複數的極式
Author: 藍邦偉  老師

$ O $

$ 直角座標轉換為極座標 $

$%v$

$ P $

(\frac{\sqrt6+\sqrt2}{4},\frac{\sqrt6-\sqrt2}{4})

(\frac{\sqrt6+\sqrt2}{2},\frac{\sqrt6-\sqrt2}{2})

(\frac{3\sqrt6+3\sqrt2}{4},\frac{3\sqrt6-3\sqrt2}{4})

(\sqrt6+\sqrt2,\sqrt6-\sqrt2)

(\frac{5\sqrt6+5\sqrt2}{4},\frac{5\sqrt6-5\sqrt2}{4})

(\frac{\sqrt6+\sqrt2}{4},\frac{\sqrt6-\sqrt2}{4})

(\frac{\sqrt6+\sqrt2}{2},\frac{\sqrt6-\sqrt2}{2})

(\frac{3\sqrt6+3\sqrt2}{4},\frac{3\sqrt6-3\sqrt2}{4})

(\sqrt6+\sqrt2,\sqrt6-\sqrt2)

(\frac{5\sqrt6+5\sqrt2}{4},\frac{5\sqrt6-5\sqrt2}{4})

(-\frac{\sqrt6-\sqrt2}{4},\frac{\sqrt6+\sqrt2}{4})

(-\frac{\sqrt6-\sqrt2}{2},\frac{\sqrt6+\sqrt2}{2})

(-\frac{3\sqrt6-3\sqrt2}{4},\frac{3\sqrt6+3\sqrt2}{4})

(-\sqrt6+\sqrt2,\sqrt6+\sqrt2)

(-\frac{5\sqrt6-5\sqrt2}{4},\frac{5\sqrt6+5\sqrt2}{4})

(-\frac{\sqrt6+\sqrt2}{4},\frac{\sqrt6-\sqrt2}{4})

(-\frac{\sqrt6+\sqrt2}{2},\frac{\sqrt6-\sqrt2}{2})

(-\frac{3\sqrt6+3\sqrt2}{4},\frac{3\sqrt6-3\sqrt2}{4})

(-\frac{5\sqrt6+5\sqrt2}{4},\frac{5\sqrt6-5\sqrt2}{4})

(-\sqrt6-\sqrt2,\sqrt6-\sqrt2)

(-\frac{\sqrt6+\sqrt2}{4},-\frac{\sqrt6-\sqrt2}{4})

(-\frac{\sqrt6+\sqrt2}{2},-\frac{\sqrt6-\sqrt2}{2})

(-\frac{3\sqrt6+3\sqrt2}{4},-\frac{3\sqrt6-3\sqrt2}{4})

(-\sqrt6-\sqrt2,-\sqrt6+\sqrt2)

(-\frac{5\sqrt6+5\sqrt2}{4},-\frac{5\sqrt6-5\sqrt2}{4})

(-\frac{\sqrt6-\sqrt2}{4},-\frac{\sqrt6+\sqrt2}{4})

(-\frac{\sqrt6-\sqrt2}{2},-\frac{\sqrt6+\sqrt2}{2})

(-\frac{3\sqrt6-3\sqrt2}{4},-\frac{3\sqrt6+3\sqrt2}{4})

(-\frac{5\sqrt6-5\sqrt2}{4},-\frac{5\sqrt6+5\sqrt2}{4})

(-\sqrt6+\sqrt2,-\sqrt6-\sqrt2)

(\frac{\sqrt6-\sqrt2}{4},-\frac{\sqrt6+\sqrt2}{4})

(\frac{\sqrt6-\sqrt2}{2},-\frac{\sqrt6+\sqrt2}{2})

(\frac{3\sqrt6-3\sqrt2}{4},-\frac{3\sqrt6+3\sqrt2}{4})

(\frac{5\sqrt6-5\sqrt2}{4},-\frac{5\sqrt6+5\sqrt2}{4})

(\sqrt6-\sqrt2,-\sqrt6-\sqrt2)

