fospathi

Consider a standard coordinate system C in three dimensions. Let a linear transformation R be any rotation or reflection
around or through the origin respectively.

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

fospathi

₀₁₂₃₄₅₆₇₈₉ ⁰¹²³⁴⁵⁶⁷⁸⁹ ⁺⁻⁼⁽⁾ ₐₑₕᵢⱼₖₗₘₙₒₚᵣₛₜᵤᵥₓᴬᴮᴰᴱᴳᴴᴵᴶᴷᴸᴹᴺᴼᴾᴿᵀᵁⱽᵂ
⋅±→∠⨯ √  ‥
αβγδεζηθικλμνξοπρςστυφχψω
Α α, Β β, Γ γ, Δ δ, Ε ε, Ζ ζ, Η η, Θ θ, Ι ι, Κ κ, Λ λ, Μ μ, Ν ν, Ξ ξ, Ο ο, Π π, Ρ ρ, Σ σ/ς, Τ τ, Υ υ, Φ φ, Χ χ, Ψ ψ, Ω
ᵦ ᵧ ᵨ ᵩ ᵪ ᵅ ᵝ ᵞ ᵟ ᵋ ᶿ ᶥ ᶲ ᵠ ᵡ

fospathi

Find the values of the parameter t (in the range 0..1 inclusive) at the intersection points of a quadratic Bezier curve 
and a plane in three dimensions.

Let P₀, P₁, and P₂ be the control points of a quadratic Bezier curve B such that

B{t} = (1-t)²P₀ + 2(1-t)(t)P₁ + (t²)P₂

Let A be a unit normal vector to the plane where Q is any position vector on the plane such that

A⋅Q = d

fospathi

Produce a new function which is a horizontal reflection of the function f{x} in the vertical line x=k.

There are three steps: (1) horizontally translate the function in such a way that the reflection can be applied across the 
Y-axis (i.e. all points on the translated function's curve should be the same distance from the Y-axis as they originally
were from the line x=k); (2) perform a reflection across the Y-axis; (3) reverse the translation made in step 1.

1. Consider the translation necessary to make the line x=k coincide with the Y-axis. Apply this translation to f{x} 
   producing f₁{x}.

   f₁{x} = f{x+k}

fospathi

/// Print the calculation which multiplies two 3x4 affine transformation
/// matrices.
///
/// The matrices store their elements in a flat array in column order.
void main() {
  StringBuffer buffer = new StringBuffer();
  String l = "a1";
  String r = "a2";
  for (int col in new Iterable.generate(4)) {
    for (int row in new Iterable.generate(3)) {

fospathi

/// Print the calculation which multiplies two 4x4 matrices.
///
/// The matrices store their elements in a flat array in column order.
void main() {
  StringBuffer buffer = new StringBuffer();
  String l = "m1";
  String r = "m2";
  for (int col in new Iterable.generate(4)) {
    for (int row in new Iterable.generate(4)) {
      buffer.writeAll([

fospathi

/// Print the calculation which multiplies two 3x3 matrices.
///
/// The matrices store their elements in a flat array in column order.
void main() {
  StringBuffer buffer = new StringBuffer();
  String l = "m1";
  String r = "m2";
  for (int col in new Iterable.generate(3)) {
    for (int row in new Iterable.generate(3)) {
      buffer.writeAll([