Travelling was a waste of time in the estimation of the head of the Lannister Clan, and long-distance traveling was an already obnoxious process taken to its worst extreme.
At the very least he was glad that the carriage was stable enough to provide him a relatively useful working space, even if it was still subject to the odious nature of unpaved roads and their typically bumpy manner.
At one point in his youth he had considered simply paving all of the major routes in the Westerlands, but even for the Rock that would be a considerable expense, and at the time not one worth pursuing.
The carriage was a better investment, wasteful as it was in some ways. It served not only as a portable study of sorts for long journeys in peacetime but also as one more ostentatious display of the wealth and power of his house. Crimson Ironwood planks were gilded in gold and decorated with lion statuettes and shields of arms, and it was supported on six tall wheels that gave it a tall and wide profile, occupying the entire road.
He preferences horseback of course, but that sent… unfortunate signals in peacetime. Horses were the mounts of conquerors, and while his guard were certainly allowed their steeds it wouldn't due to ride into King's landing on one for a wedding, especially given his history doing so.
He imagined that was why Tyrion had insisted on riding, some attempt to spite him no doubt from the malformed child.
All of his children seemed intent to spite him, whether intentionally or not.
Still, they were the future of his dynasty, whether he wished it or not.
He grimaced as he felt the carriage shudder to a stop, the wheels, durable as they were, protesting under the sudden shift.
"What is going on up there?" The Lannister party was gigantic, and outriders were sent ahead to ensure that all knew of their coming. Few would dare cross them. One would have to be colossal stupid and egotistical to-
"It's the Northern party, with the Crown Prince and Lord Stark at its head."
"Ah." Tywin pulled back his anger. Of all the parties in Westerns that was perhaps one of the few that outranked his own. Heading down the Kingsroad no doubt, they were as early as he was. "Have someone fetch my horse, I will go out and meet my Grandson and the Lord Stark."
The implicit 'with my guard' went unanswered. The Northerners were certainly not plotting anything, but precautions were things to maintain in any and every situation.
"Yes, my Lord."
It wasn't long at all before he was riding out, the Mountain behind him. The Clegane towering over the rest of his guard with seeming indifference. He had met Lord Stark before, just after the Rebellion, and the man who was fostering his grandson seemed largely unchanged, save for the children that followed him, Joffrey included. Cold and harsh in his features, he looked the foil to King Robert in the rebellion, but on his own, he was little threat. The Starks were more interested in the North than elsewhere. Still, there were far worse choices to foster his grandson, and he overall approved of the King's decision to have the boy fostered.
It was important that a crown prince understand the realities of the world beyond his palace walls. A lack of personal knowledge had been Rhaegar's undoing.
Behind him, alongside the Stark heir, rode a boy was obviously Joffrey. He looked every bit the Lannister that Tywin expected, and he eyed his grandfather with a suspicious look as he approached, probably a good sign, that.
"Greetings, Lord Tywin."
"And you as well, Lord Eddard." Tywin smiled, though it was a cold thing. "I had not thought to meet you on the road given how early I had left."
"We have had fair weather and good fortunes on our journey south, save for the neck." The Stark admitted. "I had factored in the possibility of inconvenience."
'Leave it to a Stark to plan for the weather.' Tywin observed, nodding slightly. "Would you care to make a camp with my party this evening? I would like the opportunity to speak with my grandson."
The Wolf looked at him for a long moment before nodding. "Of course."
Tywin nodded, turning his gaze to the boy. "Well, your Grace? Have nothing to say to your grandfather?"
The boy looked up at him, the suspicion on his gaze only increasing, before moving to considered neutrality. "Certainly, Grandfather, only we should not linger on the road. If we move quickly then we might well make Harrenhal by this evening from what Lord Stark has said.
Tywin looked at the boy, before nodding. "I will ride with you then. There is much I would like to ask about. I regret having not visited you in King's landing before your departure for the War in Essos."
The boy made a face as if he had bitten into something sour before shaking his head ever so slightly. "It… is good that you meet me now, I should think then." The boy turned to his tutor. "May I ride with my Grandfather Lord Stark."
"Yes." The wolf-lord spoke, turning to lead his own party, a vastly smaller thing than the Lannister host behind him. No doubt as much a practical affair as a measure of the humility of the Northerners.
"Thank you." Joffrey nodded, riding over to Tywin's side. He eyed the boy more closely, spotting the shortsword on his hip.
Perhaps he was more martially inclined than Cersei's letters claimed, or perhaps he was simply playing at it in the Starks company.
He would know soon enough.