linear algebra thus far

src/linear_algebra.jl

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+using GeometryBasics
+using Plots
+using LinearAlgebra
+using RationalRoots
+
+"""
+    distance_2_points(p::Point, q::Point) -> Norm(v)
+Distance between two points
+"""
+function distance_2_points(p::Point, q::Point)
+    v = q - p
+    norm(v)
+end
+
+"""
+    center of gravity(p::Point, q::Point, t) → r
+Creates center of gravity of points `p` and `q`using parametric equation of a line
+"""
+function center_of_gravity(p::Point, q::Point, t)
+    v = q - p
+    r = p + (t * v)
+end
+
+"""
+    barycentric_coord(p::Point, q::Point, r::Point) -> t::Float64
+Calculate the barycentric co-ordinate or parater of points p and q with center of gravity point r
+"""
+function barycentric_coord(p::Point, q::Point, r::Point)
+    t = norm(r - p) / norm(q - r)
+end
+
+"""
+    plot_param_line(p::Point, q::Point, n::Int64) → [Point]
+Creates `n` points on a line defined by `p` and `q`, using the parametric equation of a line, then plot
+"""
+
+function plot_param_line(p::Point, q::Point, n::Int64)
+    Ps = Point[]
+    for i = 1:n
+        t = i / n
+        r = center_of_gravity(p, q, t)
+        s = scatter!(r, legend=false)
+        push!(Ps, r)
+    end
+    s = plot!(Ps, legend=false)
+    display(s)
+    Ps
+end
+
+"""
+    vector_angle_cos(p::Vector, q::Vector) -> cos θ
+Calculate cosine of angle between 2 points
+"""
+function vector_angle_cos(p::Vector, q::Vector)
+    s = dot(p, q) / (norm(p) * norm(q))
+end
+
+"""
+    is_orthogonal(p::Point, q::Point) -> bool
+Check if two vectors are is_orthogonal
+"""
+function is_orthogonal(p::Point, q::Point)
+    dot(p, q) == 0
+end
+
+"""
+    function polar_unit(y::Vector) -> Vector{Float64}
+Return unit vector in polar form for vector `y`
+"""
+function polar_unit(y::Vector)
+    [(y[1] / norm(y)), (y[2] / norm(y))]
+end
+
+"""
+    orthproj(v::Vector, w::Vector) -> Vector
+Orthogonol projection of vector `w` onto `v`
+"""
+function orthproj(v::Vector, w::Vector)
+    u = (dot(v, w) / dot(v, v)) * v
+end
+
+"""
+    function reflection(v::Vector, w::Vector) -> Vector
+The projection P from 'w' onto 'v' is the midpoint of the reflection of w around v
+"""
+function reflection(v::Vector, w::Vector)
+    P = orthproj(v,w)
+    (2 * P ) - X
+end
+
+"""
+function rotation(θ::Number, v::Vector) -> Vector
+    θ : degrees to rotate v
+"""
+function rotation(θ::Number, v::Vector)
+    x′ = (cos(deg2rad(θ)) * v[1]) - (sin(deg2rad(θ)) * v[2])
+    y′ = (sin(deg2rad(θ)) * v[1]) + (cos(deg2rad(θ)) * v[2])
+    [round(x′), round(y′)]
+end
+
+"""
+    point_in_implicit_line(p::Point, q::Point, x::Point) -> Float64
+The orthogonal vector α is calculated as:
+    v = Vector(q - p)
+    α = [v[2], -v[1]]
+The implicit equation of the line is:
+    α * (x -p) = 0
+    a = α[1]
+    b = α[2]
+    c = -(a * p[1]) - (b * p[2])
+Then calculate the distance of point `x` from the line using the formula:
+    (a * x[1] + b * x[2] + c) / norm([a, b]
+If 0 the point is in the line!
+"""
+function point_in_implicit_line(p::Point, q::Point, x::Point)
+    a = q[2] - p[2]
+    b = p[1] - q[1]
+    c = -(a * p[1]) - (b * p[2])
+    (a * x[1] + b * x[2] + c) / norm([a, b])
+end
+
+"""
+    parametric_to_implicit_line(p::Point, v::Vector) -> Tuple{Number, Number, Number}
+The parametric equation is:
+    l : l(t) = p + tv
+Use p and v to calculate:
+    l : ax1 + bx2 + c = 0.
