The Tetrahedron Quality Factors of CSDS

Transcription

1 MAX PLANCK INSTITUT FÜR AERONOMIE D Katlenburg-Lindau, Federal Republi of Germany MPAE W The Tetrahedron Quality Fators of CSDS PATRICK W. DALY 1994 June 7 This report available from Abstrat The four Cluster spaeraft will form a tetrahedron, whih ideally should be a regular one: equal spaing between all pairs of verties. In reality, this will not be the ase. A number of parameters exist to speify how badly off the true figure is. This paper presents some mathematis of tetrahedrons, desribes the two parameters that are to be used in the Cluster Siene Data System, and gives a Fortran program for their alulation. Introdution Four points in spae define a tetrahedron. If the separations between eah pair of points are equal, then it is a regular tetrahedron. The four Cluster spaeraft will form a tetrahedron, whih in general is not regular. How an we speify the degree to whih regularity is ahieved? The Glassmeier Parameter The parameter proposed by vom Stein, Glassmeier, and Dunlop (1992) is defined as Q G = True Vol. True Surf. + Ideal Vol. Ideal Surf. + 1 (1) and takes on values between 1 and 3. It tends to desribe the dimensionality of the figure, as listed in Table 1. The ideal volume and surfae are alulated for a regular tetrahedron with a side length equal to the average of the 6 distanes between the 4 points. Table 1. Speial values of the Glassmeier parameter Q G Meaning 1.0 The four points are olinear 2.0 The points all lie in a plane 3.0 A regular tetrahedron is formed The Robert/Roux Parameter In their paper on tetrahedron shape, Robert and Roux (1993) present 17 different parameters, as ratios of various volumes, sizes, areas. Of these, the CSDS ommunity has deided to adopt one as its seond quality parameter for the auxiliary data. It is defined as ( ) 1 True Vol. 3 Q R = N (2) Sphere Vol. where the sphere is that irumsribing the tetrahedron (all four points on its surfae) and N is a 1

2 2 P. W. DALY normalization fator to make Q R = 1 for a regular tetrahedron. The range of values is between 0 and 1. Mathematis of a Tetrahedron Consider four points in spae and the figure formed by joining them with lines (Figure 1). The points are numbered 0 to 3, and their vetors are r 0, r 1, r 2, r 3. Without any loss of generality, we may onsider only the differenes S d 2 d 1 Figure 2. Area S of a triangle. in desribing the points. Area of a Side d i = r i r 0 The area of a parallelogram bounded by two vetors d 1 and d 2 is given by the magnitude of their ross produt; any triangle is half of a parallelogram, so its area is S = 1 2 d 1 d 2 where d 1 and d 2 are the vetors for any two sides of the triangle (Figure 2). For the four sides of the tetrahedron, speify side n to be that one that does not ontain point n at any of its verties. Thus: S 1 = 1 2 d 2 d 3 (3) S 2 = 1 2 d 1 d 3 (4) Figure 1. A tetrahedron and its four verties. S 3 = 1 2 d 1 d 2 (5) S 0 = 1 2 (d 2 d 1 ) (d 3 d 1 ) = 1 2 d 1 d 2 + d 2 d 3 + d 3 d 1 (6) The total surfae S is the sum 3 n=0 S n. Volume of a Tetrahedron The volume of a figure bounded by three vetors in spae is the triple produt of those vetors. Any tetrahedron is 1/6 of suh a figure, hene V = 1 6 d 1 d 2 d 3 (7) = 1 d 1x d 1y d 1z 6 d 2x d 2y d 2z (8) d 3x d 3y d 3z Center of Cirumsribed Sphere To find the irumsribed sphere, we need the point that is equidistant from all four verties, i.e. we want r suh that (r r n ) (r r n ) = ρ 2 ; n = 0, 3 r 2 2r r n + r 2 n = ρ 2 If we take point 0 as the origin, that is, if we use the d n vetors in plae of the r n, then r 2 = ρ 2, the sphere radius, and the above 4 equations redue to 2r d n = d 2 n ; n = 1, 3

3 THE CSDS TETRAHEDRON FACTORS 3 Table 2. Values for regular tetrahedron Quantity Value S 0 = 3/4 S = 3 V = 2/12 ρ = 6/4 V = 4 3 π ( 3 8 This yields the matrix equation for the enter of the sphere d 1x d 1y d 1z x d d 2x d 2y d 2z y = d 2 2 (9) d 3x d 3y d 3z z whih an be solved for the vetor (x, y, z) and the radius of the sphere ρ 2 = x 2 +y 2 +z 2. Note that the leftmost matrix in equation 9 is the same as the one whose determinant yields the volume of the tetrahedron (equation 8). The volume of the irumsribed sphere is then ) 3 2 d 2 3 V = 4 3 πρ2 (10) The Regular Tetrahedron The regular tetrahedron of unit side is the ideal against whih the true figure of the four spaeraft is to be measured. We may take d 0 = (0, 0, 0) d 1 = (1, 0, 0) ( ) 1 3 d 2 = 2, 2, 0 ( ) d 3 = 2, 6, 3 Values for the regular tetrahedron of unit side length are listed in Table 2. Calulating the Quality Fators The quality fators in equations 1 and 2 an now be found with the help of these formulas. For Q G, we average the 6 distanes between the 4 points to get the side L of the ideal regular tetrahedron, with volume L 3 2/12 and surfae L 2 3. The true volume and surfae are found from equations 7 and 3 6. For Q R, the radius of the irumsribing sphere is alulated from equation 9. The atual volume of the sphere need not be alulated, for all the fators just go into the normalizing N. Q R = ( 9 ) V ρ 1 A Fortran program at the end of this paper alulates both these parameters. Referenes Robert, P. and Roux, A. (1993). Influene of the Shape of the Tetrahedron on the Auray of the Estimate of the Current Density. Proeedings of ESA START Conferene, Aussois, Frane, Centre de Reherhes en Physique de l Environment, Issy les Moulineaux, Frane. vom Stein, R., Glassmeier, K.-H., and Dunlop, M. (1992). A Configuration Parameter for the Cluster Satellites. Teh. Rep. 2/1992, Institut für Geophysik und Meteologie der Tehnishen Universität Braunshweig.

