Rotations in 3 Dimensions

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Points: 12
Time limit: 1.0s
Memory limit: 256M

Problem type

Your task is simple: given some points in 3D, rotate them around an axis of rotation.

Input Specification

The first line of input will contain T, the number of test cases (1 \le T \le 1\,000).
The next T lines will each contain 7 real numbers to 6 decimal places, x\ y\ z\ rx\ ry\ rz\ \theta. You are to rotate the point (x, y, z) around the axis of rotation (rx, ry, rz) such that if you look at the origin from the axis of rotation, it will be rotated \theta radians counterclockwise. All coordinates will have absolute value of at most 1\,000 and \theta will be such that 0 \le \theta < 2\pi. It is guaranteed at least one of rx, ry, rz is nonzero.

Output Specification

Output T lines, each line should have the (x', y', z'), the result of rotation as three space-separated real numbers. Your answer will be judged as correct if it is within an absolute or relative error of 10^{-6}.

Sample Input

0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 3.141593
1.000000 0.000000 0.000000 1.000000 1.000000 1.000000 1.570796

Sample Output

0.000000 0.000000 0.000000
0.333334 0.910683 -0.244017


  • -1
    AStrangePotato  commented on Aug. 20, 2022, 2:57 p.m.

    Hi, I've been reading this problem for a while now, and I don't seem to get how you return an exact coordinate value. Can't you look at the origin from r at any rotation? That is, there are more than one ways of looking at o from r, so how do I know which rotation to look at the origin? If anybody could clarify for me, that would be great! Thanks in advance.

    • -1
      Josh  commented on Aug. 23, 2022, 6:35 a.m.

      It shouldn't matter which way we look from r, since we are rotating (x, y, z) around an axis (a line), rather than around a point.

  • 2
    Jerry_Gu  commented on March 3, 2018, 12:07 p.m.

    should the first line in the sample output be 2.000000 0.000000 0.000000 since if you look at the origin from (1, 0, 0) and rotate the point (0, 0, 0) 180 degrees counterclockwise then it will end up at (2, 0, 0)?

  • 0
    neal_nicdao  commented on Jan. 15, 2016, 12:00 a.m.

    Sample input and output work perfectly. I tried printf("%f %f %f\n", x, y, z), printf("%.6f %.6f %.6f\n", x, y, z), %e, and many other combinations and all of them return WA... Im using g++ as well. What did I do wrong?

    • 0
      Xyene  commented on Jan. 15, 2016, 12:18 a.m.

      Your code suffers from floating point precision issues: if you change your variables to doubles, your solution passes.

      • 0
        neal_nicdao  commented on Jan. 15, 2016, 12:37 a.m.

        Thanks mate, some of math was already in doubles but after changing data to doubles from float, everything worked.

  • 2
    sinxcosx  commented on Oct. 26, 2015, 8:18 p.m.

    First of all, I'm wondering what's the order of rotation?

    Secondly, say if I want to rotate pie/4 (45 degree) by the z-axis, and then for the next rotation, let say by the x-axis, is it rotating along the old x-axis or the new x-axis, which is the axis produced after the previous rotation with a 45 degree offset?

    Can someone explain this program to me?

    Thank you

    • 1
      bobhob314  commented on Dec. 2, 2015, 9:45 p.m.

      Now that you've solved it, can you explain it to us?

      ^ serious question. Thanks!

      • 5
        omysho  commented on Dec. 7, 2016, 12:21 a.m.

        Hi, I notice that you convert the angle to radian value in your code.

        But the input specification has said that the angle is already radian