There is something satisfyingly sensible about his approach to the game — no point in doing it if you don’t give it all you have; no point in being on court if you don’t seek to win every point; no point in hitting a tennis ball unless you hit it so hard it may never come back.

This is the approach, this is the nature that has allowed Agassi to win eight grand-slam tournaments, to win all four titles on all four different surfaces; a player for all seasons, a player for all sorts and conditions of spectator. All this and Steffi, too. So let us salute the passing of a man of credibility, dignity and power.

But I also remember Agassi when he was silly. He was for some years the silliest athlete in sport, a man who dedicated himself to silliness with the same single-mindedness he later brought to being sensible.

For a start, he refused to play at Wimbledon because they wouldn’t allow him on court in cut-off jeans. That was Agassi: the long, long bleached hair, all artful tousle, the grunge shirt, the cut-offs worn over neon-pink or vomit-green Lycra.

Eventually, he decided that his young-rebel principles would, after all, allow him to play in SW19, not least because he’d get more money that way. I was there on Centre Court when he made his entrance in a throat-to-ankle tracksuit, in which he did his warm-up. He then performed the first striptease to be seen at the All England Club, finally revealing himself in a three-tiered outfit in virginal white — the same grunge outfit bled of all its colour. He looked as if he was going to a fancy dress ball as a wedding cake. And he played a few flashy rounds and then got knocked out, and we laughed and we smiled and we sneered.

This was not a serious sportsman. This was a winsome boy whose self-elected task was to decorate an occasion, a peripheral figure there to have a bit of fun and make a few quid from endorsements on the way. He was swept up by a camera company for the most ill-advised advertising campaign any sportsman has taken on. “Image is everything,” he declared from hoardings and television screens, hanging off the back of a London bus with a camera round his neck.

It seemed that his mission was to prove that you don’t have to win stuff to be a sporting millionaire, any more than a pop star needs to create good music. Agassi’s talent was so immense that he won Wimbledon one year more or less by accident, and that reorganised him a bit. But at heart he was still a talented waster, one who was content to sell out to his lesser talent.

But the point about Agassi is that he is one of the great shape-shifters of world sport. He reinvented himself. The dyed mane became a cue-ball, the waster became a remorseless champion, the boy of image became a man of substance, the player who messed about became a winner who just got on with the job. He has been a marvellous player to watch.

Shape-shifting has been the entire point of his career. He stands for a principle seldom accepted by anyone in any walk of life: that you can be one kind of person and then, if you have the ability, if you have the desire, you can be another kind of person entirely. A wild and adventurous woman can settle down to domestic bliss; an unreconstructed lad can find a taste for responsibility and order; a waster can become a person of substance; a loser can become a winner.

Sport tells us this elementary truth about life in its usual vivid way and Agassi is just one majestic example of the principle. Other shape-shifters include the three most revered figures active in British sport.

Let us start with Andrew Flintoff, last summer’s hero. He was one of those colossally brilliant young things who was content to get by on talent alone. Why bust a gut when you can slob along at the second level so agreeably?

Then, in an often-told story, Flintoff travelled a well-trodden route. He was given a serious bollocking, met the love of his life, had a child, settled down, grew up, became a champion, pared down, ripped physique, ditto sporting mind. Hard on himself, generous with others, especially (beaten) opponents. He changed his physical shape, went for the committed-jock buzz-cut and became the finest cricketer in the world, as I was privileged to write in the present Wisden.

Another classic shape-shifter is David Beckham, the Prince Hal of football, the fop-prince who became a king of power and substance. I know there is a bit of revisionism going on about Beckham — people now like to say that he never was any good in the first place — but that’s not how it happened.

The silly sent-off twit of the World Cup of 1998 became the leader who took England to qualification for the 2002 finals by means of that staggering performance against Greece. He then took the penalty that gave England victory against Argentina in Japan when — briefly — not just the World Cup but the world itself seemed to be Beckham’s for the asking.

Then, there is Paula Radcliffe, herself undergoing a spot of revisionism after her 2004 Olympic Games disaster. But she is a shape-shifter nonetheless: the serial fourth-placer, who led everywhere except at the tape, upgraded to the marathon to become a radiant, all-conquering champion.

I remember her first marathon and the television commentators imploring her to slow down. Instead, she went quicker. She passed, in long and lovely strides, from doubt into certainty and as a direct result she became beautiful. That’s shape-shifting for you.

Can a man change his stars? The eternal question asked in the delightful A Knight’s Tale, a film that gives the ultimately cheering answer of “yes”.

Is it a matter of complete personal reinvention, then, or is it a strange process of setting free the person within? Perhaps the shedding of the hot-lava shorts, the laddish gut, the Brylcreem flop, the bewildered, put-upon expression was enough to give us Andre, Freddie, Becks, Paula Unbound.

Was the champion always there, hidden within, a potential that was yearning to be made actual? Well, we all have all kinds of unrealised bits of potential concealed inside ourselves. Let us salute the shape-shifters who realised their own. And perhaps the curious and circuitous route they took was the only one that could have taken them to this destination, to the time and place in which they became — however briefly — truly themselves.