Hello There miss_yuko,
Well, before I begin to answer your question, allow me to give you what "The Rising Sign/Ascendant" means (atleast to me).
The Ascendant-the degree of the ecliptic rising over the eastern horizon at the moment of birth - is the starting point of an astrological chart. It indicated the foundation of the personality, and shows how a subject adapts to his or her environment. The effect of the Ascendant has been compared to that of dawn - the approaching light that begins to illuminate the complexities of a human landscape. Just as one's birthday represents and individual "New Year's Day", so the Ascendant represents the start of that day.
The Ascendant shows us as we really are, for the traits suggested by the Ascendant emerge only when one really knows the person concerned. Some astrologers argue that it is the characteristics of the Ascendant that are immediately noticeable. But to take a superficial example, A Sun Sign Leo woman with a Sagittarius Ascendant us far more likely to go to the supermarket in her smartest winter coat (Leo Image) than to a formal occasion dressed very casually (Sagittarius motivation and personality trait). If Sag were the sun sign and Leo the rising, the image would be causal, but those who knew the person well would probably be aware that, inspite of an excellent sense of humor and an informal approach, she could be autocratic and bossy (Leo Rising).
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Now to answer your queries
My mother used to do astrology charts and she told me my ascendant is Sagittarius, just like my sun sign. However, a while ago, i tried to do my own chart using those sites that offer free astrology stuff (my plan was to use myself as a guinea pig to start learning astrology). But some sites said my ascendant was Sagittarius, and others said it was Capricorn.
Well, there are many different kinds of astrological practices (Vedic, Tropical/Western, Chinese, Mayan..etc etc) Some use sidereal signs (Based on the fixed constellations), some use Tropical Signs (Based on the ecliptic circle that surrounds the earth). So I would suggest It's best to ask your mother what system she used, and then compare with the website as to which system they use.
For site IMO for creating accurate charts (It's very important to know the exact time of birth because the ascending degree changes every 4 minutes, and the ascendant remains in one sign for roughly 2 hours daily) is
www.astro.com . It offers a variety of choices regarding house systems and even a choice between creating tropical and sidereal horoscopes.
What does it mean to have a sun sign or an ascendant on a cusp?
And does it make any difference? I feel it does, but then again i'm a beginner and i need the opinion of the people who know more than i.
Well personally, I feel the idea of "cusps" (I.e. when a planet is along a 5 degree borderline between two signs) is not really valid. With all the precise mathematical calculations that take place in an astrological chart, it's a bit immature to consider that a planet contains the characteristics of 2 signs just because it comes on a cusp. Example, Someone born on the 20th of April claims that he's born on the cusp of Aries and Taurus, hence feels he possesses both qualities of Aries and Taurus is sheer rubbish. Depending upon the degree of the sun, even if it is in 27 degrees of Aries (30 degrees to each sign) the man's sun has nothing to do with taurus. If he feels he has taurean qualities, it's only because that the sign of taurus is either his ascendant, or moon sign, or that he has numerous planets in that sign that are powerful, hence creating the impression of Taurean qualities. But his sun shall be Aries.
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I hope this has been helpful. If you wish to know anything more or wish to discuss anything at all.. do let me know... I'll be happy to help