Yoo, Kyunghwan

2 D Equation

Screen& Point

Q = ArcTan[A_x - B_x, A_y - B_y] ;

D = (x - A_x) * Cos[Q] + (y - A_y) * Sin[Q] ;

{x, y} = {A_x + D * Cos[Q], A_y + D * Sin[Q]} ;

Reflection Line

FQ = 2 * ArcTan[A_y - B_y, B_x - A_x] - ArcTan[S_x - x, S_y - y] ;

FD = (E_x - x) * Cos[FQ] + (E_y - y) * Sin[FQ] ;

{E_x, E_y} = {x + FD * Cos[FQ], y + FD * Sin[FQ]} ;

Point

P_a = {x_a, y_a, 1}

P_b = {x_b, y_b, 1}

Line

(x - x_a)/(x_b - x_a) == (y - y_a)/(y_b - y_a)

y = (y_b - y_a)/(x_b - x_a) (x - x_a) + y_a ;

y = Tan[Q] * (x - x_a) + y_a ;

2 Point

R = ((x_b - x_a)^2 + (y_b - y_a)^2)^(1/2) ;

Q = ArcTan[x_b - x_a, y_b - y_a] = ArcTan[(y_b - y_a)/(x_b - x_a)] ;

x_a = x_b + R * Cos[θ] ;

y_a = y_b + R * Sin[θ] ;

({{x_a}, {y_a}, {1}}) = ({{1, 0, R Cos[θ]}, {0, 1, R Sin[θ]}, {0, 0, 1}}) . ({{x_b}, {y_b}, {1}})

({{x_a}, {y_a}, {1}}) = ({{Cos[θ], -Sin[θ], x_b}, {Sin[θ], Cos[θ], y_b}, {0, 0, 1}}) . ({{R}, {0}, {1}})//MatrixForm

({{R}, {0}, {1}}) = ( {{Cos[θ] (x_a - x_b) + Sin[θ] (y_a - y_b)}, {Sin[θ] (-x_a + x_b) + Cos[θ] (y_a - y_b)}, {1}} ) ;

Tan[θ] = Sin[θ]/Cos[θ] = (y_a - y_b)/(x_a - x_b) ;

θ = ArcTan[(y_a - y_b)/(x_a - x_b)] ;

R = Cos[θ] (x_a - x_b) + Sin[θ] (y_a - y_b) ;

Det[({{ax, ay, 1}, {bx, by, 1}, {cx, cy, 1}})] == ax * (by - cy) + bx * (cy - ay) + cx * (ay - by)