Parallel to Cartesian Axes Transformation

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Guide Line

Point 2

Point 1

Point 3

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y

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Data

$ y=x $

A = Point(yAxis)
B = Intersect(xAxis, yAxis)
C = Point(yAxis)
D = Point(yAxis)
E = Point(yAxis)
F = Point(yAxis)
G = Point(yAxis)
H = Point(yAxis)
I = Point(yAxis)
J = Point(yAxis)
K = Point(yAxis)
a = Segment(A, B)
L = Point(xAxis)
u = Vector(B, L)
A' = Translate(A, u)
B' = Translate(B, u)
a' = Segment(A', B')
C' = Translate(C, u)
D' = Translate(D, u)
E' = Translate(E, u)
F' = Translate(F, u)
G' = Translate(G, u)
H' = Translate(H, u)
I' = Translate(I, u)
J' = Translate(J, u)
K' = Translate(K, u)
M = Midpoint(B, L)
b = Line(M, a)
c = Segment(G, G')
Q = Intersect(b, c)
N = Midpoint(A, A')
d = Segment(N, M)
R = Point(a)
e = Line(R, Q)
S = Intersect(e, a')
f = Segment(R, S)
h = Line(L, A')
l = Line(L, A')
p = Ray(B, L)
Z = Point(p)
v = Vector(B, Z)
A'_1 = Translate(A, v)
B'_1 = Translate(B, v)
a'_1 = Segment(A'_1, B'_1)
C'_1 = Translate(C, v)
D'_1 = Translate(D, v)
E'_1 = Translate(E, v)
F'_1 = Translate(F, v)
G'_1 = Translate(G, v)
H'_1 = Translate(H, v)
I'_1 = Translate(I, v)
J'_1 = Translate(J, v)
K'_1 = Translate(K, v)
A_1 = Rotate(A'_1, 315*°, Z)
β = Angle(A_1, Z, A'_1)
q = Line(Z, A_1)
B_1 = Mirror(A'_1, q)
C_1 = Mirror(C'_1, q)
D_1 = Mirror(D'_1, q)
E_1 = Mirror(E'_1, q)
F_1 = Mirror(F'_1, q)
G_1 = Mirror(G'_1, q)
H_1 = Mirror(H'_1, q)
I_1 = Mirror(I'_1, q)
J_1 = Mirror(J'_1, q)
K_1 = Mirror(K'_1, q)
r = Segment(Z, B_1)
s = OrthogonalLine(B_1, r)
t = OrthogonalLine(A'_1, a'_1)
a_1 = OrthogonalLine(K'_1, a'_1)
b_1 = OrthogonalLine(J'_1, a'_1)
c_1 = OrthogonalLine(I'_1, a'_1)
d_1 = OrthogonalLine(H'_1, a'_1)
e_1 = OrthogonalLine(G'_1, a'_1)
f_1 = OrthogonalLine(F'_1, a'_1)
g_1 = OrthogonalLine(E'_1, a'_1)
h_1 = OrthogonalLine(D'_1, a'_1)
i_1 = OrthogonalLine(C'_1, a'_1)
j_1 = OrthogonalLine(C_1, r)
k_1 = OrthogonalLine(D_1, r)
l_1 = OrthogonalLine(E_1, r)
m_1 = OrthogonalLine(F_1, r)
n_1 = OrthogonalLine(G_1, r)
p_1 = OrthogonalLine(H_1, r)
q_1 = OrthogonalLine(I_1, r)
r_1 = OrthogonalLine(J_1, r)
s_1 = OrthogonalLine(K_1, r)
L_1 = Intersect(t, j_1)
M_1 = Intersect(t, k_1)
N_1 = Intersect(t, l_1)
O_1 = Intersect(t, m_1)
P_1 = Intersect(t, n_1)
Q_1 = Intersect(t, p_1)
R_1 = Intersect(t, q_1)
S_1 = Intersect(t, r_1)
T_1 = Intersect(t, s_1)
U_1 = Intersect(s, t)
V_1 = Intersect(s, a_1)
W_1 = Intersect(s, b_1)
Z_1 = Intersect(s, c_1)
A_2 = Intersect(s, d_1)
B_2 = Intersect(s, e_1)
C_2 = Intersect(s, f_1)
D_2 = Intersect(s, g_1)
E_2 = Intersect(s, h_1)
F_2 = Intersect(s, i_1)
t_1 = Segment(A'_1, U_1)
a_2 = Segment(K'_1, V_1)
b_2 = Segment(J'_1, W_1)
c_2 = Segment(Z_1, I'_1)
d_2 = Segment(H'_1, A_2)
e_2 = Segment(G'_1, B_2)
f_2 = Segment(F'_1, C_2)
g_2 = Segment(E'_1, D_2)
h_2 = Segment(D'_1, E_2)
i_2 = Segment(C'_1, F_2)
j_2 = Segment(B_1, U_1)
k_2 = Segment(T_1, K_1)
l_2 = Segment(J_1, S_1)
m_2 = Segment(R_1, I_1)
n_2 = Segment(H_1, Q_1)
p_2 = Segment(P_1, G_1)
q_2 = Segment(F_1, O_1)
r_2 = Segment(N_1, E_1)
s_2 = Segment(D_1, M_1)
t_2 = Segment(L_1, C_1)
rL: y(R) - y(B)
rR: y(A') - y(S)
rGrad: rR / rL
G_2 = Mirror(A'_1, r)
