h a l f b a k e r y
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I can't claim to be the first to have come up with this, but I feel there are some additions to be made.
First of all, the flat, octagonal surface (see link) is not ideal for an umbrella, as it does not deflect water as well, and is also harder to control in windy conditions. My change would be
to have a more typical, semi-spherical shape. This would fix the aforementioned problems and also put a part of the umbrella's surface perpendicular to the sun's rays even when the umbrella itself wasn't facing directly into the sun.
Also, it should be able to fold. These two requirements could be met either by having numerous individual cells hinged together, or by using a few large, flexible cells.
Various sizes would be nice, too. Large umbrellas for stationary activities, like shading picnic tables. Mid-sized ones for normal use, and small ones (probably with telescoping handles) for camping or carrying around in unpredictable weather.
Finally, have a port in the handle for the device which is to be powered by the cells.
(?) Photovoltaic Umbrella
It's a good start. [nick_n_uit, Jun 05 2001]
SOLARBrella incorporated photovoltaic strips into tension membranes & umbrellas [americanph, Sep 24 2009]
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||The one on the link is not actually an umbrella in the sense of carrying around to deflect rain, its a stationary sunshade, and not a bad idea.
||Problem is, if you need an umbrella, you're not likely to get much solar power. And if you just want the solar power, a folding solar array would probably be less cumbersome.