Wednesday, February 28, 2007

Back to School

On my last trip, I relied on "muscle memory" to get me safely down, with a rather mixed outcome (injuries and indignities mixed with occasional flashes of competence). This time, nothing was to be left to chance: I signed up for another round of ski school, claiming an "intermediate" skill level.

Cultural differences between Andorran and French ski-schools became apparent pretty rapidly; my instructor (and C's) were fairly ruthless critics. Although the class was full of moderately experienced skiers (moi very definitely the neophyte), it seemed that we couldn't make a single turn to our instructor's satisfaction.

Je fais le "windscreen-wiper turn" - très mal!

Ah, yes. Turns. While we could all descend moderately steep pistes with a fair degree of control, we all seemed to have the same très mal habit - the "windscreen-wiper" turn. Demonstrated in the above picture by myself, this turn involves allowing the skis to slip downhill as a turn is completed - throwing photogenic clouds of powder about, and, crucially, slowing one down.

I thought that I was skiing quite confidently; I was wrong. Although my weight was (just about) forward enough to make parallel turns possible, it wasn't nearly far enough forward to give really good control, or to allow the turn to happen with the minimum of effort. My turns were also unnecessarily defensive - on most pistes, most of the time, I don't need to brake so dramatically; instead, my instructor exhorted me to cut neat "rainbow" turns (sine wave might be a better description). Many pole-planting exercises followed.

You might think that since we could all turn and control our speed after a fashion, further tuition was simply gilding the lily - fancy techniques we would never actually use. Not so; later in the week, after a fairly considerable fall of fresh power snow, Thomas (our acerbic French instructor) took us a short but very instructive distance off-piste. He started us on relatively gentle inclines - broad, virgin powder fields between the marked pistes. One fast, straight run and a dramatic head-planting incident later (marked for a few days by a huge me-shaped snow crater) was enough to convince me of the need for further work on those turns.

Practice payed off; before the day's lesson ended, Thomas told us he had something special for us; turning off a perfectly good piste, we paused in sequence at the lip of a gully, ski tips protruding into thin air. I did the (sort of) sensible thing, and launched immediately, before reason had a chance to consider the possibilities... There followed a second or two of severe misgivings, obliterated an instant later by a flash-flood of exhiliration as my momentum carried me well up the far wall of the gully. Turning easily at the "hang point" was a kind of skiing epiphany - and then gravity kicked in, and I was accelerating again, ricocheting down that beautiful gully, skis hissing through deep, fresh powder; it was like flying.

Pure winter magic.

Monday, February 19, 2007

In the Haute Tarauntaise

Flew to Lyon on a Saturday evening; boarded a coach which took us east, slowly wending its way into that part of the Alps which curls south to define the French/Italian border. The bus made increasingly heavy work of the journey as the road became ever narrower and more precipitous. Eventually, it was a narrow two-lane switch back, with a rock wall on our left, and a deep void on our right, softly falling snow adding an extra element of interest to the driver's efforts to maintain traction.

We arrived at midnight in the village of Val d'Isere, towing our cases through streetscapes made magical by the simply alchemy of fresh powder. Our accommodation jarred a bit though - cramped, shoddy fittings, and a strong smell of vomit. Too late to move that night; but, the next day, I chewed my way through four unfortunate reps, finally getting us into a very nice chalet in on the main street very close to the lift stations.

Snowy lane in Val d'Isere

This was our first time skiing in France, and we found our resort lived up to its reputation - instructors who didn't flinch from telling us the unvarnished truth about our technique, and pistes that routinely turned out to be a grade harder than our maps modestly suggested. The ski area (Espace Killy) is truly enormous, and runs the gamut from very high (year-round) glacier skiing to gentle forest runs low in the valleys. The scenery, hidden in clouds for the first few days, made a pretty wonderful backdrop to our holiday when it eventually emerged - maybe not quite a match for the Swiss Alps mentioned a post or two back, but...

I still haven't gotten over the sheer wonder of standing on a high ridge and seeing a winter landscape that stretches to the horizon, deep snow blanketing the land from the peaks to the valleys - even the seracs on the glaciers almost invisible beneath it. Thirty miles away, the (ferocious) southern (Italian) face of Mont Blanc was clearly visible - the air so clear one might have thought it to be in the next valley.

To be continued...

Thursday, February 08, 2007

Finessing, with Trig

Adventure in a rather flatter environment today; the cartesian plane. An interesting problem came my way this morning, and I found it so exquisitely satisfying to solve that I decided to set out my journey across said plane for the edification of my readership. I realise that many of you will be starting out rather sceptical that words like “elegant” belong on even the same page as a mathematical formula; what I want to give you is sense of my own wonder that such fearsome beasts as trigonometric functions can be whispered and coaxed into perfect co-operation, solving problems with no apparent relation to geometry – and the sheer pleasure of mathematical tinkering.

For anyone still reading, my problem was this: a certain object is to be drawn by a computer using two very different devices. This object has an opacity ranging between 1 (opaque) and zero (invisible); now, the artist has found that his favoured opacity value looks nice on one output device, but a little too solid on the other. He doesn't want to set the opacity twice - he just wants his picture to "look right" on the second device. Clearly, I need a "creative" way to interpret my picky artist's choice of opacity.

Perhaps I could simply halve the value used for the "too solid" device? No, that wouldn't do; a setting of "fully opaque" would give a solid rendering on one device, and a semi-transparent from the other.

Clearly, I needed a more elegant function; some formula that would "preserve" the start and end values (1 and 0), yet interpret intermediate values in a way that improves (increases) the transparency for the second device. Since the default(x) forms a straight line, I realised that I need a curve to get “under” x while still starting and ending at the same points. Below, you can see a chart like the one I used to hunt for my function; x and x/2 form straight lines, while the functions sine, cosine and tan form various curves. The tan (purple) gave me an “aha!” moment. True, it did not start precisely at 1; but it “dived” beautifully at first, before leveling out and settling gently onto zero.

Nearly there; to make my formula start at 1, I needed to “squash” it to fit my range: I scaled it by adding an extra term: now, I had tan(x) * 1 / tan(x). Much better! Now, my curve started and finished with the line for x – but I wasn't happy yet. Wouldn't an even steeper gradient be better? It occurred to me that squaring x might be the answer: numbers between 1 and 0 get smaller when squared. For example, 0.8 squares to 0.64, and 0.5 squares to 0.25 – while, handily, 1 squares to 1 and 0 to 0 – perfect! Adjusting our formula one last time, we arrive at tan(x^2) * 1 / tan(x) – or, visually, a curve which precisely fits the range we needed, swooping gracefully towards full transparency. If the (elegantly!) fudged transparency values still don't quite suit our artist, higher or lower powers of the first x in our formula can be used to “tune” the dip of our curve to his taste.