Centers of a Simplex. Contents

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1 Ceters of a Simplex David Eberly, Geometric Tools, Redmod WA This work is licesed uder the Creative Commos Attributio 4. Iteratioal Licese. To view a copy of this licese, visit or sed a letter to Creative Commos, PO Box 866, Moutai View, CA 9442, USA. Created: October 3, 2 Last Modified: February 2, 28 Cotets Ceter of Mass 2 2 Circumscribed Hypersphere 3 3 Iscribed Hypersphere 4

2 Let the vertices of a -dimesioal simplex be v i IR for i. The edge legths are L ij v i v j. Defie M to be the matrix whose i th row is v i v. The hypervolume of the simplex is V det M /!. Ceter of Mass The ceter of mass of a simplex of homogeeous material is the average of the vertices of the simplex, C v i. + The proof is as follows. From calculus, the ceter of mass for the volume bouded by the simplex S is S C i x i dx dx S dx. dx The umerator of the fractio is a momet about the hyperplae whose ormal is i the x i directio. The deomiator of the fractio is the mass of the object. Let x Ay + b be the affie trasformatio which maps the vertices of the simplex to the + poits ad e k, k, where e k has a zero i all compoets except for a oe i the k compoet. Thus, b v ad v k Ae k + b for k. Let T be the simplex defied by these ew poits. Chage variables i the itegrals: C i i xi dx dx S S dx dx T j aijyj+bi deta) dy dy T deta) dy dy T j aijyj+bi dy dy T dy dy yj dy dy T j a ij T dy dy + b i. If γ is the ceter of mass of T, the the above shows that C Aγ + b. That is, the affie trasformatio of the ceter of mass is the ceter of mass of the affiely trasformed object. The symmetry of T requires that the ceter of mass is γ K,..., K) for some K, ). The value K I)/I) where Ip ) T xp dx dx x xp x x x x 2 x x p dx dx )! w) dx dx βp, ), the beta fuctio Γp)Γ) )! Γp+), Gamma fuctios p )! )! )! p+ )! p+ ) p+)p Each itegral i the iteratio ivolves itegratio of a sigle polyomial term, successive terms icreasig i degree by. Therefore, K [ ]/[ + ) 2] / + ). As a result, γ ) e i i

3 which is the average of the + vertices of T. Moreover, C Aγ + b ) A + i e i + b A i e i++)b + Aei+b)+b i + i v i+v +, so the ceter of mass C of simplex S is the average of the vertices of S. 2 Circumscribed Hypersphere A circumscribig hypersphere for the simplex is that hypersphere passig through all the vertices of the simplex. The ceter of this hypersphere, C, is equidistat from the vertices, say of distace r. The costraits are C v i r, i. Squarig the equatios, expadig the dot products, ad subtractig the equatio for i yields v i v ) C v ) 2 v i v 2 2 L2 i, i. This is a system of liear equatios i the compoets of C. Let M be the matrix defied earlier. Let b be the vector whose i th row is L 2 i /2. The equatio defiig the ceter is MC v ) b ad has solutio C v + M b. The radius of the circumscribed hypersphere is For 2 the simplex is a triagle. The area is The ceter x, y) ad radius r are r C v M b. A x x )y 2 y ) x 2 x )y y ) /2. x x + 4A +y 2 y )L 2 y y )L 2 2) y y + 4A x 2 x )L 2 + x x )L 2 2) r x x ) 2 + y y ) 2. For 3 the simplex is a tetrahedro. Defie X i x i x, Y i y i y, ad Z i z i z for i 3. The volume is X V Y Z det X 6 2 Y 2 Z 2. X 3 Y 3 Z 3 3

4 The ceter x, y, z) ad radius r are x x + 2V y y + 2V z z + 2V +Y2 Z 3 Y 3 Z 2 )L 2 Y Z 3 Y 3 Z )L Y Z 2 Y 2 Z )L 2 3) X2 Z 3 X 3 Z 2 )L 2 + X Z 3 X 3 Z )L 2 2 X Z 2 X 2 Z )L 2 3) +X2 Y 3 X 3 Y 2 )L 2 X Y 3 X 3 Y )L X Y 2 X 2 Y )L 2 3) r x x ) 2 + y y ) 2 + z z ) 2 3 Iscribed Hypersphere A iscribig hypersphere for the simplex is that hypersphere which is cotaied etirely withi the simplex ad is taget to all faces of the simple. The ceter of the hypersphere, C, is equidistat from the faces, say of distace r. The costraits are N i C v i ) r, i, where N i is the ier uit ormal to the hyperface determied by the vertices v i mod, v i+) mod,..., v i+ ) mod. This is a liear system of + equatios i the + ukows C, r). The system ca be writte as N i, ) C, r) N i v i. Defie the + ) + ) matrix M to be that matrix whose i th row is the vector N i, ). Defie the + ) vector b to be that vector whose i th row is the scalar N i v i. The liear system is the MC b ad has solutio C, r) M b. The radius of the iscribed hypersphere is r N C v ). For 2 the ormals are N y y),+x x)) L N y2 y),+x2 x)) L 2 N 2 y y2),+x x2)) L 2 The system of equatios is y y ) x x ) L x y 2 y ) x 2 x ) L 2 y y y 2 ) x x 2 ) L 2 r Applyig a symbolic iversio yields solutio x y x y x 2 y x y 2 x y 2 x 2 y. x xl2+xl2+x2l L +L 2+L 2 y yl2+yl2+y2l L +L 2+L 2 r xy xy+xy2 x2y+x2y xy2 L +L 2+L 2 4

5 For 3 the ormals are N L v v ) v 2 v ) N L v 3 v ) v v ) N 2 L 2 v 2 v ) v 3 v ) N 3 L 3 v 3 v ) v 2 v ) where the deomiators L k are the legths of the correspodig vectors i the umerators. The 4 4 system of equatios ca be solved symbolically to yield x xl3+xl2+x2l+x3l L +L +L 2+L 3 y yl3+yl2+y2l+y3l L +L +L 2+L 3 z zl3+zl2+z2l+z3l L +L +L 2+L 3 r γ L +L +L 2+L 3 where γ is the absolute value of the sum of the compoets of the geeralized cross product e e e 2 e 3 x x x 2 x 3 y y y 2 y 3 z z z 2 z 3 5