1.      the major axis’s direction is perpendicular to the y-axis when it rotates around y-axis.

2.     the base circle have the same radius with the unit circle.

3.     this ellipse passing through another axis point z.

Thus this ellipse can be constructed from compress method.

1.     That is, if point A→A’ and B→B’ rotate the same angle in their plane, then the points on base circle respect to these points passing through the same angle.

2.     Or we can say that the rotation of a point on the base circle respect to a point of ellipse follows uniform angle motion.

    Thus the construction need to find the point on base circle respect to the point of ellipse, and make uniform angle rotation on base circle, then project it onto the ellipse again.

     Given a point P on ellipse, first find its corresponding point P’ on base circle, make an uniform angle rotation on base circle to the point Q’, and then project it to the point Q of ellipse. This could make the rotation alike the real rotation in three-dimensions.

1.     As the figure on the right, if we want to rotate the direction of x-direction to the direction of z, we need to rotate counterclockwise.

2.     But when base axis may changes and the new axes of x and z would change, too. Like this new figure of its below, if we still want to rotate from x-direction to z-direction, we need to rotate clockwise now.

Thus we may bisect the ellipse into two parts with major axis, observing two directions x and z. One of them behaves well that when its direction passes through the opposite side of major axis, rotation direction changes from counterclockwise to clockwise.

Base on this direction we can form a criterion resulting in {1} and {-1} when rotate direction becomes clockwise or counterclockwise respectively. This prevent the construction for repeating when it behaves two rotational directions.