Re: affine geometry question

From: Nick Singer (anonymous_at_mathforum.org)
Date: 08/19/04 

Date: Thu, 19 Aug 2004 12:08:44 +0000 (UTC)

On 19 Aug 2004, Lynn Kurtz wrote:
>On Thu, 19 Aug 2004 03:56:03 GMT, "Wilson McGramer"
><loargy_13032@yahoo.com> wrote:
>
>>A set of notes by Kenneth I. Joy states:
>>
>>-------
>>Consider three points P1, P2, P3. A point P is defined by
>>
>>P = a1P1 + a2P2 + a3P3
>>
>>where a1 + a2 + a3 = 1 gives a point in the triangle P1P2P3. we note that
>>the definition of affine combination defines this point to be
>>
>>P = P1 + a2(P2 - P1) + a3(P3 - P1)
>>---------
>>
>>How can I rework the last equation so I can find a2 and a3, if all other
>>values are given.
>>
>>Also, assuming that I retrieve them, how do I test for a1 + a2 + a3 = 1, (P
>>is within the triangle) when a1 is not available (where did it go in the
>>last equation?)
>
>Put a1 = 1 - a2 -a3 in the first equation to get the second, that's
>where it went.
>
>--Lynn

And don't forget that you must also have a1, a2, and a3 *nonnegative* if you want the barycentric sum to stay within the triangle (= convex hull of P1, P2, P3).

Nick

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