Calculating a rotation matrix based on location/ta

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ｺ WWH            ｺ Calculating a rotation matrix based on location/target     ｺ
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ｺ Text version   ｺ 1.0                                                        ｺ
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ｺ Written by     ｺ Paul Nettle (midnight@grafix3d.dyn.ml.org)                 ｺ
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ｺ Last Modified  ｺ May 15, 1997                                               ｺ
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ｺ Prerequisites  ｺ Basic vector and matrix mathematics                        ｺ
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ｳ WWH                                 WHAT                                WWH ｳ
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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++.

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ｳ WWH                                 WHY                                 WWH ｳ
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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.

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ｳ WWH                                 HOW                                 WWH ｳ
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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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ｳ WWH                                 END                                 WWH ｳ
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Date this article was posted to GameDev.net: 7/16/1999
(Note that this date does not necessarily correspond to the date the article was written)
See Also:
Matrices