Barycentric Coordinates

Transcription

1 rentri oordintes,, re the distnes from the verties to the ontt points of the eires with the sides. + = + = + from whih = s - = s - The three rs in this pitures re Gergonne r, medin, nd Ngel r. This piture n e used to show tht the Gergonne nd Ngel points re isotomes. ontents rentri oordintes The inenter fmil of points onw on rentris onw ompres vrious oordinte sstems

2 Two oordintes to desrie point in the plne. For severl resons it is etter to desrie points in two dimensionl tringle geometr three oordintes. Tringles re the fundmentl shpe nd tend to see things in threes. rentri oordintes (,, ) re suh oordinte sstem. Sine three numers re used when onl two re needed, we dd one other ondition to these oordintes. If + + = 1, we s tht the oordintes re normlied. lterntel onl rtios re used with (,, ) = (k, k, k) nd k > 0. In this se we write the oordintes s ( ). oordintes vlid up to rtio re lled homogeneous oordintes. Mn properties of tringles re independent of sle so rtio independene n e n dvntge. rentri oordintes P The rentri oordintes ( ) of point P n e red from the evin rtios of P or the res used P (see piture). It is surprisingl es to find the oordintes this w. If,, nd re the vetors representing the verties of the tringle then the rentri oordintes re used to write P s P = + + where + + = 1. + The verties re (1, 0, 0), (0, 1, 0), nd + + (0, 0, 1). + + The signs of the oordintes re shown in the digrm to the right. The oordinte's sign (0,1,0) depends on its position reltive to side. It is + (1,0,0) = 0 on side + not possile for ll three oordintes to e + + negtive. omputtions in homogeneous oordintes re it unusul sine we re free to multipl the oordintes n ftor. s n emple ( ) = (1/ 1/ 1/) sine we n divide eh oordinte the ftor. Often oordintes re smmetrill epressed in terms of the sides or ngles of the tringle. For emple the oordintes of the Ngel point re ( ). In this se shorthnd nottion n e used ( +- ), showing onl the middle oordinte. The importne of smmetri forms smmetri epressions in the ngles or sides of tringle hve speil importne. The re, the semiperimeter, nd the rdii of the insried nd irumirles re ll smmetri in the sides of the tringle. Using the rtio independene of the oordintes the inenter ( ) n e written in seond form s follows ( ) = ( /2R ) = ( sin ), where R is the irumrdius. = 0 on side (0,0,1) = 0 on side The entroid s n emple of the use of rentri oordintes, we n see tht the oordintes of G, the entroid, re (111). This is seen sine the entroid is the onurrene of the medins, whih divide eh side equll in 11 rtio.

3 rentris of the Inenter fmil of points The Inenter n ngle isetor stisfies the formul / = /. ppling this reltionship to ll three isetors we otin these rtios. We red the rentri oordintes from the evin rtios. Inenter oordintes ( ) = ( ) This finl form is onw's shorthnd nottion. For points where the oordintes hve the sme form onl the middle oordinte is shown. The Gergonne Point The Gergonne point Go is the onurrene of the evins to the ontt points of the inirle with the sides. We first find some properties of the tringle with its inirle. Let,, nd e the distnes from the verties of to the ontt points of the inirle with the tringle. The Ngel Point The Ngel point No is the onurrene of the evins to the ontt points of the eirle with. Ngel point oordintes ( s ) nd re the distnes from the verties to the ontt points of the eires with the sides = perimeter + + = s = s ( + ) = s We use the shorthnd s = s -. s - s - s - G o Gergonne point oordintes ( s s s s s s ) = ( 1/s 1/s 1/s ) = ( 1/s ) Divide ll oordintes s s s to get this form. The three rs in this pitures re Gergonne r, medin, nd Ngel r. This piture n e used to show tht the Gergonne nd Ngel points re isotomes. s - + = + = + from whih = s - = s - s - s - We djust the evin rtios so we n see the rentri oordintes. (s - )(s - ) (s - )(s - ) (s - )(s - ) G o (s - )(s - ) (s - )(s - ) (s - )(s-) How to red the rentri rtios from mesurements on the tringle. oordinte orresponds to verte. Notie the position of the 's ompred to their verte. The nd oordintes re equivlentl pled. Rememering tht we re free to djust rtios multipling oth elements the sme numer, we dnust the rtios so tht the rentri oordintes n e red. P