(\frac{\sqrt6+\sqrt2}{4},-\frac{\sqrt6-\sqrt2}{4})

(\frac{\sqrt6+\sqrt2}{2},-\frac{\sqrt6-\sqrt2}{2})

(\frac{3\sqrt6+3\sqrt2}{4},-\frac{3\sqrt6-3\sqrt2}{4})

(\frac{5\sqrt6+5\sqrt2}{4},-\frac{5\sqrt6-5\sqrt2}{4})

(\sqrt6+\sqrt2,-\sqrt6+\sqrt2)

u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
list4 = Sequence(Sequence[(r * cos(15° * k), r * sin(15° * k)), k, 0, 23], r, 1, 7)
list5 = Sequence(Sequence[(sqrt(2) * r * cos(45° * k), r * sqrt(2) * sin(45° * k)), k, 1, 7], r, 1, 7)
list6 = Union(list4, list5)
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
O = (0, 0)
Q_2: 0.62 * Corner[1] + 0.38 * Corner[2]
Q_3: (x(Q_2), y(Corner[4]))
poly1, q_1, q_2, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
P = Point(list6)
list1 = Sequence(x^(2) + y^(2) = r^(2), r, 1, 10)
a = Segment(O, P)
B: (x(P), 0)
b = Segment(P, B)
θ = Angle((1, 0), O, P)
list2 = Sequence((15 * k)°, k, 0, 23)
list3 = Sequence(Ray[(0, 0), (cos(Element[list2, k]), sin(Element[list2, k]))], k, 1, 24)
r = Distance(O, P)
t1: "$P\;(" + (LaTeX[If[x(P) < 0, "-"]]) + "" + (LaTeX[SurdText[abs(x(P))]]) + "," + (LaTeX[If[y(P) < 0, "-"]]) + "" + (LaTeX[SurdText[abs(y(P))]]) + ")$
$r=\sqrt{(" + (LaTeX[SurdText[abs(x(P))]]) + ")²+(" + (LaTeX[SurdText[abs(y(P))]]) + ")²}=" + (LaTeX[SurdText[r]]) + "$
$⇒(" + (LaTeX[SurdText[r]]) + "×" + (LaTeX[If[cos(θ) < 0, "(-"]]) + "" + (LaTeX[SurdText[abs(cos(θ))]]) + "" + (LaTeX[If[cos(θ) < 0, ")"]]) + "," + (LaTeX[SurdText[r]]) + "×" + (LaTeX[If[sin(θ) < 0, "(-"]]) + "" + (LaTeX[SurdText[abs(sin(θ))]]) + "" + (LaTeX[If[sin(θ) < 0, ")"]]) + ")$
$=(" + (LaTeX[SurdText[r]]) + "\;cos" + (LaTeX[θ]) + "," + (LaTeX[SurdText[r]]) + "\;sin" + (LaTeX[θ]) + ")$
$=[" + (LaTeX[SurdText[r]]) + "\;," + (LaTeX[θ]) + "]$"
A: Corner[4] + (0.2, -1)
