Understanding Isometries in R2 and R3: A Comprehensive Breakdown

Understanding Isometries in R2

It retains the standard distance i = A · A^T = Id. Det = +1 or -1.

Vector fix: v ∈ P associates at v ∈ P 1SI F ∈ G = T_v, then F o G also. Det H = Det G · Det F.

Equations: T_v(x, y) = (x, y). Base service: T_v = C_ne · T_n · C_ne^-1

Note: A * I must be symmetrical.

Find a point on the axis or plane (one that satisfies the equation).
From the equation equal to 0, if you are caught, separate subjects. If not equal to 0, they are roofed and left all depending on the sign and normalize.
u₂ will be looking to normalize.
Find u₃ to (i, j, k) and normalize.
The three u are CNE.
A base change (x, y, z).
The image of a point is A (vertical extent).

Calculate the determinant.
Det = +1 or -1.
1. Det = -1 (axial symmetry). Axis: Ker(A – Id), f(x) = (1, 0), (0, -1).
2. Det = +1, angle = +1.
3. 1. Identity 0.
  2. Symmetry (-Id), angle = 180.
  3. Any rotation, angle = arccos(Trace / 2), f(x, y) = (cos, sin)(-sin, cos).

Conserve only if F and G are distances. If F is isometric, the image is also isometric. +v = I (symmetric) and F_v = A * v.

Calculate the determinant.
Det = +1 or -1.
1. Symmetry (-Id).
2. Calculate Rg(A – Id) = 1 or 3.
3. 1. Rg(A – Id) = 1. Specular symmetry, plane: Ker(A – Id), f(x, y, z) T_n = (-1, 0, 0)(0, 1, 0)(0, 0, 1).
  2. Rg(A – Id) = 3. Rotational symmetry, angle = arccos((Trace + 1) / 2), axis: Ker(A + Id), plane: Ker(A + Id) [orthogonal] i.e. (x, y, z)(Id)(Axis) and gives 2 vectors in the plane, f(x, y, z) = T_n(-1, 0, 0), (0, cos, -sin), (0, sin, cos).
Det = +1.
1. (Id) 0.
2. Any rotation, angle = arccos((Trace – 1) / 2), axis: Ker(A – Id), angle = 180 = Symmetry.

Calculate the determinant.
Det A = +1 or -1.
Det = -1.
1. Axisymmetric fixed points if you have a solution, (Id – A)(x, y)v = (a, b)v.
2. Glide reflection fixed points if you do not have a solution, (Id – A)(x, y)v = (a, b)v – v.
Det = +1.
1. Translation. T = (a, b) A = Id.
2. Symmetry, center p = 1 / 2(a, b), A = -Id.
3. Rotation, angle = arccos(trace / 2), center (Id – A)(x, y)v = (a, b)v.

Calculate the determinant.
Det A = +1 or -1.
Det = -1.
Symmetry, A = -Id, center p = 1 / 2(a, b, c).
Calculate the rank (1 or 3).
Rank = 1.
1. Mirror symmetry, plane (Id – A)(x, y, z)^v = (a, b, c)^v. It must have a solution but is a glide reflection.
2. Glide reflection, plane (Id – A)(x, y, z)^v = (a, b, c)^v – V, V (glide vector) = 1 / 2(Id + A)(a, b, c)^v.
Rank = 3.
1. Rotational symmetry, axis x = p + λu, p: (Id – A)(x, y, z)^v = (a, b, c)^v, u belongs to Ker(A + Id), angle = arccos((trace + 1) / 2), plane: x = p + αv + βw, where w belongs to Ker(A + Id) [orthogonal].
Det = +1.
1. Translation A = Id, t(a, b, c).
2. Rotation, angle = arccos((trace – 1) / 2), axis: (Id – A)(x, y, z)^v = (a, b, c)^v. If there are helical movement solutions, α = 180° axial symmetry.
3. Helical movement, angle = arccos((trace – 1) / 2), axis: (Id – A)²(x, y, z)^v = (Id – A)(a, b, c)^v, glide vector v = (A – Id)p + (a, b, c)^v.

Axial Symmetry: (a) Pick up point P on the line (b) A = I (c) t = (Id – A)p.
Glide Reflection: (a) Calculate p (b) A = I (c) t = (Id – A)p + v.
Translation: (a) A = Id (b) t is the translation vector.
Symmetry: Center p (a) A = -Id (b) t = 2p.
Rotation: Center p, A = I, t = (Id – A)p.

Symmetry: Center p (a) A = -Id (b) t = 2p.
Specular Symmetry Plane π: (a) Calculate p (b) A = I (c) t = (Id – A)p + v.
Glide Reflection Plane π and Vector V: (a) Calculate p (b) A = I (c) t = (Id – A)p + v.
Rotational Symmetry: (a) P = π contained in r (b) A = I (c) t = (Id – A)p.
Translation: A = Id, t is the translation vector.
Rotation Axis r and Angle α: (a) Calculate p of the axis (b) A = I (c) t = (Id – A)p.
Helical Movement Axis r, Angle α, and Glide Vector V: (a) Calculate p of the axis (b) A = I (c) t = (Id – A)p + v.