2D Euclidean vector rotations

问题:

I have a euclidean vector a sitting at the coordinates (0, 1). I want to rotate a by 90 degrees (clockwise) around the origin: (0, 0).

If I have a proper understanding of how this should work, the resultant (x, y) coordinates after the rotation should be (1, 0). If I were to rotate it by 45 degrees (still clockwise) instead, I would have expected the resultant coordinates to be (0.707, 0.707).

theta = deg2rad(angle);

cs = cos(theta);
sn = sin(theta);

x = x * cs - y * sn;
y = x * sn + y * cs;

Using the above code, with an angle value of 90.0 degrees, the resultant coordinates are: (-1, 1). And I am so damn confused. The examples seen in the following links represent the same formula shown above surely?

What have I done wrong? Or have I misunderstood how a vector is to be rotated?

回答1:

you should remove the vars from the function:

x = x * cs - y * sn; // now x is something different than original vector x
y = x * sn + y * cs;

create new coordinates becomes, to avoid calculation of x before it reaches the second line:

px = x * cs - y * sn; 
py = x * sn + y * cs;

回答2:

Rotating a vector 90 degrees is particularily simple.

(x, y) rotated 90 degrees around (0, 0) is (-y, x).

If you want to rotate clockwise, you simply do it the other way around, getting (y, -x).

回答3:

Rotate by 90 degress around 0,0:

x' = -y
y' = x

Rotate by 90 degress around px,py:

x' = -(y - py) + px
y' = (x - px) + py

回答4:

You're calculating the y-part of your new coordinate based on the 'new' x-part of the new coordinate. Basically this means your calculating the new output in terms of the new output...

Try to rewrite in terms of input and output:

vector2<double> multiply( vector2<double> input, double cs, double sn ) {
  vector2<double> result;
  result.x = input.x * cs - input.y * sn;
  result.y = input.x * sn + input.y * cs;
  return result;
}

Then you can do this:

vector2<double> input(0,1);
vector2<double> transformed = multiply( input, cs, sn );

Note how choosing proper names for your variables can avoid this problem alltogether!

回答5:

Sounds easier to do with the standard classes:

std::complex<double> vecA(0,1);
std::complex<double> i(0,1); // 90 degrees
std::complex<double> r45(sqrt(2.0),sqrt(2.0));
vecA *= i;
vecA *= r45;

Vector rotation is a subset of complex multiplication..