These are the three mitzvoth de'oraisa that we are given with respect to the shalosh regalim in general , according to the braisa in Chagiga 6B. (Each one has its own specific mitzvot as well, but these are the three general ones true of all of the shalosh regalim.)

There is a masechet in the Talmud called Chagiga which deals with these three mitzvot as they are applied to the shalosh regalim. But the masechet begins with the words Hakol Chayavin Be-Re'iya - a discussion of re'iah - being seen at the Beit Hamikdash during the holiday. The Rambam in Hilchot Chagiga, asks: Since these are the opening words of the masechet, why is it not called Masechet Re'iya? Why specifically Chagiga?

In addition, in the Tosefta we see that they attributed to the Olat Re'iyah and the Shalmei Chagiga the overall name of "Chagiga". Why?

Furthermore, in Shmot 23, there are 2 psukim - "Shalosh Regalim Tachog Li Bashana", and then "Shalosh Peamim Bashana Year'eh Kol Zechurcha Et Pnei Ha'adon Hashem." What does the word "Tachog" mean, and what is the difference between it and "yera'eh"?

The Rambam explains in Hilchot Beit Habechira 1:1 that there is a mitzvah to build a Beit Hamikdash (1) that will be there to bring Korbanot, and (2) where the people will celebrate (chogegin) three times a year. In parallel with this, in Sefer Hamitzvot Mitzvah 20: It is a mitzvah to build a Beit Hamikdash (1) as a place for bringing korbanot and burning the eish tamid, and (2) where people should go and gather there three times a year.

The Rambam did not waste words. We must look at every word he wrote very carefully in order to understand his thinking and insights. In my humble opinion, the explanation here is that "chag" means circle (as the word for an implement that draws is a circle is a "Chogeh"). Since people who gather for an event tend to gather in a circle, hence the mitzvah is to gather in a circle, "tachog", three times a year. This reconciles the two versions in the Rambam.

We can now better understand the Rambam in Hilchot Chagiga, where he says that all of the mitzvoth of the chagim are anchored in re'iah. Everyone who is obligated in Re'iyah is obligated in Chagiga as well, and everyone who is exempt from Re'iyah is exempt from Chagiga as well. The Rambam even goes so far as to include the mitzvah of Hakhel in this rule as being included in hilchot Chagiga, such that everyone who is obligated to do re'iya is also obligated to do Hakhel, as a form of gathering (other than cheresh, shoteh vekatan.)

The question is raised of the tumtum and whether he is chayav on the mitzvah of re'iyah, and to what level. The tumtum is not chayav on the korbanot even though safek de'oraisa lechumra, because if it turns out that he is not male, then any animal he may bring would be chulin and would profane the Beit Hamikdash. The opinion is then voiced that he should at least be chayav on re'iah, since that is a mitzvat asei that is doche the lo ta'asei of chulin. The answer is that "lo yera'u panai rekam". If there is no korban, then there is no issue of re'iah at all. The korban is an integral part of the mitzvah of Re'iya. If a person comes to the Beit Hamikdash and gathers there, but comes without the Olat Re'iya, not only has he not fulfilled the mitzvah of re'iya, but he has committed an avaira (lo yera'u panai rekam). Therefore, since the tumtum is not chayav on a korban, he is not chayav on re'iah either.

The Mishna discusses a dispute between Beit Shamai and Beit Hillel over the question of a person who has only a very limited amount of money and needs to figure out which of these two korbanot he should invest more in.

Beit Shammai says he should invest two thirds of the money in the olat re'iah and one third in the shalmei chagiga, and Beit Hillel says that he should invest two thirds in the shalemi chagiga and one third in the olat re'iah.

The Rogatchover provides a very important insight into the over-all dispute between Shammai and Hillel that can be applied to every issue in which they argue. Beit Shammai is concerned with the ends. He emphasizes the achievement of the ends as the most important element in the mitzvah. Beit Hillel is more concerned with the means - the path to completing the mitzvah - and views the process as the main element.

Thus, in the dispute over whether the heavens were created first or whether the land was created first, Beit Shammai says that the heavens were created first, since the ultimate purpose of our existence in this world is to achieve a higher level of spirituality and bring ourselves closer to Heaven, while Beit Hillel says that the earth was created first since it is through our existence on this earth that we achieve our higher spirituality and more heavenly aim.

And in the dispute over how many Chanuka candles to light each night, Beit Shammai rules that you begin with 8 - the ultimate aim of the mitzvah - and reduce the number by one each night, while Beit Hillel rules that you begin with one and increase the number each night until reaching the ultimate aim on the 8th night - concerned more with the process of reaching the aim.

With this background, we can more fully understand the dispute over which korban to invest more money in. The Olat Re'iah is the ultimate aim of the holiday mitzvot. The olah is a korban that is given in its entirety to Hashem and the owner has no part in it after it is given. Thus, Beit Shammai rules that it is the korban that must be invested in more heavily. On the other hand, the Shalmei Chagiga represent more of the process - the shlamim were korbanot that the giver has a stake in, since the giver will ultimately eat from that korban. Thus, since it is part of the process of the owner's celebration of the holiday, Beit Hillel rules that it is to be invested in more heavily.

The Tosefta therefore holds that according to Beit Shamai, the Olah is not called "Chagiga" because the Chagiga is not the main part of the Mitzvah. It is only the means. But according to Beit Hillel, both are called "Chagiga" since the main part of the mitzvah is the process, what we actually do rather than where we actually are. Hence the masechta is called Chagiga and not Re'iya.