Is a full-wave rectifier really required with a 7810 regulator?
In this situation: 12 V at 500 mA:

After HWR (half-wave rectification) the peak-to-peak voltage is \$12\cdot \sqrt{2} - 0.7 = 16.27\,\mathrm{V}\$. The 10V regulator needs a voltage greater than 12.5V to work, so the maximum ripple shouldn't exceed 3.77V.

This means that the capacitor value should be: \$C = \dfrac{I}{fV_p} = 2650\,\mu\mathrm{F}\$ minimum.

But with a center tap transformer you can use this kind of full wave rectifier (not bridge!):



Source: Wikipedia - Center tap

So the minimum value capacitor has the same value because the maximum available current and frequency are double now. But if you only need max. 500 mA a capacitor of minimum \${2650 / 2} = 1325\mu \mathrm{F}\$ will be enough.