As you may know already, there is a special type of Elite rank that appears exclusively on Legendary difficulty and only one per level is encountered. Enjoy this little guide:

Pillar of Autumn: Right after you meet with Captain Keyes, head your way back to the Cryo Bay until you end up in the hallway were you first met with an Elite Ultra, then head for the room he retreated in.

Halo: Mandatory encounter. He is waiting for you on the small balcony where the light bridge switch is located in the underground tunnel.

Truth & Reconciliation: Appears right when you are about to finish the level. He will attack right after you defeat the last wave of elites in the shuttle bay.

The Silent Cartographer: Mandatory encounter. He is waiting for you where the Elite Zealot normally ambushes you after you discovered the Map Room.

Assault on the Control Room: (Thanks to tpd1864blake for finding this one out!) Can only be found if you steal a Banshee in the final canyon of the level, where the Control Room is located, then skip to it. Notice that instead of the regular encounter, the hallway to the control room will be full of Spec Ops and Camo Spec Ops supporting the Vanguard.

343 Guilty Spark: Right after you encounter the Flood for the first time, head back to the lift where you first entered the facility and press the switch.

The Library: Once you get near the end of the level, head right instead of rushing towards the index in the last room. There is a small Covenant group camping there with the Vanguard among them. (Pun intended. You'll understand what I mean if you proceed further after the Covenant camp)

Two Betrayals: Once you cleared the Covenant at the bottom of the Control Room, instead of taking the banshee, head for the door on the left. Proceed in the corridors until you reach the main room on the upper floor.

Keyes: When you reach the shuttle bay after boarding the cruiser again, you'll be forced to drop down on the first floor before taking a hallway leading back up to the second floor. Once there, make a stop at the balcony where a group of Covenant holds their ground with a Shade turret.

The Maw: Mandatory encounter. In the corridor before the armory and the engine room, fighting off the Flood.