Not a member of Pastebin yet?
Sign Up,
it unlocks many cool features!
- Consider a standard coordinate system C in three dimensions. Let a linear transformation R be any rotation or reflection.
- Let R be applied to the orthonormal basis vectors that define the axis directions of C. Let R be applied to any points in
- C, say P₀ and P₁.
- R: C → C'
- R: P₀, P₁ → P'₀, P'₁
- A new coordinate system C' results whose axis directions are the transformed axis directions of C under the transformation
- R. Observe that the points P'₀, P'₁ will have the same coordinates in C' as they had in C since the transformed points did
- not move relative to the transformed basis vectors.
- To distinguish between coinciding points in different coordinate systems let P'₀, P'₁ in C be designated Q₀, Q₁ in C'. Also
- let P₀, P₁ in C be designated Q'₀, Q'₁ in C'.
- Suppose we wish to discover the coordinates of the points P₀, P₁ in C', that is we wish to find Q'₀, Q'₁.
- Let R⁻¹ be the inverse of the transformation R. We note that R⁻¹ transforms P'₀, P'₁ to P₀, P₁ and therefore transforms
- Q₀, Q₁ to Q'₀, Q'₁. Since Q₀, Q₁ have the same coordinates in C' as P₀, P₁ in C, to find Q'₀, Q'₁ we simply need to
- apply R⁻¹ to P₀, P₁. Let P and Q' represent the same point but in C and C' respectively, then
- R⁻¹: P → Q' (Trivially, we can also say R: Q' → P)
- The columns of the matrix R are the basis vectors of C' in C
- let R = |a d g| then |1| |1| |a|, |0| |0| |d|, |0| |0| |g|
- |b e h| |0|→ R|0|=|b| |1|→ R|1|=|e| |0|→ R|0|=|h|
- |c f i| |0| |0| |c| |0| |0| |f| |1| |1| |i|
- A matrix whose columns constitute an orthonormal basis is an orthogonal matrix. The inverse of an orthogonal matrix is its
- transpose. That is
- R⁻¹ = Rᵀ , and so
- Rᵀ: P → Q'
Add Comment
Please, Sign In to add comment