Geometry2D.jl

Geometry2D models basic geometric shapes and common geometric formulae. This is a flat module with entities grouped into similar files under src.

See online documentation at https://mechanomy.github.io/Geometry2D.jl/dev/ and test for examples of each function.

Type union accepting Angle or Real.

Type union accepting Unitful.Length or Real.

Type union accepting Pitch or Real.

Convenience unit vector in the x direction.

Convenience unit vector in the y direction.

Convenience unit vector in the z direction.

A circle having a center::Point and a radius::Unitful.Length.

Circle( x::Unitful.Length, y::Unitful.Length, r::Unitful.Length )

A circle having a center at x::Unitful.Length and y::Unitful.Length with radius::Unitful.Length.

A difference between two points on the cartesian plane, measured in dx and dy from the plane's origin. This is introduced as a separate type to avoid using Vector{}s with inexplicit length.

A point on the cartesian plane, measured in x and y from the plane's origin.

Describes a spiral progressing from a0,r0 to a1,r1. If thickness is not specified it is calculated such that each layer touches neighbors, a tightly wound spiral, if it is specified then an open spiral is modeled.

• a0::AngleOrReal – start angle
• a1::AngleOrReal – stop angle
• r0::LengthOrReal – start radius
• r1::LengthOrReal – stop radius
• pitch::PitchOrReal – pitch

Spiral( a0::AngleOrReal, a1::AngleOrReal, r0::LengthOrReal, r1::LengthOrReal ) Create a Spiral, calculating the pitch from a0, a1, r0, and r1.

A UnitVector type is unitless, expressing only relative magnitude. It has fields x, y, and z

• x

  Component in the x direction.

• y

  Component in the y direction.

• z

  Component in the z direction.

UnitVector(vec::AbstractVector)

Given some subtype of AbstractVector, return a UnitVector.

A UnitVector2D type is unitless, expressing only relative magnitude. It has fields x and y.

UnitVector2D(dl::Delta)

Constructs a UnitVector2D in the direction of dl.

A 2D vector from origin to tip. Functions are provided to convert from the struct's cartesian format to polar.

• origin

  the origin Point

• tip

  the tip Point

(*)(p::Point, f::Real) :: Point

Multiplies p by the given factor f.

(+)(p::Point, d::Delta) :: Point

Adds a d Delta to Point p.

(-)(p::Point, d::Delta) :: Point

Subtracts a Delta d from the Point p.

(-)(a::Point, b::Point) :: Delta

Finds the Delta between Points a and b.

(-)(a::Real, uv::UnitVector) :: UnitVector

Provides negation of UnitVectors.

(/)(d::Delta, f::Real) :: Delta

Divides d by the given factor f.

(/)(p::Point, f::Real) :: Point

Divides p by the given factor f.

angle(d::Delta) :: Real

Calculate the angle of Delta d relative to global x = horizontal.

angle(v::Vector2D) :: Angle
angled(v::Vector2D) :: typeof(1.0u"°")

Calculate the angle of Vector2D v relative to global x = horizontal, via atan().

isapprox(a::Delta, b::Delta; atol=0, rtol=√eps()) :: Bool

Approximately compare Deltas a and b via absolute tolerance atol and relative tolerance rtol, as in isapprox.

isapprox(p::Point, q::Point; atol=0, rtol=√eps()) :: Bool

Approximately compare Points p and q via absolute tolerance atol and relative tolerance rtol, as in isapprox.

isapprox(a::UnitVector, b::UnitVector; atol=0, rtol=√eps()) :: Bool

Approximately compare UnitVectors a to b via absolute tolerance atol and relative tolerance rtol, as in isapprox.

isapprox(a::UnitVector2D, b::UnitVector2D; atol=0, rtol=√eps()) :: Bool

Approximately compare UnitVector2Ds a and b via absolute tolerance atol and relative tolerance rtol, as in isapprox.

isapprox(a::Vector2D, b::Vector2D; atol=0, rtol=√eps()) :: Bool

Approximately compare Vector2Ds a to b via absolute tolerance atol and relative tolerance rtol, as in isapprox.

Rz(p::Point, a::Angle)

Rotate p in the plane an angle a.

Rz(angle::Real)

Create a 2D rotation matrix effecting a rotation of angle. Homogeneous transformation matrices are multiplicative and therefore should be unitless.

Rz(angle::Angle)

Create a 2D rotation matrix effecting a rotation of angle. Homogeneous transformation matrices are multiplicative and therefore should be unitless.

Tx(p::Point, d::Unitful.Length)

Translate p a distance of d in the X direction. Homogeneous transformation matrices are multiplicative and therefore should be unitless.

