X == Id - Inverse @ M D .A.Transpose @ M D .M + 4 M - Inverse @ M D .A.M.M X Š Id + 4 M - Inverse @ M D .A.M.M - Inverse @ M D .A.Transpose @ M D .M

1

Remove @ "Global`*" D

M. H Id - X L .Inverse @ M D + M - A.M == A.Transpose @ M D - 3 M M - A.M + M. H Id - X L .Inverse @ M D Š -3 M + A.Transpose @ M D Id - M.X.Inverse @ M D + M - A.M == A.Transpose @ M D - 3 M Id + M - A.M - M.X.Inverse @ M D Š -3 M + A.Transpose @ M D Id - A.Transpose @ M D + 3 M - A.M + M == M.X.Inverse @ M D Id + 4 M - A.M - A.Transpose @ M D Š M.X.Inverse @ M D X == Inverse @ M D . H Id - A.Transpose @ M D + 4 M - A.M L .M X Š Inverse @ M D . H Id + 4 M - A.M - A.Transpose @ M DL .M

X == Id - Inverse @ M D .A.Transpose @ M D .M + 4 M - Inverse @ M D .A.M.M X Š Id + 4 M - Inverse @ M D .A.M.M - Inverse @ M D .A.Transpose @ M D .M

2

Remove @ "Global`*" D r0 = 8 1, 2, -1 < ; a = 8 2, 1, 1 < ; q = 8 3, 10, 14 < ; rq = q - r0 8 2, 8, 15 <

gq = Norm @ Cross @ a, rq DD • Norm @ a D

7 $%%%%%%% 7

€€€€€

2

N @ % D

13.0958

3

Remove @ "Global`*" D r0 = 8 1, 2, -1 < ; a = 8 2, 1, 1 < ; b = 8 1, - 1, -2 < ; q = 8 3, 10, 14 < ; rq = q - r0 8 2, 8, 15 <

Ÿ a

gq = -Det @8 a, b, rq <D • Norm @ Cross @ a, b DD

$%%%%%%% 7

€€€€€

5

N @ % D 1.18322

Ÿ b

v @l_, m_ D := r0 + l a + m b; v @l, m D

8 1 + 2 l + m, 2 + l - m, -1 + l - 2 m <

w @ t_ D := q + t Cross @ a, b D ; w @ t D 8 3 - t, 10 + 5 t, 14 - 3 t <

solv1 = Flatten @ Solve @ v @l, m D == w @ t D , 8l, m , t <DD

9l ® 18

€€€€€€€

5 , m ® - 27

€€€€€€€

5 , t ® 1

€€€€€

5 = t0 = t • . solv1

€€€€€ 1 5

w @ t0 D 9 14

€€€€€€€

5 , 11, 67

€€€€€€€

5 = N @ % D

8 2.8, 11., 13.4 <

p1 = 8 1, 1, 0 < ; p2 = 8-1, 2, 2 < ; p3 = 8-3, -2, 3 < ; p4 = 8 1, 1, 4 < ; m = 8 v1 = p2 - p1, v2 = p3 - p2, v3 = p4 - p3 <

88-2, 1, 2 < , 8- 2, -4, 1 < , 8 4, 3, 1 <<

m + 1 • 2 m

99-3, 3

€€€€€

2 , 3 = , 9-3, -6, 3

€€€€€

2 = , 9 6, 9

€€€€€

2 , 3

€€€€€

2 ==

s = Sum @ 1 • 2 ^ k, 8 k, 0, 99 <D

1267650600228229401496703205375

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

633825300114114700748351602688

s •• N 2.

