TXM5: Tripolar and Trilinear Coordinate Relation
In this notebook, we introduce a simple connection between tripolar coordinates and trilinear
coordinates.
1
.
Note​
​TXM5 is the author’s formula code
2
.
Caption​
LettriangleABCandanarbitraryM.Thenwehave:​​​​
d
a
=
a
S
A
2S
+
-
2
a
2
MA
+
S
B
2
MC
+
S
C
2
MB
2Sa
​​
d
b
=
b
S
B
2S
+
-
2
b
2
MB
+
S
C
2
MA
+
S
A
2
MC
2Sb
​​
d
c
=
c
S
C
2S
+
-
2
c
2
MC
+
S
A
2
MB
+
S
B
2
MA
2Sc
​​​​Where​​
S,
S
A
,
S
B
,
S
C
areConwaytrianglenotations​​
d
a
,
d
b
,
d
c
aresignedalgebraicdistancesfromMtoBC,CA,AB,respectively.
3
.
Code
In[]:=
ManipulateModule{Mtocentroid,crossAB,crossBC,crossCA,altAB,altBC,altCA,Mdist,a,b,c,MA,MB,MC,SA,SB,SC,S},​​Mtocentroid=Line[{M,Mean[{AA,BB,CC}]}];​​crossAB=If[RegionIntersection[Mtocentroid,InfiniteLine[{AA,BB}]]===EmptyRegion[2],1,-1];​​crossBC=If[RegionIntersection[Mtocentroid,InfiniteLine[{BB,CC}]]===EmptyRegion[2],1,-1];​​crossCA=If[RegionIntersection[Mtocentroid,InfiniteLine[{AA,CC}]]===EmptyRegion[2],1,-1];​​altBC=TriangleConstruct[{BB,M,CC},"Altitude"];​​altCA=TriangleConstruct[{AA,M,CC},"Altitude"];​​altAB=TriangleConstruct[{AA,M,BB},"Altitude"];​​Mdist=EuclideanDistance@@First[#]&/@{altBC,altCA,altAB};​​a=EuclideanDistance[BB,CC];​​b=EuclideanDistance[CC,AA];​​c=EuclideanDistance[AA,BB];​​MA=EuclideanDistance[M,AA];​​MB=EuclideanDistance[M,BB];​​MC=EuclideanDistance[M,CC];​​SA=
2
b
+
2
c
-
2
a
2
;SB=
2
c
+
2
a
-
2
b
2
;SC=
2
a
+
2
b
-
2
c
2
;S=
SASB+SBSC+SCSA
;​​Column​​Text@Grid​​{Style["
d
a
",Red],Style["
d
b
",Black],Style["
d
c
",Blue]},​​{Style[crossBCMdist[[1]],Red],Style[crossCAMdist[[2]],Black],Style[crossABMdist[[3]],Blue]},​​Style"
2
a
(
S
A
-
2
MA
)+
S
C
2
MB
+
S
B
2
MC
2aS
",Red,"
2
b
(
S
B
-
2
MB
)+
S
A
2
MC
+
S
C
2
MA
2bS
",Style"
2
c

S
C
-
2
MC
+
S
B
2
MA
+
S
A
2
MB
2cS
",Blue,​​Style
2
a
(SA-
2
MA
)+SC
2
MB
+SB
2
MC
2aS
,Red,
2
b
(SB-
2
MB
)+SA
2
MC
+SC
2
MA
2bS
,Style
2
c
(SC-
2
MC
)+SB
2
MA
+SA
2
MB
2cS
,Blue,Frame->All,FrameStyle->Hue[.6,.4,.8],Background->{None,{{None,GrayLevel[.9]}},{{1,1}->Hue[.6,.3,1],{1,2}->Hue[.6,.3,1],{1,3}->Hue[.6,.3,1],{1,4}->Hue[.6,.3,1],{1,5}->Hue[.6,.3,1],{1,6}->Hue[.6,.3,1]}},BaseStyle->{FontFamily->Times,FontSize->20},​​Graphics[{PointSize[.02],​​Blue,InfiniteLine/@Subsets[{AA,BB,CC},{2}],​​Gray,{altAB,altBC,altCA},​​Green,Line[{M,AA}],Line[{M,BB}],Line[{M,CC}],​​Black,​​Text[Style["A",20,"Label"],AA+{0,.4}],​​Text[Style["B",20,"Label"],BB+{0,.4}],​​Text[Style["C",20,"Label"],CC+{0,.4}],​​Text[Style["M",20,"Label"],M+{0,.4}]},​​ImageSize{600,350},PlotRange{{-4.5,4.5},{-4.5,4.5}},PlotRangePadding.25],​​{{AA,{-3.`,-2.25`}},{-4,-4},{4,4},Locator},​​{{BB,{-2.705`,2.685`}},{-4,-4},{4,4},Locator},​​{{CC,{3.`,-0.75`}},{-4,-4},{4,4},Locator},​​{{M,{2,2}},{-4,-4},{4,4},Locator},SynchronousUpdatingFalse,SynchronousInitializationFalse
Out[]=
​
d
a
d
b
d
c
-1.8401
2.91043
4.73749
2
a
(
S
A
-
2
MA
)+
S
C
2
MB
+
S
B
2
MC
2aS
2
b
(
S
B
-
2
MB
)+
S
A
2
MC
+
S
C
2
MA
2bS
2
c
(
S
C
-
2
MC
)+
S
B
2
MA
+
S
A
2
MB
2cS
-1.8401
2.91043
4.73749
4
.
Related Links
​https://community.wolfram.com/groups/-/m/t/2510942?p_p_auth=JsdU7w9j
5
.
Author information​
Trịnh Xuân Minh