Arch-based Reciprocal Nomogram

1

2

3

4

5

6

7

8

9

10

11

12

24

22

20

18

16

14

12

10

8

6

4

2

24

22

20

18

16

14

12

10

8

6

4

2

where

Operation

A = Point(xAxis)
A' = Mirror(A, yAxis)
c = Circle(A, A')
d = Circle(A', A)
B, C = Intersect(d, c)
e = Circle(A, 0.3)
f = Circle(A', 0.3)
D, E = Intersect(f, xAxis)
F, G = Intersect(e, xAxis)
g = Circle(A, D)
h = Circle(A', G)
H, I = Intersect(h, g)
k = CircleArc(A, B, A')
p = CircleArc(A, H, D)
q = CircleArc(A', A, B)
r = CircleArc(A', G, H)
J = (0, 1)
K = (0, 2)
L = (0, 3)
M = (0, 4)
N = (0, 5)
O = (0, 6)
P = (0, 7)
Q = (0, 8)
R = (0, 9)
S = (0, 10)
T = (0, 11)
U = (0, 12)
a = Ray(A, J)
b = Ray(A, K)
i = Ray(A, L)
j = Ray(A, M)
l = Ray(A, N)
m = Ray(A, O)
n = Ray(A, P)
s = Ray(A, Q)
t = Ray(A, R)
a_1 = Ray(A, S)
b_1 = Ray(A, T)
c_1 = Ray(A, U)
V = Intersect(p, c_1, 1)
W = Intersect(p, b_1)
Z = Intersect(p, a_1)
A_1 = Intersect(p, t)
B_1 = Intersect(p, s)
C_1 = Intersect(p, n)
D_1 = Intersect(p, m)
E_1 = Intersect(p, l)
F_1 = Intersect(p, j)
G_1 = Intersect(p, i)
H_1 = Intersect(p, b)
I_1 = Intersect(p, a)
J_1 = Intersect(k, a)
K_1 = Intersect(k, b)
L_1 = Intersect(k, i)
M_1 = Intersect(k, j)
N_1 = Intersect(k, l)
O_1 = Intersect(k, m)
P_1 = Intersect(k, n)
Q_1 = Intersect(k, s)
R_1 = Intersect(k, t)
S_1 = Intersect(k, a_1)
T_1 = Intersect(k, b_1)
U_1 = Intersect(k, c_1)
d_1 = Segment(I_1, J_1)
e_1 = Segment(H_1, K_1)
f_1 = Segment(G_1, L_1)
g_1 = Segment(F_1, M_1)
h_1 = Segment(E_1, N_1)
i_1 = Segment(D_1, O_1)
j_1 = Segment(C_1, P_1)
k_1 = Segment(B_1, Q_1)
l_1 = Segment(A_1, R_1)
m_1 = Segment(Z, S_1)
n_1 = Segment(W, T_1)
p_1 = Segment(V, U_1)
q_1 = Ray(A', U)
r_1 = Ray(A', T)
s_1 = Ray(A', S)
t_1 = Ray(A', R)
a_2 = Ray(A', Q)
b_2 = Ray(A', P)
c_2 = Ray(A', O)
d_2 = Ray(A', N)
e_2 = Ray(A', M)
f_2 = Ray(A', L)
g_2 = Ray(A', K)
h_2 = Ray(A', J)
V_1 = Intersect(r, q_1, 1)
W_1 = Intersect(q, q_1)
Z_1 = Intersect(q, r_1)
A_2 = Intersect(r, r_1)
B_2 = Intersect(r, s_1)
C_2 = Intersect(q, s_1)
D_2 = Intersect(q, t_1)
E_2 = Intersect(r, t_1)
F_2 = Intersect(r, a_2)
G_2 = Intersect(q, a_2)
H_2 = Intersect(q, b_2)
I_2 = Intersect(r, b_2)
J_2 = Intersect(r, c_2)
K_2 = Intersect(q, c_2)
L_2 = Intersect(q, d_2)
M_2 = Intersect(r, d_2)
N_2 = Intersect(q, e_2, 1)
O_2 = Intersect(r, e_2)
P_2 = Intersect(r, f_2)
Q_2 = Intersect(q, f_2)
R_2 = Intersect(q, g_2)
S_2 = Intersect(r, g_2)
T_2 = Intersect(r, h_2)
U_2 = Intersect(q, h_2)
i_2 = Segment(W_1, V_1)
j_2 = Segment(Z_1, A_2)
k_2 = Segment(C_2, B_2)
l_2 = Segment(D_2, E_2)
m_2 = Segment(G_2, F_2)
n_2 = Segment(H_2, I_2)
p_2 = Segment(K_2, J_2)
q_2 = Segment(L_2, M_2)
r_2 = Segment(N_2, O_2)
s_2 = Segment(Q_2, P_2)
t_2 = Segment(R_2, S_2)
a_3 = Segment(U_2, T_2)
V_2 = Intersect(xAxis, yAxis)
b_3 = Segment(H, V_2)
c_3 = Segment(D, A)
d_3 = Segment(A, G)
W_2 = Point(p)
Z_2 = Point(r)
e_3 = Segment(A', Z_2)
f_3 = Segment(A, W_2)
A_3 = Intersect(e_3, f_3)
B_3 = Intersect(f_3, b_3)
C_3 = Intersect(e_3, b_3)
y1: y(B_3)
y2: y(C_3)
y1_2: 2 * y1
y2_2: 2 * y2
text1: "" + y2_2 + ""
text2: "" + y1_2 + ""
g_3 = OrthogonalLine(A_3, b_3)
D_3 = Intersect(g_3, b_3)
h_3 = Segment(A_3, D_3)
yRec: y(D_3)
text3: "" + yRec + ""
recyRec: 1 / yRec
text4: "$ \frac{1}{" + (LaTeX[y1_2]) + "} $ + $\frac{1}{" + (LaTeX[y2_2]) + "}$ = $\frac{1}{" + (LaTeX[yRec]) + "}$"
o = false
text5: "$ \frac{1}{" + (LaTeX[yRec]) + "} $ = $ " + (LaTeX[recyRec]) + "$"

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')