We have seen that an important property of analytic functions is that they are conformal transformations everywhere they are defined, except where the derivative vanishes.  More precisely, if two curves intersect at a point and their tangents make a certain angle there, then the angle between the images curves under any analytic function (with non-vanishing derivative) will be the same in both magnitude and sense. A consequence of conformality, namely the preservation of orthogonality of intersecting coordinate lines, is visible in the  following animation where we use the power function , which is conformal everywhere except at the origin. Here we apply the map to a square grid in the first quadrant, with the power a varying from 1 to 3.9. Notice how all the right angle intersections remain right angles throughout the transformation, except the angle at the origin, where the power maps are not conformal: