that includes
  // all rotations

  cMatrix result = roll * rot;

  // All done

  return result;
}

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Date this article was posted to GameDev.net: 8/23/1999
(Note that this date does not necessarily correspond to the date the article was written)

See Also:
WWH Series