4 Dte From To 12/2/ PM John onw, Steve Sigur, Unfortuntel I red few introdutor ooks mself, nd the onl ones I know tht use rentris re ver old nd in n se, s ver little. ut there is in ft ver little tht needs to e sid! The NORMLIZED rentris of P with respet to,, re the unique numers (X,Y,Z) dding up to 1 for whih the vetor eqution P = X + Y + Z holds - UNNORMLIZED ones re n triple proportionl to these. For the ltter I write (XYZ) with olons to indite tht onl their rtios mtter. Sine ou re fmilir with orthogonl triliners, whih I write [,,] or [], ou'll need to know the onversion rules [] onverts to (XYZ) onverts to () [X/Y/Z/]. So there's ver little differene, ut the differene is vlule! If ou drw our tringles the w round I do, XYZ re esil red off from the evin rtios or re-rtios Z / \ X / \ / \ / \ * * / \ Y / P \ Y / X P Z \ / \ / Y \ -----* *----- X Z (sorr I n't drw the seprting lines P,P). So plinl G = (111), or, when normlied (1/3,1/3,1/3), mening tht G = (++)/3 in vetor terms. The suordinte nd superior of P re 3G/2 - P/2 nd 3G-2P (normlied) or (Y+ZZ+XX+Y) nd (-X+Y+ZX-Y+ZX+Y-Z) (unnormlied) s we see using G = (X+Y+ZX+Y+ZX+Y+Z). nd The isotomi nd isogonl onjugtes of P re (/X/Y/Z) (onjugl) (1/X1/Y1/Z) (isotome).

5 Sometimes, of ourse, the OT formul is the simpler one. For instne the onjugl of [] = () is (///) = (///) = [1/1/1/], simpler thn the rentri formul, while its isotome is [1/1/1/] (more omple). ut THIS simpliit osures the sitution. The isotomi onjugte is n ffinel invrint onept (in other words, if ou linerl streth the tringle, this reltion remins the sme) - we n see this euse the rentri formul (1/X1/Y1/Z) mentions no speil onstnts like,,. The isogonl onjugl isn't ffinel invrint (it depends on ngles); so its rentri formul SHOULD mention the shpe of the tringle. If ou look upstirs, ou'll see tht the formule for suordintes nd superiors re lso onstnt-free in rentris, gin euse these re ffinel invrint - gin tht is osured in OTs. nother instne of the sme phenomenon is tht G = (111) in rentris, ut [1/1/1/] in OTs, while I = () in rentris, ut [111] in OTs. gin, the simpliit of G in rentris tells us something - tht the entroid is n ffinel invrint onept - while tht of I in OTs tells us nothing. The swith to rentris should e quite pinless if t first ou just get into the hit of lling the tpil point [X/Y/Z/] rther thn [,,] - in other words, regrd the rentris just s renmed triliners. s I sid, the're ver muh the sme, ut the slight differene is importnt. The numers,, re mere nuisneftors tht lutter up most of the OT theor nd often osure the sitution. John onw I'm sending this to severl people interested in tringles, nd hope the'll onfirm reeipt (eept for rkg, who might e out of emil touh), nd send n further sustntil messges out tringles to everone else on the list. There re three stndrd sstems tht use three oordintes to represent point in the plne of give tringle, nmel RYENTRIS (), RELS (), nd ORTHOGONL TRILINERS (OT). Often the lst is shortened to "TRILINERS", ut I prefer the longer nme sine in ft ll three sstems re triliner. Eh m or m not e normlied. For the not-neessril normlied versions I'll use (XYZ) for nd, nd [] for OT nd for the normlied ones (X,Y,Z) for, nd [,,] for OT. ll three (unnormlied) sstems re ver similr - indeed the oinide for rentris nd rels, so I'll usull ll these jointl, nd the onversion from TO to these is ver simple [] eomes (). So for mn purposes it hrdl mtters whih sstem one uses. However, there RE ws in whih one or other of the sstems is etter thn nother, nd it is the purpose of this note to point out tht when one tkes ll these into ount the rentri sstem emerges s the ler winner. The letters,,,ot efore eh numered point elow show how this deision ws rehed. letter N indites tht normlied oordintes re involved.