H_2 = Mirror(U_1, r)
a_3 = Segment(G_2, H_2)
I_2 = Mirror(K'_1, r)
J_2 = Mirror(V_1, r)
b_3 = Segment(I_2, J_2)
K_2 = Mirror(J'_1, r)
L_2 = Mirror(W_1, r)
c_3 = Segment(K_2, L_2)
M_2 = Mirror(Z_1, r)
N_2 = Mirror(I'_1, r)
d_3 = Segment(M_2, N_2)
O_2 = Mirror(H'_1, r)
P_2 = Mirror(A_2, r)
e_3 = Segment(O_2, P_2)
Q_2 = Mirror(G'_1, r)
R_2 = Mirror(B_2, r)
f_3 = Segment(Q_2, R_2)
S_2 = Mirror(F'_1, r)
T_2 = Mirror(C_2, r)
g_3 = Segment(S_2, T_2)
U_2 = Mirror(E'_1, r)
V_2 = Mirror(D_2, r)
h_3 = Segment(U_2, V_2)
W_2 = Mirror(D'_1, r)
Z_2 = Mirror(E_2, r)
i_3 = Segment(W_2, Z_2)
A_3 = Mirror(C'_1, r)
B_3 = Mirror(F_2, r)
j_3 = Segment(A_3, B_3)
C_3 = Mirror(A'_1, r)
D_3 = Mirror(B'_1, r)
k_3 = Segment(C_3, D_3)
E_3 = Mirror(L_1, r)
F_3 = Mirror(C_1, r)
l_3 = Segment(E_3, F_3)
G_3 = Mirror(D_1, r)
H_3 = Mirror(M_1, r)
m_3 = Segment(G_3, H_3)
I_3 = Mirror(N_1, r)
J_3 = Mirror(E_1, r)
n_3 = Segment(I_3, J_3)
K_3 = Mirror(F_1, r)
L_3 = Mirror(O_1, r)
p_3 = Segment(K_3, L_3)
M_3 = Mirror(P_1, r)
N_3 = Mirror(G_1, r)
q_3 = Segment(M_3, N_3)
O_3 = Mirror(H_1, r)
P_3 = Mirror(Q_1, r)
r_3 = Segment(O_3, P_3)
Q_3 = Mirror(R_1, r)
R_3 = Mirror(I_1, r)
s_3 = Segment(Q_3, R_3)
S_3 = Mirror(J_1, r)
T_3 = Mirror(S_1, r)
t_3 = Segment(S_3, T_3)
U_3 = Mirror(T_1, r)
V_3 = Mirror(K_1, r)
a_4 = Segment(U_3, V_3)
W_3 = Mirror(B_1, r)
Z_3 = Mirror(U_1, r)
b_4 = Segment(W_3, Z_3)
xTrans: x(Z)
c_4: y = rGrad * (x - xTrans)
A_4 = Intersect(t_1, c_4)
d_4 = Segment(Z, A_4)
B_4 = Intersect(t_1, j_2)
C_4 = Intersect(b_4, a_3)
f_4 = Segment(B_4, C_4)
g = Segment(B, A')
O = Point(g)
P = Point(a)
i = Line(P, O)
pL: y(P) - y(B)
j = Line(L, A')
k = Line(P, O)
T = Intersect(k, j)
m = Segment(T, L)
n = Segment(P, T)
pR: y(A') - y(T)
pGrad: pR / pL
e_4: y = pGrad * (x - xTrans)
U = Intersect(e_4, f_4)
V = Intersect(e_4, t_1)
g_4 = Segment(Z, V)
h_4 = Segment(Z, U)
o = false
i_4 = Line(L, A')
W = Point(i_4)
j_4 = Segment(L, W)
k_4 = Segment(B, W)
D_4 = Point(k_4)
E_4 = Point(a)
l_4 = Line(E_4, D_4)
m_4 = Line(L, A')
F_4 = Intersect(l_4, m_4)
n_4 = Segment(E_4, F_4)
p_4 = Segment(L, F_4)
gCut: y(A') - y(W)
dy = Distance(D_4, W)
dx = Distance(B, D_4)
gGrad: dy / dx
q_4: y = gGrad * (x - xTrans) + gCut
r_4 = Segment(A'_1, C_3)
G_4 = Intersect(r_4, q_4)
H_4 = Intersect(t_1, q_4)
I_4 = Intersect(f_4, q_4)
s_4 = Segment(G_4, I_4)
t_4 = Segment(G_4, H_4)
w = false
z = false
o_1 = false
xp: y(P) - y(B)
yp: y(A') - y(T)
xg: y(E_4) - y(B)
yg: y(A') - y(F_4)
xr: y(R) - y(B)
yr: y(A') - y(S)
text48: "\begin{array}{} x=" + (LaTeX[xr]) + " \\ y=" + (LaTeX[yr]) + " \end{array} "
text49: "\begin{array}{} x= " + (LaTeX[xp]) + " \\ y= " + (LaTeX[yp]) + " \end{array}"
text50: "\begin{array}{} x= " + (LaTeX[xg]) + " \\ y= " + (LaTeX[yg]) + " \end{array}"
rRatio: yr / xr
pRatio: yp / xp
gRatio: yg / xg
text51: "y:x=" + (LaTeX[rRatio]) + ":1"
text52: "y:x =" + (LaTeX[pRatio]) + " :1"
text53: "y:x= " + (LaTeX[gRatio]) + ":1"
u_1 = false
text55: "y=" + (LaTeX[pGrad]) + " x "
cut = If(gCut ≥ 0, "+", "-")
aCut: abs(gCut)
text56: "y=" + (LaTeX[gGrad]) + "x" + (LaTeX[cut]) + "" + (LaTeX[aCut]) + ""
xP_{H2}: y(P)
P2: (xP_{H2} + xTrans, 10 - y(T))
xP_{H3}: y(E_4)
P3: (xP_{H3} + xTrans, 10 - y(F_4))
xP_{H1}: y(R)
P1: (xP_{H1} + xTrans, 10 - y(S))

