x_0'/2

A'

y_0

y_0

x_0

ty

x'-Achse

y'-Achse

z'-Achse

R_z:

R_y:

R_x:

x_0'

y_0'

z_0'

X_0

Y_0

$X→Achse$

Y↑Achse

r_0

 \scalebox{0.9}\text{Raumillusion: Abbildung drehender Raumpunkte in die 2-dimensionale Ebene}

x'-Achse

y'-Achse

z'-Achse

\text {3D  Illusions-Darstellung}

abzulesende 2D Koordinaten : X_0 und Y_0

x-Achse

z-Achse

y-Achse

z_0

y_0

x_0

entsprechende 2D Koordinaten werden definiert:

Koordinatensysteme:

$\scalebox{1.4} \text{\copyright Roman Chijner, Heidelberg 2012}$
$\scalebox{1.3} \text{    Gestaltungshilfe:}\scalebox{1.3}\text{ Werner Knobloch}$

aij-Koeffizienten der Drehmatrix R_{zyx} im R³:

P-Koeffizienten

Änderungen der Winkellage:

3.  X und Y-Achse als Koordinatensystem  der Projektionsebene       
mit X_0 und Y_0 als entsprechende Punktkoordinaten   

$ \vec{r'_{0}}=R_{zyx}*\vec{r_{0}} $

$ \text{1. \fgcolor{0000ff}{ x}, \fgcolor{009900}{ y} und \fgcolor{ff0000}{ z} als drehbares Koordinatensystem$
$\vec{r_0} =( \fgcolor{0000ff}{\; x_{0},\fgcolor{009900}{ \; y_0} \;,  \fgcolor{ff0000}{  z_0}) -Ortsvektor \; des\; P\; $   

$ 2. \;    \fgcolor{009999}{x'}, \fgcolor{999900}{ y'}, \fgcolor{ff00ff}{ z'} \; als \; ruhendes \;  Koordinatensystem ,$
$  \fgcolor{999999}{festgelegt\; durch\; folgende\; Punkte (Polarkoordinaten)   }$   
$ \scalebox{0.8} \text{\fgcolor{009999}{Ax'=(0.5;180°+45°)}   \fgcolor{999900}{ Ay' = (1; 0°)}    \fgcolor{ff00ff}{ Az' = (1;90°)}}$   
$ \vec{r'_0}=(\fgcolor{009999}{x_0'},\fgcolor{999900}{ y_0'},\fgcolor{ff00ff}{ z_0'})-Ortsvektor\; des\;Punktes  \;  P'\;$   

Drehungen um feste Raumpunkte können als Koordinatentransformationen interpretiert werden.
Es sind spezielle lineare  Abbildungen, die durch 'orthogonale' Drehmatrizen beschrieben werden.

Resultate:

x,y,z-drehbares / x',y',z'-ruhendes System

$ \vec{r'_0} $

$ \vec{r_0} $

Koordinatensystem:

P(x_0,y_0,z_0) :

P'(x_0',y_0',z_0') :

P_2(X_0,Y_0) :

