Curves

A curve is meant to be a smooth, non-self-intersecting curve in the extended complex plane. The curve parameter is a real number in the range $[0,1]$ that is used to traverse the curve.

All curve types are parameterized by a base floating-point type, such as Float64, that is the numeric type of the curve parameter as well as (possibly complexified) its points.

Abstract interface

AbstractCurve

Every realization of AbstractCurve is expected to implement the following methods. (Here C represents a value of type AbstractCurve and z is a number.)

Method	Description
point(C, t::Real)	Complex point on C at parameter value t in [0,1].
tangent(C, t::Real)	Complex tangent to C at t.
reverse(C)	Reverse the direction of traversal.
isfinite(C)	True if the curve does not pass through infinity.
conj(C)	Complex conjugate of the curve.
C+z	Translate of the curve by z.
-C	Negate the curve.
C*z	Multiply the curve C by complex number z; i.e., scale and rotate it about the origin.
inv(C)	Invert the curve pointwise.

There are also default implementations of the following methods:

Method	Description
point(C, t::AbstractArray{T<:Real})	Vectorization of the point method.
z+C, C-z, z-C, z*C, C/z, z/C	Translate/rotate/scale by a complex value.
unittangent(C, t::Real)	Normalized tangent to C at t.
normal(C, t::Real)	Unit (leftward) normal to C at t.
arclength(C)	Arc length of C.

AbstractClosedCurve

The AbstractClosedCurve subtype is used to signify that the starting and ending points of the curve are (approximately) identical. In addition to the methods of AbstractCurve, it provides the following:

Method	Description
winding(C, z::Number)	Winding number of C about z.
isinside(z::Number, C)	Detect whether z lies inside the curve.
isoutside(z::Number, C)	Detect whether z lies outside the curve.

Generic types

Curve

A Curve represents an implementation of AbstractCurve that requires only an explicit parameterization of the curve. Given the (bounded) complex-valued function $f$ defined on $[0,1]$, then Curve(f) represents the curve $z=f(t)$. If $f$ is defined on $[a,b]$ instead, then Curve(f,a,b) is appropriate, but all future work with the curve object uses the standard interval $[0,1]$ for the parameter. All Curve values are expected to be finite; i.e., isfinite(C) will always be true.

By default, a tangent to the curve is computed when needed using automatic differentiation. If a function df is available for the complex-valued tangent $z^{\prime }(t)$, it can be supplied via Curve(f, df).

ClosedCurve

A ClosedCurve implements AbstractClosedCurve and is similar to a Curve, but at construction the parameterization is checked for $f(0)\approx f(1)$, up to a tolerance.

Specific subtypes

The following particular types of curves are provided. The default floating-point type is Float64, but you can specify another type explicitly, as in Line{Float32}(0, 1im) or Segment(BigFloat(0), 1).

In addition to the minimal methods set by the AbstractCurve definition above, each of these types provides the following methods. (C is a value of one of these types, and z is a number.)

Method	Description
arg(C, z)	Curve parameter value of a given point on the curve.
isapprox(C1, C2)	Determine whether two values represent the same curve.
isleft(z, C), isright(z, C)	Determine whether a point lies "to the left" or "to the right" of a line, ray, or segment in its given orientation.
dist(z, C)	Distance from a point to the curve.
closest(z, C)	Point on the curve nearest to a given number.

Line

Like other curves, a line is parameterized over $[0,1]$, with L(0) and L(1) both being infinity.

• Line(a, b) creates the line through the values $a$ and $b$.

• Line(p, direction=s) creates a line through the point p in the direction of the complex number s.

• Line(p, angle=θ) creates a line through the point p at the angle θ.

Use reflect(z, L) to find the reflection of a point z across line L.

Ray

• Ray(z, θ) constructs a ray starting at z and extending to infinity at the angle θ.

• Ray(z, θ, true) constructs a ray starting at infinity and extending to z at the angle θ.

Segment

• Segment(a, b) constructs the line segment from a to b.

Circle

• Circle(z, r) constructs a circle centered at z with radius r, oriented counterclockwise.

• Circle(z, r, false) constructs the circle with clockwise orientation.

• Circle(a, b, c) constructs the circle through the points a, b, and c. The ordering of the points determines the orientation of the circle. If the points are collinear, a Line is returned instead.

Use reflect(z, C) to reflect a point z through the circle C.

Arc

• Arc(a, b, c) constructs the circular arc through the given three points. If the points are collinear, a Segment is returned.

• Arc(C, start, Δ) constructs an arc from a Circle C, starting at the given start value and extending an amount Δ. The values are expressed as fractions of a full rotation starting from the real axis.

Examples

julia> ℓ = Line(1/2, 1/2+1im)    # line through 0.5 and 0.5+1i
Line{Float64} in the complex plane:
   through (0.5 + 0.0im) parallel to (0.0 + 1.0im)

julia> c = 1 / ℓ    # a circle
Circle{Float64} in the complex plane:
   centered at (1.0 - 2.7755575615628914e-17im) with radius 1.0, negatively oriented

julia> winding(c, 1.5), winding(c, -1)
(-1, 0)

julia> tangent(c, 0.75)
6.283185307179586 + 0.0im

julia> reflect(-1, c)
0.5 - 2.0816681711721685e-17im

julia> 2c - 2
Circle{Float64} in the complex plane:
   centered at (0.0 - 5.551115123125783e-17im) with radius 2.0, negatively oriented