s m

99- 1267650600228229401496703205375

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

316912650057057350374175801344 , 1267650600228229401496703205375

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

633825300114114700748351602688 , 1267650600228229401496703205375

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

316912650057057350374175801344 = , 9 - 1267650600228229401496703205375

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

316912650057057350374175801344 , - 1267650600228229401496703205375

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

158456325028528675187087900672 , 1267650600228229401496703205375

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

633825300114114700748351602688 = , 9 1267650600228229401496703205375

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€

158456325028528675187087900672 , 3802951800684688204490109616125

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

633825300114114700748351602688 , 1267650600228229401496703205375

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

633825300114114700748351602688 ==

s m •• N

88-4., 2., 4. < , 8-4., -8., 2. < , 8 8., 6., 2. <<

TotalPunkte =

H 3 * 100 + 1 L Punkte H* je 3 Anfangspunkte von l

bis l

plus Endpunkt *L 301 Punkte

p301 = p1 + Apply @ Plus, s m D •• N 8 1., 1., 8. <

5

Remove @ "Global`*" D

Solve @8 x + y + z Š 1, x - y + z Š 1, x - y - z Š 0 < , 8 x, y, z <D

99 x ® 1

€€€€€

2 , y ® 0, z ® 1

€€€€€

2 ==

6

Remove @ "Global`*" D

A = 88 1, 2, 3 < , 8 2, 3, 1 < , 8 0, 1, 1 << ; A •• MatrixForm i

k jjjjj j

1 2 3 2 3 1 0 1 1 y { zzzzz z

B = 88 1, 0, 3 < , 8 2, 3, 1 < , 8 0, 1, 1 << ; A •• MatrixForm i

k jjjjj j

1 2 3 2 3 1 0 1 1 y { zzzzz z

A.B •• MatrixForm i

k jjjjj j

5 9 8 8 10 10 2 4 2

y { zzzzz z

Inverse @ A D •• MatrixForm i

k jjjjj jjjjjj

€€€€

€€€€

- €€€€

- €€€€

€€€€

€€€€

€€€€

- €€€€

- €€€€

y

{ zzzzz zzzzzz

Inverse @ A D •• N •• MatrixForm i

k jjjjj j

0.5 0.25 -1.75

-0.5 0.25 1.25

0.5 -0.25 -0.25

y { zzzzz z

Inverse @ B D •• MatrixForm i

k jjjjj jjjjjj

€€€€

€€€€

- €€€€

- €€€€

€€€€

€€€€

€€€€

- €€€€

€€€€

y

{ zzzzz zzzzzz

Inverse @ B D •• N •• MatrixForm i

k jjjjj j

0.25 0.375 -1.125

-0.25 0.125 0.625

0.25 -0.125 0.375

y { zzzzz z

Transpose @ Inverse @ A.B DD •• MatrixForm i

k jjjjj jjjjjj

- €€€€

€€€€

€€€€

€€€€€€

- €€€€€€

- €€€€€€

€€€€€€

€€€€€€

- €€€€€€

y

{

zzzzz

zzzzzz

Transpose @ Inverse @ B D .Inverse @ A DD == Transpose @ Inverse @ A.B DD True

7

M = 88 1, -1 < , 8 2, 1 << ; p0 = 8 4, 7 < ; M •• MatrixForm

J 1 -1

2 1 N p1 = M.p0 8-3, 15 <

a = 32 Degree; d32 = 88 Cos @aD , -Sin @aD< , 8 Sin @aD , Cos @aD<< ; d32 •• N •• MatrixForm J 0.848048 -0.529919

0.529919 0.848048 N p2 = d32.p1

8 -3 Cos @ 32 ° D - 15 Sin @ 32 ° D , 15 Cos @ 32 ° D - 3 Sin @ 32 ° D<

p2 •• N

8-10.4929, 11.131 <

p3 = M.p2

8-18 Cos @ 32 ° D - 12 Sin @ 32 ° D , 15 Cos @ 32 ° D + 2 H- 3 Cos @ 32 ° D - 15 Sin @ 32 ° DL - 3 Sin @ 32 ° D<

p3 •• N

8-21.6239, -9.8549 <

8

Remove @ "Global`*" D

p1 = 8 1, 1, 0 < ; p2 = 8 1, 0, 2 < ; p3 = 8 0, 2, 3 < ; S = 1 • 3 H p1 + p2 + p3 L