4 4 P. W. DALY A Fortran Subroutine to Calulate the Parameters subroutine TETRAQ(r,qg,qr) To alulate the Glassmeier (QG) and Robert/Roux (QR) quality fators for a tetrahedron. Appliation: CLUSTER SCIENCE DATA SYSTEM (These are to be two auxiliary parameters) Input: R(3,4) = positions of the 4 points Output: QG = Glassmeier fator QR = Robert/Roux fator (their fator number 10) Inputs and outputs are single preision, internal alulations are double preision True volume True surfae QG = Ideal vol Ideal surf where the ideals are volume and surfae of a regular tetrahedron of side length equal to the mean of the 6 sides. True volume (1/3) QR = Fator * Sphere vol where the sphere is that irumsribing the tetrahedron (all 4 points on the surfae), and the fator is suh that QR=1 for a regular tetrahedron. ********************************************************************* Patrik W. Daly daly@linmpi.mpg.de Max-Plank-Institut fuer Aeronomie D Katlenburg-Lindau Germany 1994 June 7 ********************************************************************* impliit none real r(3,4),qg,qr double preision d(3,3),(3,3),s1,s2,s3,s0,s,l1,l2,l3,vol double preision v(3),w,smean,vmean,lmean,r double preision dot integer n,m,k Find the differenes w=dble(r(n,1)) do m=1,3

5 THE CSDS TETRAHEDRON FACTORS 5 d(n,m)=dble(r(n,m+1))-w Find the average side length of all 6 sides l1=dsqrt(dot(d(1,1),d(1,1))) l2=dsqrt(dot(d(1,2),d(1,2))) l3=dsqrt(dot(d(1,3),d(1,3))) w= l1 + l2 + l3 v(n)=d(n,2)-d(n,1) w=w + dsqrt(dot(v,v)) v(n)=d(n,3)-d(n,1) w=w + dsqrt(dot(v,v)) v(n)=d(n,3)-d(n,2) w=w + dsqrt(dot(v,v)) lmean=w/6.d0 Find the ross produts m=mod(n,3) + 1 k=mod(m,3) + 1 all ross(d(1,m),d(1,k),(1,n)) Find the volume of the tetrahedron vol=dabs(dot(d(1,1),(1,1)))/6.d0 Find the area of the 4 surfaes and their sum s1=0.5d0*dsqrt(dot((1,1),(1,1))) s2=0.5d0*dsqrt(dot((1,2),(1,2))) s3=0.5d0*dsqrt(dot((1,3),(1,3))) v(n)=(n,1) + (n,2) + (n,3) s0=0.5d0*dsqrt(dot(v,v)) s=s0 + s1 + s2 + s3 Find the volume and total area for reg tetrahedron with mean side vmean=dsqrt(2.d0)*lmean*lmean*lmean/1.2d1 smean=dsqrt(3.d0)*lmean*lmean

6 6 P. W. DALY Calulate the Glassmeier fator w= vol/vmean + s/smean + 1.d0 qg=sngl(w) Find the enter of the irumsribed irle w=1.d0/(1.2d1*vol) v(1)=w*((1,1)*l1*l1 + (1,2)*l2*l2 + (1,3)*l3*l3) v(2)=w*((2,1)*l1*l1 + (2,2)*l2*l2 + (2,3)*l3*l3) v(3)=w*((3,1)*l1*l1 + (3,2)*l2*l2 + (3,3)*l3*l3) r=dsqrt(dot(v,v)) Calulate the Robert/Roux fator w=9.d0*dsqrt(3.d0)/8.d0 w=(w*vol)**(1.d0/3.d0)/r qr=sngl(w) return end funtion DOT(v,w) To return the dot produt of vetors V and W impliit none double preision dot,v(3),w(3) dot=v(1)*w(1) + v(2)*w(2) + v(3)*w(3) return end subroutine CROSS(v,w,x) To return X as the ross produt of vetors V and W impliit none double preision x(3),w(3),v(3) x(1)=v(2)*w(3) - v(3)*w(2) x(2)=v(3)*w(1) - v(1)*w(3) x(3)=v(1)*w(2) - v(2)*w(1) return end