β = 6 (slider from 0 to 6.283185307179586, step 0.1)
α = 2.1 (slider from 0 to 6.283185307179586, step 0.1)
a11: cos(α) * cos(β)
γ = 0 (slider from 0 to 6.283185307179586, step 0.1)
a13: cos(α) * sin(β) * cos(γ) + sin(α) * sin(γ)
a12: cos(α) * sin(β) * sin(γ) - sin(α) * cos(γ)
x_0 = 2.8 (slider from -5 to 5, step 0.1)
y_0 = 5 (slider from -5 to 5, step 0.1)
z_0 = 3.2 (slider from -5 to 5, step 0.1)
x': a11 * x_0 + a12 * y_0 + a13 * z_0
Pr_{x'}: (x' / 2; 225°)
O = (0, 0)
l2z: {0, 0, 0, z_0, z_0}
l2y: {0, y_0, y_0, 0, y_0}
l2x: {x_0, x_0, 0, 0, x_0}
r_z = 1 (slider from -5 to 5, step 0.1)
r_y = 1 (slider from -5 to 5, step 0.1)
r_x = 1 (slider from -5 to 5, step 0.1)
const = 0.353553390593274
P11: (sin(α) * cos(β), -sin(β)) - cos(α) * cos(β) * const * (1, 1)
P13: (sin(α) * sin(β) * cos(γ) - cos(α) * sin(γ), cos(β) * cos(γ)) - (cos(α) * sin(β) * cos(γ) + sin(α) * sin(γ)) * const * (1, 1)
P12: (sin(α) * sin(β) * sin(γ) + cos(α) * cos(γ), cos(β) * sin(γ)) - (cos(α) * sin(β) * sin(γ) - sin(α) * cos(γ)) * const
l2 = Sequence(P11 * r_x * Element[l2x, i] + P12 * r_y * Element[l2y, i] + P13 * r_z * Element[l2z, i], i, 1, 5)
a21: sin(α) * cos(β)
a22: sin(α) * sin(β) * sin(γ) + cos(α) * cos(γ)
a23: sin(α) * sin(β) * cos(γ) - cos(α) * sin(γ)
a31: -sin(β)
a32: cos(β) * sin(γ)
a33: cos(β) * cos(γ)
y': a21 * x_0 + a22 * y_0 + a23 * z_0
z': a31 * x_0 + a32 * y_0 + a33 * z_0
P0: ((-x') / sqrt(8) + y', (-x') / sqrt(8) + z')
P_2: P11 * x_0 + P12 * y_0 + P13 * z_0
l1x: {1, 0, 0}
l1y: {0, 1, 0}
l1z: {0, 0, 1}
Str1 = Segment((0, 0), Element[l2, 1])
Str2 = Segment((0, 0), Element[l2, 2])
Vieleck1, a, b, c, d = Polygon(O, Element[l2, 1], Element[l2, 2], Element[l2, 3])
Str4 = Segment(Element[l2, 4], Element[l2, 5])
l1 = Sequence(P11 * Element[l1x, i] + P12 * Element[l1y, i] + P13 * Element[l1z, i], i, 1, 3)
Strahl1 = Ray((0, 0), Element[l1, 1])
Strahl2 = Ray((0, 0), Element[l1, 2])
Strahl3 = Ray((0, 0), Element[l1, 3])
b11: -a11 * const + a21
b13: -a13 * const + a23
b21: -a11 * const + a31
b22: -a12 * const + a32
b23: -a13 * const + a33
P11b: (b11, b21)
b12: -a12 * const + a22
P12b: (b12, b22)
P13b: (b13, b23)
A: (sin(α) * sin(β) * cos(γ) - cos(α) * sin(γ), cos(β) * cos(γ)) - (cos(α) * sin(β) * cos(γ) + sin(α)) * const * (1, 1)
Az' = (6.123233995736766E-17, 1)
Ay' = (1, 0)
Ax' = (-0.35355339059327384, -0.35355339059327373)
Strahlx2 = Ray(O, Ax')
w = Vector(O, Az')
wy = Vector(O, Ay')
P_{x'}: (x'; 225°)
P_{y'}: (y'; 0°)
P_{z'}: (z'; 90°)
P_X: ((-x') / sqrt(8) + y', 0)
P_Y: (0, (-x') / sqrt(8) + z')
j = Segment(P_2, P_Y)