Tx(a::Real) Create a 2D translation matrix translating along local X an angle a. Homogeneous transformation matrices are multiplicative and therefore should be unitless.

Ty(p::Point, d::Unitful.Length)

Translate p a distance of d in the X direction. Homogeneous transformation matrices are multiplicative and therefore should be unitless.

Ty(b::Real) Create a 2D translation matrix translating along local Y by b. Homogeneous transformation matrices are multiplicative and therefore should be unitless.

angleAdjacent2Opposite(angle::Angle, adjacent::Unitful.Length) :: Unitful.Length
angleAdjacent2Opposite(angle::Angle, adjacent::Unitful.Length) :: Unitful.Length

For a right triangle, find the length of the 'opposite' leg, given angle and adjacent.

angleOpposite2Adjacent(angle::Angle, opposite::Unitful.Length) :: Unitful.Length
angleOpposite2Adjacent(; angle::Angle, opposite::Unitful.Length) :: Unitful.Length

For a right triangle, find the length of the 'adjacent' leg, given angle and opposite.

angleWrap(angle::Real) :: Real

Wraps angle between 0 and 2π.

angleWrap(angle::Radian) :: Radian

Wraps angle between 0 and 2π.

calcLength(a0::AngleOrReal, a1::AngleOrReal, r0::LengthOrReal, r1::LengthOrReal)
calcLength(;a0::AngleOrReal, a1::AngleOrReal, r0::LengthOrReal, r1::LengthOrReal)
calcLength(s::Spiral)

Length along the spiral, as if it were unrolled.

calcPitch(a0::AngleOrReal, a1::AngleOrReal, r0::LengthOrReal, r1::LengthOrReal)
calcPitch(; a0::AngleOrReal, a1::AngleOrReal, r0::LengthOrReal, r1::LengthOrReal)
calcPitch(s::Spiral)

Calculate a spiral's pitch from (r1-r0)/(a1-a0). The nominal units of Pitch defined to be [m/rev], but this will return Real if the arguments or Spiral are that, or a Pitch,m/rad,m/deg depending on __.

circleArcLength( circle::Circle, angle::Angle )

Compute the circlular arc length of circle over angle.

circleArcLength( radius::Unitful.Length, angle::Angle )

Compute the circlular arc length at radius over angle.

angle = 20u"°" 
radius = 5u"mm"
len = circleArcLength(angle, radius)

circleArea(c::Circle) :: Unitful.Area

Calculate the area of a circle with radius r.

circleArea(r::Unitful.Length) :: Unitful.Area

Calculate the area of a circle with radius r.

circumference( circle::Circle )

Calculate the circumeference of circle.

circumference( r::Unitful.Length ) :: Unitful.Length

Calculate the circumeference of circle.

delta(v::Vector2D) :: Delta

Given v, return the difference in x and y as a Delta from v's tip to origin.

distance(a::Point, b::Point ) :: Unitful.Length

Finds the straight-line distance between a and b. (It is nonsensical to ask for the 'distance' of a Delta, rather Deltas have norm().)

ellipticArcLength(a::Number, b::Number, start::Number, stop::Number)

Calculates the arc length of an ellipse from major axis a towards minor axis b between start and stop angles, as measured from the major axis a.

ellipticArcLength(a::Number, b::Number, angle::Number ) :: Number

Calculates the arc length of an ellipse from major axis a towards minor axis b through angle, measured from a, via elliptic integral: L = b * elliptic_e( atan(a/b*tan(angle)), 1-a^2/b^2 )

ellipticArcLength(a::Unitful.Length, b::Unitful.Length, angle::Angle)

Unitful version. Calculates the arc length of an ellipse from major axis a towards minor axis b through angle, measured from major axis a, via elliptic integral:. L = b * elliptic_e( atan(a/b*tan(angle)), 1-a^2/b^2 )

hvec2point(v::Vector{<:Unitful.Length}) :: Point

Convert a 1-terminated vector back into a Point.

isClockwise(s::Spiral) :: Bool

Conversely, clockwise implies decreasing angle.

isCounterClockwise(s::Spiral) :: Bool

2D Spirals are 'counterclockwise' if of increasing angle, a0 to a1.

isDecreasing(s::Spiral) :: Bool

Does the radius decrease

isIncreasing(s::Spiral) :: Bool

Does the radius increase?

isSegmentMutuallyTangent(; cA::Circle, cB::Circle, thA::Radian, thB::Radian, tol::Number=1e-3) :: Bool

Given Circles circleA and circleB, tests whether the points on their edges at angles thA and thB define a line segment that is tangent to both circles. tol is the tolerance on the tangency criterion.

