custom dimmer linear AC power

[spetroce]

I am working on a dimmer circuit for a light bulb. I've implemented an analog circuit that detects the zero crossing on the AC mains and sends out a trigger pulse to the MCU on our product to handle the forward phase switching of the dimmer. This all works fine. Something I noticed though, was the brightness to the light does not increase linearly with the duty cycle of the dimming. I then realized that taking equally sized discrete steps along the sine wave does not always translate to the same area under the sine wave. That is, we should be taking discrete switching steps along the sine wave in our dimming such that area under the curve for each step is the same. So I wrote some code that takes the discrete integral of the sine wave, blah blah, and now I have steps that each have the same area under the curve. Great. I don't know if this is correct though. My theory may be off. Does this mean that the power will increase linearly? In other words, with a purely resistive load, is the the voltage/time area equal to the power (or is it energy) that the load will experience? My thought on this whole thing was with PWM on a normal DC source where the increase in duty cycle is a linear increase in the area under the square wave form. So I just wanted to mimic the same thing with the AC sine wave, but with the AC we have a varying voltage and therefore the current is varying along the sine wave... This is where my theory weakens. I'd greatly appreciate any input on this. Thank you!

This chart below shows the discrete steps. The area between any two points is the same.