The Placidus house system divides the semi-arc of each degree of the ecliptic into three equal intervals of sidereal time. Rather than following the degree through time, the construction considers the sky at a single instant.
At the moment the Ascendant lies on the eastern horizon, every ecliptic degree already occupies a particular position on its own declination circle. The problem is to determine which declination circle is currently at the 1/3, 2/3, and 3/3 points of its semi-arc between the horizon and the meridian. Where these declination circles intersect the ecliptic determines the cusps of the 12th, 11th, and 10th houses.
Because each ecliptic degree occupies its own declination circle, each has a unique semi-arc. This construction must then be repeated for every likely degree of the zodiac until the declination circles whose intersections with the ecliptic define the 11th and 12th house cusps are found.
Unlike Alcabitius, which simply trisects the semi-arc of the Ascendant, the Placidus construction must be repeated for many degrees of the zodiac until the correct declination circles are found. Because each degree occupies its own declination circle, each has a unique semi-arc. This process is long and quite tedious.
Happily, modern astrology software obtains the required house cusps almost instantaneously using interpolation between pre-calculated ephemeris positions, which frankly, is what most astrologers did, even in the middle ages.