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2 1 x GMS x/r 3 y GMS y/r 3 z GMS z/r 3 G MS GMS: km 3 /s 2 r r 2 = x 2 + y 2 + z 2 t = 0 x = x 0, y = y 0, z = z 0, v = v 0

3 AU: /*km*/; rpr2: [ , , ]*AU /*2 rpr/au [x,y,z] * - */; vr2: [ , , ] /*2 vr [vx,vy,vz] * - */; df:1201/* 01 or 16 */; dt:60; ts:60*24;

4 endc:15.5*ts /*( )15.5*ts or ( )15.5*ts or 16.5*ts or 13.5*ts */; GM: /*km,s*/; GMS: /*km,s*/; x:rpr2[1] /* *AU*/; y:rpr2[2] /* *AU*/; z:rpr2[3] /* *AU*/; vx:vr2[1] ; vy:vr2[2] ; vz:vr2[3] ; rr:[[x,y,z]]; rpr:[[x,y,z]]; vr:[[vx,vy,vz]];

5 fs(x,y,z):=-gms*(x)/sqrt((x)^2+(y)^2+(z)^2)^3; gs(x,y,z):=-gms*(y)/sqrt((x)^2+(y)^2+(z)^2)^3; hs(x,y,z):=-gms*(z)/sqrt((x)^2+(y)^2+(z)^2)^3; for i:1 thru endc do( kx1: vx*dt,numer, ku1: fs(x, y, z)*dt,numer, ky1: vy*dt,numer, kv1: gs(x, y, z)*dt,numer, kz1: vz*dt,numer, kw1: hs(x, y, z)*dt,numer, kx2: (vx+ku1/2)*dt,numer, ku2: fs(x+kx1/2, y+ky1/2, z+kz1/2)*dt,numer, ky2: (vy+kv1/2)*dt,numer, kv2: gs(x+kx1/2, y+ky1/2, z+kz1/2)*dt,numer,

6 kz2: (vz+kw1/2)*dt,numer, kw2: hs(x+kx1/2, y+ky1/2, z+kz1/2)*dt,numer, kx3: (vx+ku2/2)*dt,numer, ku3: fs(x+kx2/2, y+ky2/2, z+kz2/2)*dt,numer, ky3: (vy+kv2/2)*dt,numer, kv3: gs(x+kx2/2, y+ky2/2, z+kz2/2)*dt,numer, kz3: (vz+kw2/2)*dt,numer, kw3: hs(x+kx2/2, y+ky2/2, z+kz2/2)*dt,numer, kx4 : (vx+ku3)*dt,numer, ku4 : fs(x+kx3, y+ky3, z+kz3)*dt,numer, ky4 : (vy+kv3)*dt,numer, kv4 : gs(x+kx3, y+ky3, z+kz3)*dt,numer, kz4 : (vz+kw3)*dt,numer, kw4 : hs(x+kx3, y+ky3, z+kz3)*dt,numer,

7 x : float(x + (kx1 + 2*kx2 + 2*kx3 + kx4)/6),numer, vx : float(vx + (ku1 + 2*ku2 + 2*ku3 + ku4)/6),numer, y : float(y + (ky1 + 2*ky2 + 2*ky3 + ky4)/6),numer, vy : float(vy + (kv1 + 2*kv2 + 2*kv3 + kv4)/6),numer, z : float(z + (kz1 + 2*kz2 + 2*kz3 + kz4)/6),numer, vz : float(vz + (kw1 + 2*kw2 + 2*kw3 + kw4)/6),numer, rr:endcons([x,y,z],rr), if mod(i,ts)=0 then rpr:endcons([x,y,z],rpr), if mod(i,ts)=0 then vr:[[vx,vy,vz]] ); draw3d(points(rr/au),color=red,points(rpr/au))$ df /* */; ts;

8 endc/ts; rpr/au /* [x,y,z] rpr2 */; vr /* [vx,vy,vz] vr2 */;

9 2 4 5

10 rpr:[[ , , ],[ , , ],[ , , ],[ , , ],[ , , ]];

11 rr1:rpr[1]; rr2:rpr[5]; load(vect); rr3:rr1~rr2; rr3:express(rr3); rr2:rr3~rr1; rr2:express(rr2); rr1:rr1/sqrt(rr1.rr1); rr2:rr2/sqrt(rr2.rr2); rr3:rr3/sqrt(rr3.rr3);

12 xpr1:[rr1.rpr[1],rr2.rpr[1]]; xpr2:[rr1.rpr[2],rr2.rpr[2]]; xpr3:[rr1.rpr[3],rr2.rpr[3]]; xpr4:[rr1.rpr[4],rr2.rpr[4]]; xpr5:[rr1.rpr[5],rr2.rpr[5]]; zpr:[rr3.rpr[1],rr3.rpr[2],rr3.rpr[3],rr3.rpr[4],rr3.rpr[5]]; eq:a*x^2+b*x*y+c*y^2+d*x+e*y=1; eq1:subst([x=xpr1[1],y=xpr1[2]],eq); eq2:subst([x=xpr2[1],y=xpr2[2]],eq); eq3:subst([x=xpr3[1],y=xpr3[2]],eq); eq4:subst([x=xpr4[1],y=xpr4[2]],eq); eq5:subst([x=xpr5[1],y=xpr5[2]],eq); ans:solve([eq1,eq2,eq3,eq4,eq5],[a,b,c,d,e]),numer; eq:subst(ans,eq);

13 maxima (%i9)rr1:rr1/sqrt(rr1.rr1); (%o9) [ , , ] (%i10)rr2:rr2/sqrt(rr2.rr2); (%o10) [ , , ] (%i11)rr3:rr3/sqrt(rr3.rr3); (%o11) [ , , ] (%i12)xpr1:[rr1.rpr[1],rr2.rpr[1]]; (%o12) [ , ] (%i13)xpr2:[rr1.rpr[2],rr2.rpr[2]];

14 (%o13) [ , ] (%i14)xpr3:[rr1.rpr[3],rr2.rpr[3]]; (%o14) [ , ] (%i15)xpr4:[rr1.rpr[4],rr2.rpr[4]]; (%o15) [ , ] (%i16)xpr5:[rr1.rpr[5],rr2.rpr[5]]; (%o16) [ , ] (%i17)zpr:[rr3.rpr[1],rr3.rpr[2],rr3.rpr[3],rr3.rpr[4],rr3.rpr[5]]; (%o17)[ , , , , 0.0]

15 (%i25)eq:subst(ans,eq); (%o25) y xy y x x = 1 rr1 X rr2 Y rr3 Z zpr 0 XY eq

16 geogebra maxima xpr1 5 5 geogebra A,B,C,D,E c=conic[a,b,c,d,e] Focus[c] MajorAxis[c] MinorAxis[c] 5

17

18 geogebra 3D Graphing 3D (1) O O=(0,0,0) (2) tr tr CREATE SLIDER 3, (3) Tr Tr=(2cos(tr),2sin(tr)) (4) OTr f=segment(o,tr) (5) Maxima %o25 eq

19 c: y^ x*y y x^ x=1 (6) (4) (5) Cr=Intersect(f,c,2) (7) x rr1 Xr=( , , ) (8) y rr2 Yr=( , , ) (9) Rr=x(Cr)Xr+y(Cr)Yr

20 3D

21

1 2

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