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Calculate Euler rotation angles between 2 points


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#1 Jenko   Members   -  Reputation: 100

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Posted 22 October 2010 - 02:34 PM

I need to calculate the 3 Euler angles (yaw, pitch, roll) between 2 points.

The picture below shows the way I rotate. These are the angles I need to calculate.
(The only difference is I'm rotating the object in the order X,Y,Z and not Z,Y,X)

pic

This is the code I'm using but its not working for me:


// x, y, z represent a fractional value between -[1] and [1]
// a "unit vector" of the point I need to rotate towards

yaw = Math.atan2( y, x )
pitch = Math.atan2( z, Math.sqrt( x * x + y * y ) )



[Edited by - Jenko on October 22, 2010 10:04:08 PM]

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#2 mfiorilli   Members   -  Reputation: 100

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Posted 22 October 2010 - 06:14 PM

I have also been doing a lot a research on the internet about that topic. I'm currently looking for a way to do this with QUATERNIONS and I decided to post that here instead of starting a new thread since it's basicaly the same question but with another system.

So here's what I currently found:

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//Given two vectors, find the angle difference between them in quaternion

vector $V1 = << my first vector >>;
vector $V2 = << my second vector >>;


float $d = dot($V1, $V2);
vector $axis = cross($V1, $V2);

float $W = sqrt(norm($V1) * norm($V1) * norm($V2) * norm($V2) + $d);

float $X = ($axis.x);
float $Y = ($axis.y);
float $Z = ($axis.z);

// So my quaternion is now $X, $Y, $Z, $W ... transform that in a vector

vector $xyz = << $X, $Y, $Z >>;

vector $result = (2 * $W*$W -1)*$V1 + 2*dot($V1,$xyz)*$xyz + 2 * $W * cross($V1,$xyz);

// $result is my resulting vector

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It's weird because this is almost working, if I rotate on vector on a specific axis in relation to my other vector, the resulting vector seems to rotate correctly but all other axis won't work ! It gives awkward results. But if I move my two vectors as if they were stick one to the other, the resulting vector will react as if it was stick to my 2 vectors. So I think I'm not doing something correct here but I just don't know what.


And finally, here's the same thing with euler angles. I got it to work but I wanted a more optimized method.


// Calculate Euler rotation angles between 2 vectors .. we use Rodrigues formula


vector $V1 = << my first vector >>;
vector $V2 = << my second vector >>;


vector $axis;
float $angle;

$angle = acos($V1*$V2);
$axis = normalizeVector((cross($V1,$V2)));


matrix $axis_skewed[3][3] = <<
0, (-$axis.z), ($axis.y) ;
($axis.z), 0, (-$axis.x) ;
(-$axis.y), ($axis.x), 0 >>;

matrix $eye3[3][3] = <<
1, 0, 0;
0, 1, 0;
0, 0, 1 >>;

// here's Rodrigues
$R = $eye3 + sin($angle)*$axis_skewed + (1-cos($angle))*$axis_skewed*$axis_skewed;


matrix $vectorMatr[3][1];
$vectorMatr[0][0] = ($V1.x);
$vectorMatr[1][0] = ($V1.y);
$vectorMatr[2][0] = ($V1.z);

// $result is the resulting vector

$result = ($R * $vectorMatr);


----------------------------------------------------

In conclusion, my quaternion method is still not working but seems more promising than my other method which is working but might be slow. I'm looking for a better solution.

#3 Jenko   Members   -  Reputation: 100

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Posted 23 October 2010 - 09:15 AM

mfiorilli, thanks for the Euler method.

Do I assume correctly that the resultant vector stores the X and Y euler rotations into the vector's x and y properties?

However since I'm not programming this in C++ nor do I have your vector library, I'm having some trouble understanding what you've done here:


$R = $eye3 + sin($angle)*$axis_skewed + ([[1]]-cos($angle))*$axis_skewed*$axis_skewed;

// do you add all the properties of the eye3 matrix?
// do you multiply with all the properties of the axis_skewed matrix?
// etc..
// and what is R? a vector or matrix? or number?



and here:


$result = ($R * $vectorMatr);

// do you multiply the vector with the matrix to get the resultant vector using standard matrix multiplying?



If you could post your matrix/vector library I could use its code since my platform does not compute operators (* / + -) and call functions behind the scenes.




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