Rotating

The function rotate() rotates a Point about one or more of the main axes. It takes three real arguments, specifying the angles of rotation in degrees about the x, y, and z-axes respectively. Only the first argument is required, the other two are 0 by default. If rotation about the y-axis, or the y and z-axes only are required, then 0 must be used as a placeholder for the first and possibly the second argument.

     Point p(0, 1);
     p.rotate(90);
     p.show("p:");
     -| p: (0, 0, -1)
     p.rotate(0, 90);
     p.show("p:");
     -| p: (1, 0, 0)
     p.rotate(0, 0, 90);
     p.show("p:");
     -| p: (0, 1, 0)
     

The rotations are performed successively about the x, y, and z-axes. However, rotation is not a commutative operation, so if rotation about the main axes in a different order is required, then rotate() must be invoked more than once:

     Point A(2, 3, 4);
     Point B(A);
     A.rotate(30, 60, 90);
     A.show("A:");
     -| A: (-4.59808, -0.700962, 2.7141)
     B.rotate(0, 0, 90);
     B.rotate(0, 60);
     B.rotate(30);
     B.show("B:");
     -| B: (-4.9641, 1.43301, -1.51795)
     

Rotation need not be about the main axes; it can also be performed about a line defined by two Points. The function rotate() with two Point arguments and a real argument for the angle of rotation (in degrees) about the axis. The real argument is optional, with 180 degrees

as the default.

     Point p0 (-1.06066, 0, 1.06066);
     Point p1 (1.06066, 0, -1.06066);
     p1 *= p0.rotate(0, 30, 30);
     p0.show("p0:");
     -| p0: (-1.25477, -0.724444, 0.388228)
     p1.show("p1:");
     -| p1: (1.25477, 0.724444, -0.388228)
     p0.draw(p1);
     Point p2(1.06066, 0, 1.06066);
     p2.show("p2:");
     -| p2: (1.06066, 0, 1.06066)
     Point p3(p2);
     p3.rotate(p1, p0, 45);
     p3.show("p3:");
     -| p3 (1.09721, 1.15036, 1.17879)
     Point p4(p2);
     p4.rotate(p1, p0, 90);
     p4.show("p4:");
     -| p4: (0.882625, 2.05122, 0.485242)
     Point p5(p2);
     p5.rotate(p1, p0, 135);
     p5.show("p5:");
     -| p5: (0.542606, 2.17488, -0.613716)
     Point p6(p2);
     p6.rotate(p1, p0);
     p6.show("p6:");
     -| p6: (0.276332, 1.44889, -1.47433)
     

Fig. 2.

I have sometimes gotten erroneous results using rotate() for rotation about two Points. It's usually worked to reverse the order of the Point arguments, or to change sign of the angle argument. I think I've fixed the problem, though.