t2_1: "$P\;" + (LaTeX[If[r ≟ 1 ∧ θ ≟ 15°, text11, If[r ≟ 2 ∧ θ ≟ 15°, text12, If[r ≟ 3 ∧ θ ≟ 15°, text13, If[r ≟ 4 ∧ θ ≟ 15°, text14, If[r ≟ 5 ∧ θ ≟ 15°, text15]]]]]]) + "" + (LaTeX[If[r ≟ 1 ∧ θ ≟ 75°, text21, If[r ≟ 2 ∧ θ ≟ 75°, text22, If[r ≟ 3 ∧ θ ≟ 75°, text23, If[r ≟ 4 ∧ θ ≟ 75°, text24, If[r ≟ 5 ∧ θ ≟ 75°, text25]]]]]]) + "" + (LaTeX[If[r ≟ 1 ∧ θ ≟ 105°, text31, If[r ≟ 2 ∧ θ ≟ 105°, text32, If[r ≟ 3 ∧ θ ≟ 105°, text33, If[r ≟ 4 ∧ θ ≟ 105°, text34, If[r ≟ 5 ∧ θ ≟ 105°, text35]]]]]]) + "" + (LaTeX[If[r ≟ 1 ∧ θ ≟ 165°, text41, If[r ≟ 2 ∧ θ ≟ 165°, text42, If[r ≟ 3 ∧ θ ≟ 165°, text43, If[r ≟ 4 ∧ θ ≟ 165°, text44, If[r ≟ 5 ∧ θ ≟ 165°, text45]]]]]]) + "" + (LaTeX[If[r ≟ 1 ∧ θ ≟ 195°, text51, If[r ≟ 2 ∧ θ ≟ 195°, text52, If[r ≟ 3 ∧ θ ≟ 195°, text53, If[r ≟ 4 ∧ θ ≟ 195°, text54, If[r ≟ 5 ∧ θ ≟ 195°, text55]]]]]]) + "" + (LaTeX[If[r ≟ 1 ∧ θ ≟ 255°, text61, If[r ≟ 2 ∧ θ ≟ 255°, text62, If[r ≟ 3 ∧ θ ≟ 255°, text63, If[r ≟ 4 ∧ θ ≟ 255°, text64, If[r ≟ 5 ∧ θ ≟ 255°, text65]]]]]]) + "" + (LaTeX[If[r ≟ 1 ∧ θ ≟ 285°, text71, If[r ≟ 2 ∧ θ ≟ 285°, text72, If[r ≟ 3 ∧ θ ≟ 285°, text73, If[r ≟ 4 ∧ θ ≟ 285°, text74, If[r ≟ 5 ∧ θ ≟ 285°, text75]]]]]]) + "" + (LaTeX[If[r ≟ 1 ∧ θ ≟ 345°, text81, If[r ≟ 2 ∧ θ ≟ 345°, text82, If[r ≟ 3 ∧ θ ≟ 345°, text83, If[r ≟ 4 ∧ θ ≟ 345°, text84, If[r ≟ 5 ∧ θ ≟ 345°, text85]]]]]]) + "$
$r=" + (LaTeX[SurdText[r]]) + " $
$( " + (LaTeX[SurdText[r]]) + "·" + (LaTeX[If[θ ≟ 105° ∨ θ ≟ 165° ∨ θ ≟ 195° ∨ θ ≟ 255°, "(-"]]) + "\frac{\sqrt6" + (LaTeX[If[θ ≟ 75° ∨ θ ≟ 105° ∨ θ ≟ 255° ∨ θ ≟ 285°, "-"]]) + "" + (LaTeX[If[θ ≟ 195° ∨ θ ≟ 165° ∨ θ ≟ 15° ∨ θ ≟ 345°, "+"]]) + "\sqrt2}{4}" + (LaTeX[If[θ ≟ 105° ∨ θ ≟ 165° ∨ θ ≟ 195° ∨ θ ≟ 255°, ")"]]) + "," + (LaTeX[SurdText[r]]) + " ·" + (LaTeX[If[θ ≟ 195° ∨ θ ≟ 255° ∨ θ ≟ 285° ∨ θ ≟ 345°, "(-"]]) + "\frac{\sqrt6" + (LaTeX[If[θ ≟ 75° ∨ θ ≟ 105° ∨ θ ≟ 255° ∨ θ ≟ 285°, "+"]]) + "" + (LaTeX[If[θ ≟ 15° ∨ θ ≟ 165° ∨ θ ≟ 195° ∨ θ ≟ 345°, "-"]]) + "\sqrt2}{4} " + (LaTeX[If[θ ≟ 195° ∨ θ ≟ 255° ∨ θ ≟ 285° ∨ θ ≟ 345°, ")"]]) + "  )$
$=(" + (LaTeX[SurdText[r]]) + "\;cos " + (LaTeX[θ]) + ", " + (LaTeX[SurdText[r]]) + "\;sin\;" + (LaTeX[θ]) + " )$
$⇒[ " + (LaTeX[SurdText[r]]) + "\;," + (LaTeX[θ]) + " ]$"