+And return co-efficient as tuple
+"""
+function parametric_to_implicit_line(p::Point, v::Vector)
+    a = -v[2]
+    b = v[1]
+    c = -(a * p[1]) - (b * p[2])
+    (a, b, c)
+end
+
+"""
+   implicit_to_parametric line(a::Number, b::Number, c::Number) -> Tuple(Vector, Point)
+"""
+function implicit_to_parametric_line(a::Number, b::Number, c::Number)
+    v = [b, -a]
+    if abs(a) > abs(b)
+        # the linter gives error on this constructor so shut off lint.call setting
+        p = Point(- c / a, 0)
+    else
+        p = Point(0, -c / b)
+    end
+    (v, p)
+end
+
+"""
+    explicit_line(p::Point, q::Point) -> Tuple(Float64, Float64)
+The orthogonal vector α is calculated as:
+    v = Vector(q -p)
+    α = [v[2], -v[1]]
+The explicit equation of the line requires slope & intercept
+"""
+function explicit_line(p::Point, q::Point)
+    a = q[2] - p[2]
+    b = p[1] - q[1]
+    c = -(a * p[1]) - (b * p[2])
+    slope = -a / b
+    intercept = -c / b
+    (slope, intercept)
+end
+
+"""
+   distance_to_implicit_line(a::Number, b::Number, c::Number, r::Point) -> Float64
+"""
+function distance_to_implicit_line(a::Number, b::Number, c::Number, r::Point)
+    v = [a, b]
+    d = ((a * r[1]) + (b * r[2]) + c) / norm(v)
+end
+
+"""
+   implicit_line_point_normal_form(a::Number, b::Number, c::Number) -> Tuple(RationalRoot, RationalRoot, RationalRoot)
+"""
+function implicit_line_point_normal_form(a::Number, b::Number, c::Number)
+    v = [a, b]
+    â = a/RationalRoot(norm(v))
+    b̂ = b/RationalRoot(norm(v))
+    ĉ = c/RationalRoot(norm(v))
+    (â, b̂, ĉ)
+end
+
+"""
+   distance_to_pnf_implicit_line(â::RationalRoot, b̂::RationalRoot, ĉ::RationalRoot, r::Point) -> Float64
+"""
+function distance_to_pnf_implicit_line(â::RationalRoot, b̂::RationalRoot, ĉ::RationalRoot, r::Vector)
+    d = ((â * r[1]) + (b̂ * r[2]) + ĉ)
+end
+
+"""
+   distance_to_parametric_line(p::Point, v::Vector, r::Point) -> Float64
+Definition of the line:
+   l = p + tv
+"""
+function distance_to_parametric_line(p::Point, v::Vector, r::Point)
+    # calculate the vector from p to r
+    w = Vector(r - p)
+    # calculate cos(α)
+    #cosα = dot(v,w) / (norm(v) * norm(w)) previously defined as function
+    cosα = vector_angle_cos(v,w)
+    # distance = length of w * sin(α) which is sqrt of 1 - cos(α) squared
+    d = norm(w) * sqrt(1 - (cosα ^ 2))
+end
+
+"""
+function foot_of_line(p::Point, v::Vector, r::Point) -> Tuple(Point, Float64)
+Definition of the line:
+   l = p + tv
+Find point q on l closest to r
+Calculate distance from q to A (foot of the line)
+"""
+function foot_of_line(p::Point, v::Vector, r::Point)
+    # calculate the vector from p to r
+    w = Vector(r - p)
+    t = dot(v, w) / norm(v)^2
+    # calculate foot
+    q = Point(p + t*v)
+    # calculate distance
+    d = dot(v, w) / norm(v)
+    (q, d)
+end
+
+"""
+    function foot_of_line_parallelogram(P::Point, A::Point, B::Point) -> Tuple(Point, Float64, Float64)
+Definition of the line:
+   l = p + tv
+where v is the vector from P to B
+w is the vector from P to A
+Find point q on l closest to A
+Calculate distance from q to A (foot of the line)
+Calculate area of parallelogram defined by A and B
+"""
+function foot_of_line_parallelogram(P::Point, A::Point, B::Point)
+    # calculate the vector from p to B
+    v = Vector(B - p)
+    # calculate the vector from p to A
+    w = Vector(A - p)
+    t = dot(v, w) / norm(v)^2
+    # calculate foot
+    q = Point(p + t*v)
+    # calculate distance
+    d = dot(v, w) / norm(v)
+    # calculate area of parallelogram - distance * base
+    a = d * norm(v)
+    (q, d, a)
+end
+
+"""
+    function intersection_2_parametric_lines(v::Vector, w::Vector, p::Point, q::Point) -> Vector
+l₁ = p + tv
+l₂ = q + sw
+intersection: p + t̂v = q + ŝw
+The solution is a system of 2 equations in two unknowns expressed in equation:
+    t̂*v - ŝ*w  = q - p
+"""
+function intersection_2_parametric_lines(v::Vector, w::Vector, p::Point, q::Point)
+    # calculate the vector of RHS of the solution equation
+    b = Vector(q - p)
+    # build the matrix of the LHS of the solution equation
+    A = [v[1] -w[1]; v[2] -w[2]]
+    # solve the system using left division of the matrix by the vector
+    A\b
+end
+
+"""
+    function intersection_2_implicit_lines(a₁::Number, b₁::Number, c₁::Number, a₂::Number, b₂::Number, c₂::Number) -> Vector
+l₁: a₁x̂₁ + b₁x̂₂ + c₁ = 0
+l₂: a₂x̂₁ + b₂x̂₂ + c₂ = 0
+Solve for x̂₁ and x̂₂ which is the intersection point
+"""
+function intersection_2_implicit_lines(a₁::Number, b₁::Number, c₁::Number, a₂::Number, b₂::Number, c₂::Number)
+    # calculate the c vector
+    b = [-c₁, -c₂]
+    # build the A matrix from a's and b's
+    A = [a₁ b₁; a₂ b₂]
+    # solve the system using left division of the matrix by the vector
+    A\b
+end