6 OT 0N. The distnes of P from the sides re 2X.Delt/, 2Y.Delt/, 2Z.Delt/ [,, ]. 1N. The res of P, P, P re the normlised rel oordintes X.Delt, Y.Delt, Z.Delt [ /2, /2, /2 ] 2N. If V, V, V re vetors to,,, then P = X.V + Y.V + Z.V [Of ourse these re just the definitions of the three sstems.] 3. The evin rtios re YZ, ZX, XY [,, ]. 4. The Menelen rtios of the lines PX + QY + RZ = 0 nd p + q + r = 0 re -QR -RP -PQ nd -qr -rp -pq 5. The normliing ondition is X + Y + Z = 1 () or Delt (), [ + + = 2Delt ] OT 6. The isogonl onjugte (or "onjugl") is o-p = ( ^2/X ^2/Y ^2/Z ) [ 1/X 1/Y 1/Z ] 7. The isotomi onjugte (or "isotome") is iso-p = ( 1/X 1/Y 1/Z ) [ 1/(.^2) 1/(.^2) 1/(.^2) ] 8. FFINE INVRINE. If n ffine trnsformtion tkes,,, nd P = (X,Y,Z) to 1, 1, 1 nd P1, then the rentri oordintes of P1 with respet to he new tringle re still (X,Y,Z). (It is euse isotomi onjugtion is n ffinel invrint onept tht its epression (see #7) in rentris nnot involve the edge lengths of the tringle.) 9. The onepts of suordinte nd superior points These points re the imges of P in the suordinte (or "medil") tringle, whose verties re the midpoints of the edges of, nd the superior (or "ntiomplementr") tringle, the midpoints of whose edges re,,. We hve su-p = ( Y+Z Z+X X+Y ) [ (+)/ (+)/ (+)/ ] super-p = (Y+Z-XZ+X-YX+Y-Z) [(+-)/(+-)/(+-)/] (gin the simpliit of the oordintes is due to ffine invrine.) 10. RTIONLITY. X,Y,Z re rtionl funtions of the Euliden oordintes of the points,,,p. If P is point tht's rtionll defined from,,, then its rentri oordintes re rtionl funtions of ^2, ^2, ^2. This is true, for instne, of the entroid, orthoenter, irumenter, smmedin point, rord points, nd so on. (This is n etremel importnt point, nd etends to give the ver useful propert elow.)

7 11. LGERI ONJUGTES. Mn points (suh s the inenter) n e otined solving lgeri equtions tht hve other (lgerill onjugte) solutions. Pssing to these other solutions then ields further points tht hve essentill the sme geometri properties (in this w, we get from the inenter to the eenters). We n get the rentri oordintes of suh "ompnions" s the pproprite lgeri onjugtes of those of the originl. The simplest se of this is when the oordintes re rtionl funtions of,, ut not of ^2, ^2, ^2. So for emple if point tht's rtionll onstruted from the inenter hs rentri oordintes ( X(,,), Y(,,), Z(,,) ) then the orresponding point otined from the -eenter is simpl otined "hnging the sign of ", thus ( X(-,,), Y(,-,), Z(,,-) ). For emple the Ngel point is the "super-inenter" ( ), nd so its -ompnion is ( ). The OT oordintes of these points re muh hrder to understnd [ / + / - 1 / + / - 1 / + / - 1] nd [ / + / + 1 / - / - 1 -/ + / - 1]. Well, tht will do for now. I'll just surve the "winners" OT OT In onl two ses is OT the winner, nd in ll other ses ut one (the definition of!) is t lest joint winner. I hve deliertel preferred oneptul resons for preferring one sstem to nother, rther thn mere omprisons of the simpliit of the oordintes for prtiulr points. Some points re simpler under one sstem rther thn nother, nd often it's OT tht would give the simpler ones. ut this differene n never e gret, sine[,,] trnsltes to [,,]; nd in rentris the simpliit often hs useful oneptul mening. For emple (111) = [ 1/ 1/ 1/ ] is the entroid, nd its simpliit in rentris omes from its ffine invrine. On the other hnd () = [111] is the inenter, more omplited in rentris sine it hs lgeri onjugtes (-) (-) (-). The pprent simpliit in triliners disppers when we pss to the su-inenter (Spieker point) (+++) = ( (+)/ (+)/ (+)/ ). The "Morle perspetors" look simpler in OT [ os(/3) os(/3) os(/3) ] nd [ se(/3) se(/3) se(/3) ] ut gin this pprent simpliit disppers when we wnt it more the rentri versions (.os(/3)... ) nd (.se(/3)... ) n e onjugted to get ll the "ompnion Morle perspetors" just s esil. In summr, the simpliit of oordintes for prtiulr points n go either w, nd in n se is not strong rgument. It's their theoretil properties tht mke rentris the ler winner. John onw