Tool: Complex Multiplication (command ComplexMultiplication)
Select Points at O(0, 0), P(0, -1), A and B for A*B
Input: O, P, A, B
Output: j, AB

c = Circle(O, A)
d = Circle(O, B)
D = Intersect(c, xAxis, 2)
b = Segment(O, A)
f = Segment(O, B)
G = Intersect(d, xAxis, 1)
g = Circle(D, P, G)
E = Intersect(g, yAxis, 2)
e = Circle(O, E)
a, h = AngularBisector(f, b)
D' = Mirror(D, a)
i = Ray(O, D')
AB = Intersect(e, i, 1)
j = Segment(O, AB)

Tool: Complex Division (command ComplexDivision)
Select Points O(0, 0), P(0, -1), A and B for A/B
Input: O, P, A, B
Output: l, C

A/B

c = Circle(O, A)
d = Circle(O, B)
D = Intersect(c, xAxis, 2)
e = AngularBisector(A, O, D)
B' = Mirror(B, e)
f = Ray(O, B')
E = Intersect(c, yAxis, 1)
F = Intersect(d, xAxis, 1)
g = Segment(F, E)
G = Midpoint(g)
h = OrthogonalLine(G, g)
i = Segment(P, F)
H = Midpoint(i)
j = OrthogonalLine(H, i)
I = Intersect(h, j)
k = Circle(I, F)
J = Intersect(k, xAxis, 2)
p = Circle(O, J)
C = Intersect(p, f)
l = Segment(O, C)

Tool: Complex Addition (command ComplexAddition)
Select Points O(0, 0), A and B for A + B
Input: O, A, B
Output: C', c

A + B

u = Vector(O, B)
C' = Translate(A, u)
c = Segment(O, C')

Tool: Complex Subtraction (command ComplexSubtraction)
Select Points O(0, 0), A and B for A - B
Input: O, A, B
Output: c, A'

A - B

u = Vector(B, O)
A'_1 = Translate(A, u)
A' = Translate(A, u)
c = Segment(O, A')