k = Segment(P_X, P_2)
wx = Vector(O, Ax')
u = Vector(O, P_2)
Eck13 = Corner(1, 3)
By1: (0, y(Eck13) - 1)
Eck12 = Corner(1, 2)
Bx1: (x(Eck12) - 1, 0)
m = Line(Ax', O)
n = Line(O, Az')
l = Line(O, Ay')
Eck22 = Corner(2, 2)
Eck11 = Corner(1, 1)
Eck21 = Corner(2, 1)
Eck23 = Corner(2, 3)
Bx2: (x(Eck22) - 1, 0)
By2: (0, y(Eck23) - 1)
tz1 = Segment(Element[l2, 2], Element[l2, 5])
ty1 = Segment(O, Element[l2, 3])
tx1 = Segment(O, Element[l2, 1])
vx = Vector(O, Element[l1, 1])
vy = Vector(O, Element[l1, 2])
vz = Vector(O, Element[l1, 3])
e = true
g = true
Text3: "x_0'=a11 x_0+a12 y_0+a13 z_0= " + x' + ""
Text4: "y_0'=a21 x_0+a22 y_0+a23 z_0= " + y' + ""
Text5: "z_0'=a31 x_0+a32 y_0+a33 z_0= " + z' + ""
P1 = (0, 0)
P3 = 0
P31 = 0
h = true
a11': cos(α) * cos(β)
a21': cos(α) * sin(β) * sin(γ) - sin(α) * cos(γ)
a31': cos(α) * sin(β) * cos(γ) + sin(α) * sin(γ)
a12': sin(α) * cos(β)
a22': sin(α) * sin(β) * sin(γ) + cos(α) * cos(γ)
a32': sin(α) * sin(β) * cos(γ) - cos(α) * sin(γ)
a13': -sin(β)
a23': cos(β) * sin(γ)
a33': cos(β) * cos(γ)
R_{zyx}: {{cos(α) * cos(β), cos(α) * sin(β) * sin(γ) - sin(α) * cos(γ), cos(α) * sin(β) * cos(γ) + sin(α) * sin(γ)}, {sin(α) * cos(β), sin(α) * sin(β) * sin(γ) + cos(α) * cos(γ), sin(α) * sin(β) * cos(γ) - cos(α) * sin(γ)}, {-sin(β), cos(β) * sin(γ), cos(β) * cos(γ)}}
Text722 = LaTeX(R_{zyx}, false, true)
o = true
B = Intersect(j, k)
p = Segment(B, P_{z'})
r = Line(P_{y'}, Strahlx2)
q = Line(Pr_{x'}, xAxis)
C = Intersect(r, q)
s = Segment(C, B)
a_1 = Segment(C, P_{y'})
Vieleck2, b_1, c_1, d_1, e_1 = Polygon(Pr_{x'}, C, P_{y'}, P1)
x'_Y: (0, y(Pr_{x'}))
i = Segment(Pr_{x'}, x'_Y)
x'_X: (x(Pr_{x'}), 0)
f = Segment(Pr_{x'}, x'_X)
Text7: "X_0=-\frac{\fgcolor{009999}{x_0'}}{\sqrt{8}}+\fgcolor{999900}{y_0'}=-x'_X+y_0'=" + (LaTeX[x(P_2)]) + ""
text3: "Y_0=-\frac{\fgcolor{009999}{x_0'}}{\sqrt{8}}+\fgcolor{ff0000}{z_0'}=-x'_Y+z_0'=" + (LaTeX[y(P_2)]) + ""
Ec1 = Corner(1)
Ec2 = Corner(2)
Ec3 = Corner(3)
Ec4 = Corner(4)
v = true
Text8: "$P11=" + (LaTeX[LaTeX[P11, false]]) + " $
$P12=" + (LaTeX[LaTeX[P12, false]]) + " $
$P13=" + (LaTeX[LaTeX[P13, false]]) + " $"
Text18: "Koordinaten des Punktes P im -Koordinatensystem:
1.  drehendes System:    P(x_0,y_0,z_0) = (" + x_0 + "," + y_0 + "," + z_0 + ")
2.  ständerfestes System:   P(x_0',y_0',z_0') = (" + x' + "," + y' + "," + z' + ")
3.  2D-Ebenen-System:    P(X_0,Y_0) = (" + (x(P_2)) + "," + (y(P_2)) + ")
Zusammenhang zwischen den Illusionskoordinaten in  der 2D Ebene
mit den 3D Start-Koordinaten: nach der Drehungen 
 P'(X_0,Y_0) = P11 x_0 + P12 y_0 + P13 z_0"