## construction of a point

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• Dear friends, given two points p:q:r and u:v:w in barycentrics, I wish to construct the point c^2*q*u - b^2*r*u - a^2*r*v - b^2*r*v + c^2*r*v + a^2*q*w -
Message 1 of 3 , Mar 31, 2010
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Dear friends,

given two points p:q:r and u:v:w in barycentrics, I wish to construct the point

c^2*q*u - b^2*r*u - a^2*r*v - b^2*r*v + c^2*r*v + a^2*q*w - b^2*q*w + c^2*q*w :

a^2*r*u + b^2*r*u - c^2*r*u - c^2*p*v + a^2*r*v + a^2*p*w - b^2*p*w - c^2*p*w :

-a^2*q*u + b^2*q*u - c^2*q*u - a^2*p*v + b^2*p*v + c^2*p*v + b^2*p*w - a^2*q*w

I already know one line passing through this point but cannot find another one.

Any idea ?

Thank you very much

Bernard
• [BG] ... Dear Bernard, Did you try splitting the barycentrics in terms? Generally spoken a point P with barycentrics P (P1x+P2x+P3x+P4x : P1y+P2y+P3y+P4y :
Message 2 of 3 , Apr 2, 2010
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[BG]
> given two points p:q:r and u:v:w in barycentrics, I wish to construct the point
>
> c^2*q*u - b^2*r*u - a^2*r*v - b^2*r*v + c^2*r*v + a^2*q*w - b^2*q*w + c^2*q*w :
>
> a^2*r*u + b^2*r*u - c^2*r*u - c^2*p*v + a^2*r*v + a^2*p*w - b^2*p*w - c^2*p*w :
>
> -a^2*q*u + b^2*q*u - c^2*q*u - a^2*p*v + b^2*p*v + c^2*p*v + b^2*p*w - a^2*q*w
>
> I already know one line passing through this point but cannot find another one.
>
> Any idea ?
Dear Bernard,

Did you try splitting the barycentrics in terms?
Generally spoken a point P with barycentrics
P (P1x+P2x+P3x+P4x : P1y+P2y+P3y+P4y : P1z+P2z+P3z+P4z)
can be split into:
P12 (P1x+P2x : P1y+P2y : P1z+P2z) and P34(P3x+P4x : P3y+P4y : P3z+P4z)
and also
P13 (P1x+P3x : P1y+P3y : P1z+P3z) and P24(P2x+P4x : P2y+P4y : P2z+P4z)
and also
P14 (P1x+P4x : P1y+P4y : P1z+P4z) and P23(P2x+P3x : P2y+P3y : P2z+P3z).
Now P = P12.P34 ^ P13.P24 ^ P14.P23.

When P12, P13, P14, P23, P24, P34 are difficult to construct a further split can be done.
P1(P1x: P1y : P1z), P2(P2x : P2y : P2z), P3(P3x : P3y : P3z), P4(P4x : P4y : P4z).
Now P12 is on line P1.P2,etc.

The point you mentioned has barycentrics:
(2 SB.q.w - 2 SC.r.v + c^2.q.u - b^2.r.u :
2 SC.r.u - 2 SA.p.w + a^2.r.v - c^2.p.v :
2 SA.p.v - 2 SB.q.u + b^2.p.w - a^2.q.w)
Because the barycentrics consist of 4 terms it can be split as described.

Best regards,

Chris van Tienhoven
• Dear Chris, ... That s what I did for the line I ve found which corresponds to your P1.P2 but I m still stuck ! Thanks Bernard [Non-text portions of this
Message 3 of 3 , Apr 2, 2010
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Dear Chris,

> Did you try splitting the barycentrics in terms?
> Generally spoken a point P with barycentrics
> P (P1x+P2x+P3x+P4x : P1y+P2y+P3y+P4y : P1z+P2z+P3z+P4z)
> can be split into:
> P12 (P1x+P2x : P1y+P2y : P1z+P2z) and P34(P3x+P4x : P3y+P4y : P3z+P4z)
> and also
> P13 (P1x+P3x : P1y+P3y : P1z+P3z) and P24(P2x+P4x : P2y+P4y : P2z+P4z)
> and also
> P14 (P1x+P4x : P1y+P4y : P1z+P4z) and P23(P2x+P3x : P2y+P3y : P2z+P3z).
> Now P = P12.P34 ^ P13.P24 ^ P14.P23.
>
> When P12, P13, P14, P23, P24, P34 are difficult to construct a further split can be done.
> P1(P1x: P1y : P1z), P2(P2x : P2y : P2z), P3(P3x : P3y : P3z), P4(P4x : P4y : P4z).
> Now P12 is on line P1.P2,etc.
>
> The point you mentioned has barycentrics:
> (2 SB.q.w - 2 SC.r.v + c^2.q.u - b^2.r.u :
> 2 SC.r.u - 2 SA.p.w + a^2.r.v - c^2.p.v :
> 2 SA.p.v - 2 SB.q.u + b^2.p.w - a^2.q.w)
> Because the barycentrics consist of 4 terms it can be split as described.

That's what I did for the line I've found which corresponds to your P1.P2 but I'm still stuck !

Thanks

Bernard

[Non-text portions of this message have been removed]
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