You do not send a letter to royalty, the king of Spain for example. You send it to their private secretary. Then he or she forwards to them. (The letter itself can be addressed to the monarch though. Beginning "May it please your majesty:", as per tradition.) You do not approach them in any way. That is purpose of a receiving line with royalty. Because (I was surprised to learn recently) you don't even make eye contact first. In a receiving line, everyone stands at attention, looking straight ahead. And then, let's say the king of Denmark, walks up to you, he approaches you. He makes eye contact first, and then extends his hand, showing you may touch you, and he is socially engaging him. You do not in any way social engage him. Or bring up a subject that he or she doesn't first (how does their doctor even examine them then?). One time in the 80's, Princess Diana was at some charity function with a mayor of a some English town. The mayor wanted to usher Princess Diana in some direction. So he motioned for her to go, coincidentally moving his hand towards her buttocks. "This way please..." IOW. He moved his hand towards her buttocks without her permission, he asked her to go in some direction, she didn't indicate she wanted to. He brought up the subject, she didn't. And the people in the UK were outraged. They said it was a form of lèse-majesté (Lèse-majesté - Wikipedia). Some people implied he should have been executed for treason. Princess Diana herself quipped when she found out he should be punched or slapped, just joking of course. But still very angry. All for motioning towards the princess and wanting her to go in some direction, without her indicating that's what she wanted first.