isSegmentPerpendicularToParallelVectors( vA::Vector2D, vB::Vector2D, tol::Number=1e-3) :: Bool

Tests whether a segment connecting the tips of vA and vB is perpendicular to both, which implies parallelity of vA and vB. The calculation compares the direction of the cross products of vA and the tip-connecting segment, and vB and the segment, that these are within tol of each other.

lawOfCosines(legA::Unitful.Length, legB::Unitful.Length, angleAB::Angle) :: Unitful.Length
lawOfCosines(legA::Unitful.Length, legB::Unitful.Length, angleAB::Angle) :: Unitful.Length

For any triangle, finds the length of the unknown legC from legA, legB, and angleAB between them.

lawOfSines(legA::Unitful.Length, angleBC::Angle, angleCA::Angle) :: Unitful.Length
lawOfSines(; legA::Unitful.Length, angleBC::Angle, angleCA::Angle) :: Unitful.Length

For any triangle with legs A, B, C and angles AB, BC, CA, so that legA is the open side of angleBC, finds legB corresponding to angleCA.

legHypotenuse2Leg( leg::Unitful.Length, hyp::Unitful.Length  ) :: Unitful.Length
legHypotenuse2Leg(; leg::Unitful.Length, hyp::Unitful.Length  ) :: Unitful.Length

For a right triangle, find the length of the third leg from hypotenuse hyp and leg.

legLeg2Hypotenuse( a::Unitful.Length, b::Unitful.Length ) :: Unitful.Length
legLeg2Hypotenuse(; a::Unitful.Length, b::Unitful.Length ) :: Unitful.Length

For a right triangle, find the length of the hypotenuse from the shorter legs a and b.

normalize( d::Delta ) :: UnitVector2D

Return a UnitVector2D for Delta d.

normalize( p::Point ) :: UnitVector2D

Return a UnitVector2D pointing toward Point p.

point2hvec(p::Point)

Convert p into a 1-terminated Vector for multiplication with some homogeneous transformation matrix.

pointOnCircle( c::Circle, a::Angle ) :: Point

Returns the Point on c at angle a measured from +x.

radialVector( c::Circle, a::Angle ) :: Vector2D

Returns a Vector2D with origin at c.center and tip at angle a and c.radius.

seriesCartesian(s::Spiral, n::Int=1000)
seriesCartesian(; s::Spiral, n::Int=1000)

Spread n points over spiral s, returning an (xs,ys) tuple

seriesPolar(s::Spiral, n::Int=1000)
seriesPolar(; s::Spiral, n::Int=1000)

Spread n points over spiral s, returning an (angles,radii) tuple.

toVector( uv::UnitVector ) :: AbstractVector

Convert uv to a Base.Vector.

Point(;x::Unitful.Length, y::Unitful.Length)

Keyword constructor for a point on the cartesian plane, measured in x and y from the plane's origin.

UnitVector2D(; x::Real, y::Real)

Keyword constructor.

Vector2D(;origin::Point, tip::Point)

Keyword constructor for a vector from origin to tip.

Circle(; center::Point, radius::Unitful.Length )

A circle having a center::Point and a radius::Unitful.Length.

UnitVector(; x::Real, y::Real, z::Real)

A keyword constructor of UnitVectors.

Spiral(; a0::AngleOrReal, a1::AngleOrReal, r0::LengthOrReal, r1::LengthOrReal, pitch::PitchOrReal) = Spiral(a0, a1, r0, r1, pitch)

Keyword create a Spiral with a0, a1, r0, r1, and pitch.

Spiral( a0::AngleOrReal, a1::AngleOrReal, r0::LengthOrReal, r1::LengthOrReal )

Keyword create a Spiral, calculating the pitch from a0, a1, r0, and r1.

Spiral(; a0::Real, a1::Real, r0::Real, r1::Real) = Spiral(a0, a1, r0, r1, calcPitch(a0,a1,r0,r1)::Real )

Keyword create a Spiral, calculating the pitch from a0[rad], a1[rad], r0, and r1.

norm( d::Delta; p=2 ) :: Unitful.Length

Returns the p-norm of d.

norm( pt::Point; p=2 ) :: Unitful.Length

Returns the p-norm of pt.

norm( u::UnitVector2D; p=2 ) :: Real

Returns the p-norm of u.

norm( uv::UnitVector; p=2 ) :: Real

Returns the UnitVector's 2-norm, which should always be 1.

A plot recipe for plotting Circles under Plots.jl.

c = Circle( 3mm,4mm, 5mm )
plot(c) #plot(c, linecolor=:red, ...kwArgs... )

A plot recipe for plotting Points under Plots.jl.

p = Point( 3mm,4mm )
plot(p)