9 2

€€€€€

3 , 1, 5

€€€€€

3 = N @ % D

8 0.666667, 1., 1.66667 <

Vp1p2p3 = 1 • 2 Norm @ Cross @ p2 - p1, p3 - p1 DD

$%%%%%%%%% 15

€€€€€€€

2

N @ % D 2.73861

p @ t_ D := t S;

norm @ v_ D := Sqrt @ v.v D

Vof @ t_ D := 1 • 2 H norm @ Cross @ p2 - p1, p @ t D - p1 DD + norm @ Cross @ p3 - p2, p @ t D - p2 DD + norm @ Cross @ p1 - p3, p @ t D - p3 DDL ; Vof @ t D •• ExpandAll

€€€€€ 1

2 $%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9 - 64 t

€€€€€€€€€€€€

3 + 47 t

€€€€€€€€€€€€€€

3 + 1

€€€€€

2 $%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 29 - 42 t + 49 t

€€€€€€€€€€€€€€

3 + 1

€€€€€

2 $%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 22 - 110 t

€€€€€€€€€€€€€€€

3 + 18 t

solv = Solve @ 2 Vp1p2p3 Š Vof @ t D , 8 t <D •• N •• Flatten 8 t ® 0.1902, t ® 1.80319 <

Plot @ Vof @ t D , 8 t, -1, 3 <D

-1 1 2 3

6 8 10 12

… Graphics …

p @ t D • . solv @@ 2 DD

8 1.20213, 1.80319, 3.00531 <

9

Remove @ "Global`*" D

r1 = 8 1, 2, -1 < ; r2 = 8 -1, 1, 3 < ; a = 8 2, 1, 1 < ; b = 8 3, -1, 2 < ;

v1 @ t_ D := r1 + t a; v2 @ r_ D := r2 + r b;

25

Determinante nicht 0, nicht windschief

Ÿ b (Sprachproblem: was ist der negative oder der positive Fall?)

d = V • Norm @ Cross @ a, b DD

5 $%%%%%%% 5

€€€€€

7

N @ % D 4.22577

10

Remove @ "Global`*" D

Ÿ a, b

r = 2; k @ x_, y_, z_ D := x ^ 2 + y ^ 2 + z ^ 2 - r ^ 2 f @ x_, y_, z_, a_ D := x + y + z - a

f @8 x_, y_, z_ < , a_ D := f @ x, y, z, a D k @ x, x, x D Š 0

-4 + 3 x

Š 0

solv = Solve @ k @ x, x, x D Š 0, 8 x <D •• Flatten

9 x ® - 2

€€€€€€€€€€ •!!!! 3

, x ® 2

€€€€€€€€€€ •!!!! 3 =

TPunkt = 9 2

€€€€€€€€€€€ •!!!!

3 , 2

€€€€€€€€€€€ •!!!!

3 , 2

€€€€€€€€€€€ •!!!!

3

=

9 2

€€€€€€€€€€ •!!!! 3 , 2

€€€€€€€€€€ •!!!! 3 , 2

€€€€€€€€€€ •!!!! 3 =

N @ % D

8 1.1547, 1.1547, 1.1547 <

f @ TPunkt, a D Š 0 2 •!!!! 3 - a Š 0

Solve @f@ TPunkt, a D Š 0, 8 a <D 99 a ® 2 •!!!! 3 ==

Ÿ c

e1 = 8 1, 0, 0 < ; e2 = 8 0, 1, 0 < ; e3 = 8 0, 0, 1 < ; T = TPunkt;

a = ArcCos @ T.e1 • Norm @ T D • Norm @ e1 DD

ArcCos A 1

€€€€€€€€€€ •!!!! 3 E

b = ArcCos @ T.e2 • Norm @ T D • Norm @ e2 DD ArcCos A 1

€€€€€€€€€€ •!!!! 3 E

g = ArcCos @ T.e3 • Norm @ T D • Norm @ e3 DD ArcCos A 1

€€€€€€€€€€ •!!!! 3 E

N @ % D 0.955317

% • Degree

54.7356