Re: [Algorithms] Rotation about arbitrary axis

[Thatcher U. <tu...@tu...>]

From: Pierre Terdiman <p.t...@wa...>
>
> Since I needed a piece of code to do that I searched the web and found:
> http://www.iuk.tu-harburg.de/hypgraph/modeling/mod_tran/3drota.htm
>
> I used the final matrix at the bottom of the page, but it seems to fail
when
> the arbitrary axis actually is the Z axis. The third column gets erased
> where it should at least contain a 1. This is obvious when looking at the
> provided matrix, since the third column of the third row depends on the
> rotation angle - and of course if the input axis already is the Z axis, it
> shouldn't.
>
> Now, it sounds normal regarding the underlying method (mapping the
rotation
> axis to Z, etc). But I wonder whether there's an easy way to perform a
real
> arbitrary rotation about any arbitrary axis without using different code
> paths according to the input axis.
>
> I think it can probably be done by introducing quaternions or better,
> angle-axis, in the story. But err.... maybe there's something simpler.

Here's what I use.  I'm sure it could be streamlined by expanded out the
quaternion stuff and simplifying.

class quaternion {
public:
 quaternion(const quaternion& q) : S(q.S), V(q.V) {}
 quaternion(float s, const vec3& v) : S(s), V(v) {}
 // [...]
 quaternion operator*(const quaternion& q) const;
 // [...]
 void ApplyRotation(vec3* result, const vec3& v);
private:
 float S;
 vec3 V;
};


vec3 Rotate(float Angle, const vec3& Axis, const vec3& Point)
// Rotates the given point through the given angle (in radians) about the
given
// axis.
{
 quaternion q(cos(Angle/2), Axis * sin(Angle/2));

 vec3 result;
 q.ApplyRotation(&result, Point);

 return result;
}

quaternion quaternion::operator*(const quaternion& q) const
// Multiplication of two quaternions.  Returns a new quaternion
// result.
{
 return quaternion(S * q.S - V * q.V, q.V * S + V * q.S + V.cross(q.V));
}


void quaternion::ApplyRotation(vec3* result, const vec3& v)
// Rotates the given vec3 v by the rotation represented by this quaternion.
// Stores the result in the given result vec3.
{
 quaternion q(*this * quaternion(0, v) * quaternion(S, -V)); // There's
definitely a shortcut here.  Deal with later.

 *result = q.V;
}


--
Thatcher Ulrich
http://tulrich.com