The Phase-Induced Amplitude Apodization Coronagraph (PIAAC) uses a lossless beam apodization, performed by aspheric mirrors, to produce a high contrast PSF. This concept offers a unique combination of high throughput (almost 100%), high angular resolution (λ/D), small inner working angle (IWA = 1.5 λ/D), excellent achromaticity (the apodization is performed by geometric reflection on mirrors) and low sensitivity to pointing errors or stellar angular diameter. These characteristics make the PIAAC an ideal choice for direct imaging of extrasolar terrestrial planets (ETPs) from space. We quantify the performance of the PIAAC and other coronagraph designs both in terms of "raw" coronagraphic performance (throughput, IWA etc...) and number of stars around which extrasolar terrestrial planets (ETPs) can be observed. We also identify the fundamental performance limit that can be achieved by coronagraphy, and show that no other coronagraph design is as close to this limit as the PIAAC. We find that in the photon noise limited regime, a 4m telescope with a PIAA coronagraph is able to detect Earth-like planets around 30 stars with Ihr exposure time per target (assuming 25% throughput and exozodi levels similar to our solar system). With a smaller 1 to 2-m diameter telescope, more massive rocky planets could be detected in the habitable zones of a few nearby stars, and an imaging survey of Jupiter-like planets could be performed. Laboratory results and detailed simulations confirm the large potential of this concept for direct imaging of ETPs. A prototype high contrast PIAAC system is currently being operated to demonstrate the coronagraph's performance.