Spiral Umbrella
For illustration's sake, take a piece of paper, cut a long, concentric spiral into it, then lift the middle of the spiral. That's the basis for the Spiral umbrella. Pretty useless as an umbrella so far because the water will just come right through the gap between iterations. So
take another piece of paper and cut another spiral such that, when interleaved with the first, the two overlap through their curve. Now, when you pick them up there's not as much of gap, and no gap at all from a vertical perspective.
Partnered with normal ribs and done in waterproof material, this will look nifty with its spirals and small pleats, and operate just fine as an umbrella. But wait, there's more !
Adjustable Coverage
The ribs in this spiral umbrella are normally sprung sidewards, in a slight spiral themselves. A drawstring running around the circumference of the umbrella can be tightened, curving the ribs more to shrink the umbrella from couple-sized to personal-sized, or loosened: allowing the ribs to straighten out completely, forming a 6' diameter partial-sunshade beach umbrella. And, finally...
Dragonsback Umbrella
Using the above techniques, the canopy is created as a single spiral, made up of pieces of plate mail or wood veneer. The plates will shingle outwards and also along the spiral, as the umbrella is expanded and contracted.
With that in mind, a small spring or electric motor could shift the scales around a bit, periodically, just to see if anybody's paying attention.