angle and orientation preservation

Stand at a point where two curves cross and measure the angle between them — the angle between their tangent directions. Now apply a conformal map and look at the two image curves where they cross. The astonishing fact is that they meet at exactly the same angle, and they turn the same way (a left turn stays a left turn). This is the heart of conformality: a holomorphic map with nonzero derivative is an isogonal, orientation-true map. Angles in, the same angles out.

Here is why, in one clean line. Suppose a curve passes through z_0 with tangent direction making angle alpha. Under f, the tangent to the image curve makes angle alpha + arg f'(z_0): the map adds the SAME extra turn arg f'(z_0) to every direction, because near z_0 the map acts as multiplication by the single complex number f'(z_0). If a second curve comes in at angle beta, its image leaves at beta + arg f'(z_0). The difference (beta + arg f'(z_0)) - (alpha + arg f'(z_0)) = beta - alpha is unchanged. The common twist cancels, so the angle between the two curves is preserved. And because the rotation arg f'(z_0) is an ordinary rotation (not a reflection), the sense of the angle — its sign, its handedness — is preserved too.

This is what separates holomorphic maps from their look-alikes. The conjugation map z -> z-bar also preserves the SIZE of every angle, but it flips orientation (it is a mirror reflection): a counterclockwise angle becomes clockwise. Such 'angle-size-but-not-sense' maps are called anticonformal or indirectly conformal. Genuine conformal maps preserve both magnitude and sense — and that orientation-faithfulness is exactly the geometric fingerprint of being holomorphic rather than merely 'real-differentiable'.